[{"data":1,"prerenderedAt":75801},["ShallowReactive",2],{"blog-post-expert-notes\u002Fsampling-budget-qubit-classifications":3,"sibling-dives-expert-notes\u002Fsampling-budget-qubit-classifications":5202,"learn-track-expert-notes\u002Fsampling-budget-qubit-classifications":75800},{"id":4,"title":5,"authors":6,"body":8,"breadcrumb":5166,"builders":5171,"byline":5182,"challenge":5184,"courseAuthor":5184,"courseLead":5184,"dek":5184,"description":5185,"draft":5186,"extension":5187,"eyebrow":5188,"finish":5184,"fork":5184,"hero":5184,"heroAlt":5184,"heroCta":5184,"heroImage":5184,"kind":5189,"lessonCount":5184,"meta":5190,"navigation":102,"newsItems":5184,"next":5184,"ogImage":5191,"order":5184,"outcomes":5184,"path":5192,"publishDate":5193,"readingTime":5194,"related":5195,"relatedProjects":5184,"seo":5196,"stem":5197,"tags":5198,"track":5184,"trackName":5184,"__hash__":5201},"blog\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications.md","A Sampling-Budget & Bit-String Analogy for Qubit Classifications",[7],"onri-jay-benally",{"type":9,"value":10,"toc":5162},"minimark",[11,24,4986,4994,4999,5003,5009,5015,5018,5023,5144,5148,5158],[12,13,14],"p",{},[15,16,17,18,23],"em",{},"Republished with the author's permission from ",[19,20,22],"a",{"href":21},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb","the original notebook"," and shared under CC-BY-4.0. The text, code, and figures below are Onri's, as published.",[25,26,31],"pre",{"className":27,"code":28,"language":29,"meta":30,"style":30},"language-python shiki shiki-themes one-dark-pro","# This script builds a small simulation toolkit, runs it with sensible defaults,\n# generates tables and figures, so you can iterate further.\n#\n# Contents created in the current directory:\n#  - Sampling_Budget_Qubit_Bitstrings.ipynb      (Colab notebook)\n#  - summary.csv                                 (summary table)\n#  - bitstrings_\u003CMODEL>.csv                      (raw bit-strings per model)\n#  - fig_ones_fraction.png                       (bar chart with Wilson CI)\n#  - fig_required_shots.png                      (required shots for ±ε)\n#  - simulation_config.json                      (run parameters)\n\nimport json\nimport math\nfrom dataclasses import dataclass\nfrom typing import List, Dict, Tuple\n\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom IPython.display import display, Markdown\nimport nbformat as nbf\n\nplt.rcParams['figure.dpi'] = 200\n\n# ---------- Helper math ----------\n\ndef wilson_ci(k: int, n: int, z: float = 1.96) -> Tuple[float, float]:\n    \"\"\"\n    Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n    Returns (low, high). Works for edge cases k=0 or k=n.\n    \"\"\"\n    if n == 0:\n        return (0.0, 1.0)\n    p_hat = k \u002F n\n    denom = 1.0 + (z ** 2) \u002F n\n    center = (p_hat + (z ** 2) \u002F (2 * n)) \u002F denom\n    span = (z * math.sqrt((p_hat * (1 - p_hat) \u002F n) + (z ** 2) \u002F (4 * n ** 2))) \u002F denom\n    return (max(0.0, center - span), min(1.0, center + span))\n\n\ndef shots_for_epsilon(p: float, epsilon: float = 0.01, z: float = 1.96) -> int:\n    \"\"\"\n    Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n    at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n    \"\"\"\n    p = min(max(p, 0.0), 1.0)\n    var = p * (1.0 - p)\n    if var == 0.0:\n        return 1\n    n = (z ** 2) * var \u002F (epsilon ** 2)\n    return max(1, int(math.ceil(n)))\n\n\n# ---------- Domain model ----------\n\n@dataclass(frozen=True)\nclass QubitModel:\n    name: str\n    satisfaction_prob: float  # Probability a single shot meets the analytical criterion (bit = 1)\n    description: str\n\n\nDEFAULT_MODELS: List[QubitModel] = [\n    QubitModel(\n        \"Toy working qubit\",\n        0.50,\n        \"Simple demonstrator; about half shots meet the criterion.\"\n    ),\n    QubitModel(\n        \"NISQ qubit\",\n        0.60,\n        \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n    ),\n    QubitModel(\n        \"Fault-tolerant (logical)\",\n        0.90,\n        \"Error-corrected logical qubit; high single-shot success probability.\"\n    ),\n    QubitModel(\n        \"Topologically protected\",\n        0.98,\n        \"Intrinsic protection; very high single-shot success probability.\"\n    ),\n    QubitModel(\n        \"Ideal qubit\",\n        1.00,\n        \"Theoretical perfect qubit; always successful in a single shot.\"\n    ),\n]\n\n# ---------- Simulation configuration ----------\n\nCONFIG: Dict = {\n    \"random_seed\": 123456,\n    \"shots_per_model\": 10000,\n    \"epsilons\": [0.02, 0.01, 0.005],  # ±2%, ±1%, ±0.5% targets\n    \"cost_model\": {\n        \"default_cost_per_shot_usd\": 0.00005,\n        \"fixed_overhead_usd\": 0.10\n    },\n    \"timing\": {\n        \"time_per_shot_seconds\": 2e-6\n    }\n}\n\nrng = np.random.default_rng(CONFIG[\"random_seed\"])\n\n\n# ---------- Core simulation ----------\n\ndef simulate_bitstring(p: float, shots: int, rng_: np.random.Generator) -> np.ndarray:\n    \"\"\"\n    Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n    \"\"\"\n    return rng_.binomial(1, p, size=shots).astype(int)\n\n\ndef summarize_model(model: QubitModel, bits: np.ndarray, epsilons: List[float]) -> Dict:\n    \"\"\"\n    Calculate summary statistics for a given model and its simulated bitstring.\n    \"\"\"\n    n = int(bits.size)\n    ones = int(bits.sum())\n    zeros = int(n - ones)\n    frac = ones \u002F n if n > 0 else float(\"nan\")\n    lo, hi = wilson_ci(ones, n)\n\n    row = {\n        \"Model\": model.name,\n        \"Description\": model.description,\n        \"Configured p (target)\": model.satisfaction_prob,\n        \"Shots\": n,\n        \"Ones (successes)\": ones,\n        \"Zeros (failures)\": zeros,\n        \"Ones fraction (observed)\": frac,\n        \"Wilson CI 95% low\": lo,\n        \"Wilson CI 95% high\": hi,\n        \"CI width (95%)\": hi - lo,\n    }\n    for eps in epsilons:\n        row[f\"Required shots for ±{int(eps*100)}%\"] = shots_for_epsilon(model.satisfaction_prob, eps)\n    return row\n\n\ndef apply_costs(shots: int, cost_per_shot: float, fixed_overhead: float) -> float:\n    \"\"\"\n    Calculate illustrative cost based on shots and cost model.\n    \"\"\"\n    return fixed_overhead + shots * cost_per_shot\n\n\n# ---------- Run the simulation ----------\n\nall_bitstrings: Dict[str, np.ndarray] = {}\nsummary_rows: List[Dict] = []\n\nprint(\"Running simulation...\")\nfor m in DEFAULT_MODELS:\n    bits = simulate_bitstring(m.satisfaction_prob, CONFIG[\"shots_per_model\"], rng)\n    all_bitstrings[m.name] = bits\n    summary_rows.append(summarize_model(m, bits, CONFIG[\"epsilons\"]))\n\nsummary_df = pd.DataFrame(summary_rows)\n\n# Add cost estimates\ncps = CONFIG[\"cost_model\"][\"default_cost_per_shot_usd\"]\nover = CONFIG[\"cost_model\"][\"fixed_overhead_usd\"]\nif cps > 0 or over > 0:\n    summary_df[\"Illustrative cost (USD)\"] = [\n        apply_costs(int(row[\"Shots\"]), cps, over)\n        for _, row in summary_df.iterrows()\n    ]\n\n# ---------- Save artifacts ----------\n\n# Save raw bitstrings and summary CSV\nfor name, arr in all_bitstrings.items():\n    out_path = f\"bitstrings_{name.replace(' ', '_')}.csv\"\n    pd.DataFrame({\"bit\": arr}).to_csv(out_path, index=False)\n\nsummary_csv_path = \"summary.csv\"\nsummary_df.to_csv(summary_csv_path, index=False)\n\n# Save config JSON\nwith open(\"simulation_config.json\", \"w\") as f:\n    json.dump(CONFIG, f, indent=2)\n\nprint(\"Saved data to CSV and JSON files.\")\n\n# ---------- Visualizations ----------\n\n# Figure 1: Observed ones fraction with Wilson CI\nplt.style.use('seaborn-v0_8-whitegrid')\nplt.figure(figsize=(10, 6))\nx = np.arange(len(summary_df))\ny = summary_df[\"Ones fraction (observed)\"].to_numpy()\n\n# --- FIX APPLIED HERE ---\n# Take the absolute value to prevent tiny negative numbers from floating-point\n# inaccuracies from causing a ValueError in the errorbar plot.\nlower_error = y - summary_df[\"Wilson CI 95% low\"].to_numpy()\nupper_error = summary_df[\"Wilson CI 95% high\"].to_numpy() - y\nyerr = np.vstack([np.abs(lower_error), np.abs(upper_error)])\n# --- END OF FIX ---\n\nplt.bar(x, y, color='skyblue', edgecolor='black')\nplt.errorbar(x, y, yerr=yerr, fmt=\"none\", capsize=5, color=\"black\", elinewidth=1.5)\nplt.xticks(x, summary_df[\"Model\"].tolist(), rotation=25, ha=\"right\")\nplt.ylabel(\"Observed Ones Fraction\")\nplt.title(\"Observed Single-Shot Success Rates with 95% Wilson CI\", fontsize=14)\nplt.tight_layout()\nfig1_path = \"fig_ones_fraction.png\"\nplt.savefig(fig1_path, dpi=200, bbox_inches='tight')\nplt.show()\n\n# Figure 2: Required shots at several ±epsilon targets\nplt.figure(figsize=(10, 6))\nwidth = 0.2\nx = np.arange(len(DEFAULT_MODELS))\ncolors = ['#ff9999','#66b3ff','#99ff99']\n\nfor i, eps in enumerate(CONFIG[\"epsilons\"]):\n    req = [shots_for_epsilon(m.satisfaction_prob, eps) for m in DEFAULT_MODELS]\n    plt.bar(x + i * width, req, width, label=f\"±{eps:.1%}\", color=colors[i], edgecolor='black')\n\nplt.xticks(x + width, [m.name for m in DEFAULT_MODELS], rotation=25, ha=\"right\")\nplt.ylabel(\"Required Shots (Log Scale)\")\nplt.title(\"Approximate Shots to Estimate Success Rate within ±ε\", fontsize=14)\nplt.yscale('log')\nplt.legend(title=\"Error Tolerance\")\nplt.tight_layout()\nfig2_path = \"fig_required_shots.png\"\nplt.savefig(fig2_path, dpi=200, bbox_inches='tight')\nplt.show()\n\nprint(\"Generated and saved plots.\")\n\n# ---------- Display interactive table ----------\n\ndisplay(Markdown(\"##  Qubit Sampling Budget - Summary\"))\ndisplay(summary_df.style.format({\n    \"Configured p (target)\": \"{:.2f}\",\n    \"Ones fraction (observed)\": \"{:.4f}\",\n    \"Wilson CI 95% low\": \"{:.4f}\",\n    \"Wilson CI 95% high\": \"{:.4f}\",\n    \"CI width (95%)\": \"{:.4f}\",\n    \"Illustrative cost (USD)\": \"${:.2f}\"\n}))\n\n\n# ---------- Build a ready-to-run Colab notebook ----------\n\nmd_intro = r\"\"\"\n# Sampling Budget & Bit-String Analogy for Qubit Classes\n\n**High school level (concise):** We treat each measurement (“single shot”) as a bit: `1` if the outcome matches the analytical prediction, `0` if it does not. Different qubit types have different chances of landing a `1`. We simulate many shots, show how often we get `1`s, and estimate how many shots we need for a target accuracy.\n\n**Graduate level (concise):** We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ (noisy intermediate-scale quantum), fault-tolerant (logical), topologically protected, and ideal. We report Wilson score intervals (95%) for observed proportions and estimate the required shot budget $n \\approx z^2\\,p(1-p)\u002F\\epsilon^2$ to bound absolute error by ±$\\epsilon$ at ~95% confidence.\n\"\"\"\n\ncode_config = r\"\"\"\n# @title Configuration\nfrom dataclasses import dataclass\nfrom typing import List, Dict\nimport numpy as np\n\nRANDOM_SEED = 579345  # @param {type:\"number\"}\nSHOTS_PER_MODEL = 10000  # @param {type:\"number\"}\nEPSILONS = [0.02, 0.01, 0.005]  # @param\nCOST_PER_SHOT_USD = 0.00005  # @param {type:\"number\"}\nFIXED_OVERHEAD_USD = 0.10    # @param {type:\"number\"}\n\n@dataclass(frozen=True)\nclass QubitModel:\n    name: str\n    satisfaction_prob: float\n    description: str\n\nMODELS: List[QubitModel] = [\n    QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n    QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n    QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n    QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n    QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n]\n\nrng = np.random.default_rng(RANDOM_SEED)\n\"\"\"\n\ncode_lib_and_run = r\"\"\"\n# @title Run Simulation & Display Results\nimport math\nfrom typing import Tuple, Dict\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom IPython.display import display, Markdown\n\ndef wilson_ci(k: int, n: int, z: float = 1.96) -> Tuple[float, float]:\n    if n == 0: return (0.0, 1.0)\n    p_hat, z2_n = k \u002F n, z**2 \u002F n\n    denom = 1.0 + z2_n\n    center = (p_hat + z2_n \u002F 2) \u002F denom\n    span = (z * math.sqrt(p_hat * (1 - p_hat) \u002F n + z**2 \u002F (4 * n**2))) \u002F denom\n    return (max(0.0, center - span), min(1.0, center + span))\n\ndef shots_for_epsilon(p: float, epsilon: float = 0.01, z: float = 1.96) -> int:\n    if p * (1-p) == 0: return 1\n    return max(1, int(math.ceil((z**2 * p * (1-p)) \u002F (epsilon**2))))\n\ndef summarize_model(model, bits: np.ndarray, epsilons) -> Dict:\n    n = len(bits)\n    ones = sum(bits)\n    lo, hi = wilson_ci(ones, n)\n    row = {\n        \"Model\": model.name, \"p (target)\": model.satisfaction_prob, \"Shots\": n,\n        \"Ones\": ones, \"p (observed)\": ones \u002F n if n > 0 else 0,\n        \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n    }\n    for eps in epsilons:\n        row[f\"Shots for ±{eps:.1%}\"] = shots_for_epsilon(model.satisfaction_prob, eps)\n    return row\n\n# --- Run Simulation ---\nsummary_rows = [summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\nsummary_df = pd.DataFrame(summary_rows)\nif COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n    summary_df[\"Cost (USD)\"] = FIXED_OVERHEAD_USD + summary_df[\"Shots\"] * COST_PER_SHOT_USD\n\ndisplay(Markdown(\"### Simulation Summary\"))\ndisplay(summary_df.style.format(precision=4).background_gradient(cmap='viridis', subset=['p (observed)', 'CI width']))\n\n# --- Plotting ---\nplt.style.use('seaborn-v0_8-whitegrid')\nfig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 6))\n\n# Plot 1\ny = summary_df[\"p (observed)\"].to_numpy()\nlower_error = y - summary_df[\"CI 95% low\"]\nupper_error = summary_df[\"CI 95% high\"] - y\nyerr = np.vstack([np.abs(lower_error), np.abs(upper_error)])\n\nax1.bar(summary_df[\"Model\"], y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black')\nax1.set_title(\"Observed Success Rate (95% Wilson CI)\", fontsize=14)\nax1.tick_params(axis='x', rotation=30, labelsize=10)\n\n# Plot 2\nwidth = 0.2\nx = np.arange(len(MODELS))\ncolors = ['#ff9999','#66b3ff','#99ff99']\nfor i, eps in enumerate(EPSILONS):\n    req = summary_df[f\"Shots for ±{eps:.1%}\"]\n    ax2.bar(x + i * width, req, width, label=f\"±{eps:.1%}\", color=colors[i], edgecolor='black')\n\nax2.set_xticks(x + width, summary_df[\"Model\"])\nax2.set_ylabel(\"Required Shots (Log Scale)\")\nax2.set_yscale('log')\nax2.set_title(\"Shots Needed for ±ε Tolerance\", fontsize=14)\nax2.tick_params(axis='x', rotation=30, labelsize=10)\nax2.legend(title=\"Tolerance\")\n\nplt.tight_layout()\nplt.show()\n\"\"\"\n\nnb = nbf.v4.new_notebook()\nnb[\"cells\"] = [\n    nbf.v4.new_markdown_cell(md_intro),\n    nbf.v4.new_code_cell(code_config),\n    nbf.v4.new_code_cell(code_lib_and_run),\n]\n\nnb_path = \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\nwith open(nb_path, \"w\", encoding=\"utf-8\") as f:\n    nbf.write(nb, f)\n\n# ---------- Final confirmation ----------\nprint(\"\\n\" + \"=\"*50)\nprint(\"SCRIPT COMPLETE\")\nprint(\"=\"*50)\nprint(f\"Google Colab notebook saved to: {nb_path}\")\nprint(f\"Summary data saved to: {summary_csv_path}\")\nprint(f\"Plots saved to: {fig1_path}, {fig2_path}\")\n","python","",[32,33,34,43,49,55,61,67,73,79,85,91,97,104,115,123,137,150,155,169,182,195,208,221,226,247,252,258,263,324,330,336,342,347,365,385,402,432,472,544,582,587,592,641,646,652,658,663,690,712,727,735,766,791,796,801,807,812,831,843,852,863,871,876,881,895,904,913,921,927,933,940,948,956,962,967,974,982,990,996,1001,1008,1016,1024,1030,1035,1042,1050,1058,1064,1069,1075,1080,1086,1091,1105,1118,1131,1159,1168,1181,1192,1198,1206,1217,1223,1229,1234,1261,1266,1271,1277,1282,1315,1320,1326,1331,1366,1371,1376,1409,1414,1420,1425,1438,1457,1475,1513,1526,1531,1541,1550,1559,1568,1577,1586,1595,1604,1619,1633,1647,1652,1667,1710,1718,1723,1728,1768,1773,1779,1784,1802,1807,1812,1818,1823,1840,1851,1856,1869,1885,1908,1919,1946,1951,1968,1973,1979,2003,2024,2048,2063,2082,2102,2108,2113,2119,2124,2130,2149,2188,2221,2226,2237,2256,2261,2267,2293,2318,2323,2335,2340,2346,2351,2357,2373,2402,2424,2446,2451,2457,2463,2469,2497,2525,2552,2558,2563,2594,2651,2693,2708,2733,2743,2754,2785,2795,2800,2806,2829,2840,2861,2888,2893,2917,2943,3004,3009,3050,3064,3086,3101,3120,3129,3140,3166,3175,3180,3192,3197,3203,3208,3226,3240,3261,3282,3306,3329,3349,3369,3375,3380,3385,3391,3396,3411,3417,3422,3455,3460,3542,3547,3552,3564,3570,3576,3582,3588,3593,3602,3610,3622,3630,3638,3643,3655,3661,3667,3673,3679,3684,3699,3705,3711,3717,3723,3729,3734,3739,3752,3757,3762,3774,3780,3786,3792,3798,3803,3809,3815,3820,3841,3854,3865,3876,3897,3958,3990,3995,4011,4028,4081,4086,4102,4114,4126,4139,4145,4161,4177,4183,4188,4194,4213,4219,4224,4230,4239,4252,4258,4283,4288,4304,4332,4337,4343,4355,4383,4388,4394,4408,4417,4429,4438,4443,4462,4485,4498,4503,4509,4515,4532,4541,4556,4565,4592,4597,4615,4637,4649,4662,4674,4687,4692,4700,4708,4713,4718,4734,4749,4761,4772,4782,4787,4792,4803,4831,4843,4848,4854,4880,4892,4908,4931,4954],"code",{"__ignoreMap":30},[35,36,39],"span",{"class":37,"line":38},"line",1,[35,40,42],{"class":41},"sV9Aq","# This script builds a small simulation toolkit, runs it with sensible defaults,\n",[35,44,46],{"class":37,"line":45},2,[35,47,48],{"class":41},"# generates tables and figures, so you can iterate further.\n",[35,50,52],{"class":37,"line":51},3,[35,53,54],{"class":41},"#\n",[35,56,58],{"class":37,"line":57},4,[35,59,60],{"class":41},"# Contents created in the current directory:\n",[35,62,64],{"class":37,"line":63},5,[35,65,66],{"class":41},"#  - Sampling_Budget_Qubit_Bitstrings.ipynb      (Colab notebook)\n",[35,68,70],{"class":37,"line":69},6,[35,71,72],{"class":41},"#  - summary.csv                                 (summary table)\n",[35,74,76],{"class":37,"line":75},7,[35,77,78],{"class":41},"#  - bitstrings_\u003CMODEL>.csv                      (raw bit-strings per model)\n",[35,80,82],{"class":37,"line":81},8,[35,83,84],{"class":41},"#  - fig_ones_fraction.png                       (bar chart with Wilson CI)\n",[35,86,88],{"class":37,"line":87},9,[35,89,90],{"class":41},"#  - fig_required_shots.png                      (required shots for ±ε)\n",[35,92,94],{"class":37,"line":93},10,[35,95,96],{"class":41},"#  - simulation_config.json                      (run parameters)\n",[35,98,100],{"class":37,"line":99},11,[35,101,103],{"emptyLinePlaceholder":102},true,"\n",[35,105,107,111],{"class":37,"line":106},12,[35,108,110],{"class":109},"seHd6","import",[35,112,114],{"class":113},"sn6KH"," json\n",[35,116,118,120],{"class":37,"line":117},13,[35,119,110],{"class":109},[35,121,122],{"class":113}," math\n",[35,124,126,129,132,134],{"class":37,"line":125},14,[35,127,128],{"class":109},"from",[35,130,131],{"class":113}," dataclasses ",[35,133,110],{"class":109},[35,135,136],{"class":113}," dataclass\n",[35,138,140,142,145,147],{"class":37,"line":139},15,[35,141,128],{"class":109},[35,143,144],{"class":113}," typing ",[35,146,110],{"class":109},[35,148,149],{"class":113}," List, Dict, Tuple\n",[35,151,153],{"class":37,"line":152},16,[35,154,103],{"emptyLinePlaceholder":102},[35,156,158,160,163,166],{"class":37,"line":157},17,[35,159,110],{"class":109},[35,161,162],{"class":113}," numpy ",[35,164,165],{"class":109},"as",[35,167,168],{"class":113}," np\n",[35,170,172,174,177,179],{"class":37,"line":171},18,[35,173,110],{"class":109},[35,175,176],{"class":113}," pandas ",[35,178,165],{"class":109},[35,180,181],{"class":113}," pd\n",[35,183,185,187,190,192],{"class":37,"line":184},19,[35,186,110],{"class":109},[35,188,189],{"class":113}," matplotlib.pyplot ",[35,191,165],{"class":109},[35,193,194],{"class":113}," plt\n",[35,196,198,200,203,205],{"class":37,"line":197},20,[35,199,128],{"class":109},[35,201,202],{"class":113}," IPython.display ",[35,204,110],{"class":109},[35,206,207],{"class":113}," display, Markdown\n",[35,209,211,213,216,218],{"class":37,"line":210},21,[35,212,110],{"class":109},[35,214,215],{"class":113}," nbformat ",[35,217,165],{"class":109},[35,219,220],{"class":113}," nbf\n",[35,222,224],{"class":37,"line":223},22,[35,225,103],{"emptyLinePlaceholder":102},[35,227,229,232,236,239,243],{"class":37,"line":228},23,[35,230,231],{"class":113},"plt.rcParams[",[35,233,235],{"class":234},"subq3","'figure.dpi'",[35,237,238],{"class":113},"] ",[35,240,242],{"class":241},"sjrmR","=",[35,244,246],{"class":245},"sVC51"," 200\n",[35,248,250],{"class":37,"line":249},24,[35,251,103],{"emptyLinePlaceholder":102},[35,253,255],{"class":37,"line":254},25,[35,256,257],{"class":41},"# ---------- Helper math ----------\n",[35,259,261],{"class":37,"line":260},26,[35,262,103],{"emptyLinePlaceholder":102},[35,264,266,269,273,276,280,283,286,289,292,294,296,298,301,303,306,309,312,315,317,319,321],{"class":37,"line":265},27,[35,267,268],{"class":109},"def",[35,270,272],{"class":271},"sVbv2"," wilson_ci",[35,274,275],{"class":113},"(",[35,277,279],{"class":278},"sb9H8","k",[35,281,282],{"class":113},": ",[35,284,285],{"class":241},"int",[35,287,288],{"class":113},", ",[35,290,291],{"class":278},"n",[35,293,282],{"class":113},[35,295,285],{"class":241},[35,297,288],{"class":113},[35,299,300],{"class":278},"z",[35,302,282],{"class":113},[35,304,305],{"class":241},"float",[35,307,308],{"class":241}," =",[35,310,311],{"class":245}," 1.96",[35,313,314],{"class":113},") -> Tuple[",[35,316,305],{"class":241},[35,318,288],{"class":113},[35,320,305],{"class":241},[35,322,323],{"class":113},"]:\n",[35,325,327],{"class":37,"line":326},28,[35,328,329],{"class":234},"    \"\"\"\n",[35,331,333],{"class":37,"line":332},29,[35,334,335],{"class":234},"    Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n",[35,337,339],{"class":37,"line":338},30,[35,340,341],{"class":234},"    Returns (low, high). Works for edge cases k=0 or k=n.\n",[35,343,345],{"class":37,"line":344},31,[35,346,329],{"class":234},[35,348,350,353,356,359,362],{"class":37,"line":349},32,[35,351,352],{"class":109},"    if",[35,354,355],{"class":113}," n ",[35,357,358],{"class":241},"==",[35,360,361],{"class":245}," 0",[35,363,364],{"class":113},":\n",[35,366,368,371,374,377,379,382],{"class":37,"line":367},33,[35,369,370],{"class":109},"        return",[35,372,373],{"class":113}," (",[35,375,376],{"class":245},"0.0",[35,378,288],{"class":113},[35,380,381],{"class":245},"1.0",[35,383,384],{"class":113},")\n",[35,386,388,391,393,396,399],{"class":37,"line":387},34,[35,389,390],{"class":113},"    p_hat ",[35,392,242],{"class":241},[35,394,395],{"class":113}," k ",[35,397,398],{"class":241},"\u002F",[35,400,401],{"class":113}," n\n",[35,403,405,408,410,413,416,419,422,425,428,430],{"class":37,"line":404},35,[35,406,407],{"class":113},"    denom ",[35,409,242],{"class":241},[35,411,412],{"class":245}," 1.0",[35,414,415],{"class":241}," +",[35,417,418],{"class":113}," (z ",[35,420,421],{"class":241},"**",[35,423,424],{"class":245}," 2",[35,426,427],{"class":113},") ",[35,429,398],{"class":241},[35,431,401],{"class":113},[35,433,435,438,440,443,446,448,450,452,454,456,458,461,464,467,469],{"class":37,"line":434},36,[35,436,437],{"class":113},"    center ",[35,439,242],{"class":241},[35,441,442],{"class":113}," (p_hat ",[35,444,445],{"class":241},"+",[35,447,418],{"class":113},[35,449,421],{"class":241},[35,451,424],{"class":245},[35,453,427],{"class":113},[35,455,398],{"class":241},[35,457,373],{"class":113},[35,459,460],{"class":245},"2",[35,462,463],{"class":241}," *",[35,465,466],{"class":113}," n)) ",[35,468,398],{"class":241},[35,470,471],{"class":113}," denom\n",[35,473,475,478,480,482,485,488,491,494,496,498,501,504,507,509,512,514,516,518,520,522,524,526,529,531,533,535,537,540,542],{"class":37,"line":474},37,[35,476,477],{"class":113},"    span ",[35,479,242],{"class":241},[35,481,418],{"class":113},[35,483,484],{"class":241},"*",[35,486,487],{"class":113}," math.",[35,489,490],{"class":271},"sqrt",[35,492,493],{"class":113},"((p_hat ",[35,495,484],{"class":241},[35,497,373],{"class":113},[35,499,500],{"class":245},"1",[35,502,503],{"class":241}," -",[35,505,506],{"class":113}," p_hat) ",[35,508,398],{"class":241},[35,510,511],{"class":113}," n) ",[35,513,445],{"class":241},[35,515,418],{"class":113},[35,517,421],{"class":241},[35,519,424],{"class":245},[35,521,427],{"class":113},[35,523,398],{"class":241},[35,525,373],{"class":113},[35,527,528],{"class":245},"4",[35,530,463],{"class":241},[35,532,355],{"class":113},[35,534,421],{"class":241},[35,536,424],{"class":245},[35,538,539],{"class":113},"))) ",[35,541,398],{"class":241},[35,543,471],{"class":113},[35,545,547,550,552,555,557,559,562,565,568,571,573,575,577,579],{"class":37,"line":546},38,[35,548,549],{"class":109},"    return",[35,551,373],{"class":113},[35,553,554],{"class":241},"max",[35,556,275],{"class":113},[35,558,376],{"class":245},[35,560,561],{"class":113},", center ",[35,563,564],{"class":241},"-",[35,566,567],{"class":113}," span), ",[35,569,570],{"class":241},"min",[35,572,275],{"class":113},[35,574,381],{"class":245},[35,576,561],{"class":113},[35,578,445],{"class":241},[35,580,581],{"class":113}," span))\n",[35,583,585],{"class":37,"line":584},39,[35,586,103],{"emptyLinePlaceholder":102},[35,588,590],{"class":37,"line":589},40,[35,591,103],{"emptyLinePlaceholder":102},[35,593,595,597,600,602,604,606,608,610,613,615,617,619,622,624,626,628,630,632,634,637,639],{"class":37,"line":594},41,[35,596,268],{"class":109},[35,598,599],{"class":271}," shots_for_epsilon",[35,601,275],{"class":113},[35,603,12],{"class":278},[35,605,282],{"class":113},[35,607,305],{"class":241},[35,609,288],{"class":113},[35,611,612],{"class":278},"epsilon",[35,614,282],{"class":113},[35,616,305],{"class":241},[35,618,308],{"class":241},[35,620,621],{"class":245}," 0.01",[35,623,288],{"class":113},[35,625,300],{"class":278},[35,627,282],{"class":113},[35,629,305],{"class":241},[35,631,308],{"class":241},[35,633,311],{"class":245},[35,635,636],{"class":113},") -> ",[35,638,285],{"class":241},[35,640,364],{"class":113},[35,642,644],{"class":37,"line":643},42,[35,645,329],{"class":234},[35,647,649],{"class":37,"line":648},43,[35,650,651],{"class":234},"    Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n",[35,653,655],{"class":37,"line":654},44,[35,656,657],{"class":234},"    at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n",[35,659,661],{"class":37,"line":660},45,[35,662,329],{"class":234},[35,664,666,669,671,674,676,678,681,683,686,688],{"class":37,"line":665},46,[35,667,668],{"class":113},"    p ",[35,670,242],{"class":241},[35,672,673],{"class":241}," min",[35,675,275],{"class":113},[35,677,554],{"class":241},[35,679,680],{"class":113},"(p, ",[35,682,376],{"class":245},[35,684,685],{"class":113},"), ",[35,687,381],{"class":245},[35,689,384],{"class":113},[35,691,693,696,698,701,703,705,707,709],{"class":37,"line":692},47,[35,694,695],{"class":113},"    var ",[35,697,242],{"class":241},[35,699,700],{"class":113}," p ",[35,702,484],{"class":241},[35,704,373],{"class":113},[35,706,381],{"class":245},[35,708,503],{"class":241},[35,710,711],{"class":113}," p)\n",[35,713,715,717,720,722,725],{"class":37,"line":714},48,[35,716,352],{"class":109},[35,718,719],{"class":113}," var ",[35,721,358],{"class":241},[35,723,724],{"class":245}," 0.0",[35,726,364],{"class":113},[35,728,730,732],{"class":37,"line":729},49,[35,731,370],{"class":109},[35,733,734],{"class":245}," 1\n",[35,736,738,741,743,745,747,749,751,753,755,757,760,762,764],{"class":37,"line":737},50,[35,739,740],{"class":113},"    n ",[35,742,242],{"class":241},[35,744,418],{"class":113},[35,746,421],{"class":241},[35,748,424],{"class":245},[35,750,427],{"class":113},[35,752,484],{"class":241},[35,754,719],{"class":113},[35,756,398],{"class":241},[35,758,759],{"class":113}," (epsilon ",[35,761,421],{"class":241},[35,763,424],{"class":245},[35,765,384],{"class":113},[35,767,769,771,774,776,778,780,782,785,788],{"class":37,"line":768},51,[35,770,549],{"class":109},[35,772,773],{"class":241}," max",[35,775,275],{"class":113},[35,777,500],{"class":245},[35,779,288],{"class":113},[35,781,285],{"class":241},[35,783,784],{"class":113},"(math.",[35,786,787],{"class":271},"ceil",[35,789,790],{"class":113},"(n)))\n",[35,792,794],{"class":37,"line":793},52,[35,795,103],{"emptyLinePlaceholder":102},[35,797,799],{"class":37,"line":798},53,[35,800,103],{"emptyLinePlaceholder":102},[35,802,804],{"class":37,"line":803},54,[35,805,806],{"class":41},"# ---------- Domain model ----------\n",[35,808,810],{"class":37,"line":809},55,[35,811,103],{"emptyLinePlaceholder":102},[35,813,815,818,820,824,826,829],{"class":37,"line":814},56,[35,816,817],{"class":271},"@dataclass",[35,819,275],{"class":113},[35,821,823],{"class":822},"s_ZVi","frozen",[35,825,242],{"class":241},[35,827,828],{"class":245},"True",[35,830,384],{"class":113},[35,832,834,837,841],{"class":37,"line":833},57,[35,835,836],{"class":109},"class",[35,838,840],{"class":839},"sU0A5"," QubitModel",[35,842,364],{"class":113},[35,844,846,849],{"class":37,"line":845},58,[35,847,848],{"class":113},"    name: ",[35,850,851],{"class":241},"str\n",[35,853,855,858,860],{"class":37,"line":854},59,[35,856,857],{"class":113},"    satisfaction_prob: ",[35,859,305],{"class":241},[35,861,862],{"class":41},"  # Probability a single shot meets the analytical criterion (bit = 1)\n",[35,864,866,869],{"class":37,"line":865},60,[35,867,868],{"class":113},"    description: ",[35,870,851],{"class":241},[35,872,874],{"class":37,"line":873},61,[35,875,103],{"emptyLinePlaceholder":102},[35,877,879],{"class":37,"line":878},62,[35,880,103],{"emptyLinePlaceholder":102},[35,882,884,887,890,892],{"class":37,"line":883},63,[35,885,886],{"class":245},"DEFAULT_MODELS",[35,888,889],{"class":113},": List[QubitModel] ",[35,891,242],{"class":241},[35,893,894],{"class":113}," [\n",[35,896,898,901],{"class":37,"line":897},64,[35,899,900],{"class":271},"    QubitModel",[35,902,903],{"class":113},"(\n",[35,905,907,910],{"class":37,"line":906},65,[35,908,909],{"class":234},"        \"Toy working qubit\"",[35,911,912],{"class":113},",\n",[35,914,916,919],{"class":37,"line":915},66,[35,917,918],{"class":245},"        0.50",[35,920,912],{"class":113},[35,922,924],{"class":37,"line":923},67,[35,925,926],{"class":234},"        \"Simple demonstrator; about half shots meet the criterion.\"\n",[35,928,930],{"class":37,"line":929},68,[35,931,932],{"class":113},"    ),\n",[35,934,936,938],{"class":37,"line":935},69,[35,937,900],{"class":271},[35,939,903],{"class":113},[35,941,943,946],{"class":37,"line":942},70,[35,944,945],{"class":234},"        \"NISQ qubit\"",[35,947,912],{"class":113},[35,949,951,954],{"class":37,"line":950},71,[35,952,953],{"class":245},"        0.60",[35,955,912],{"class":113},[35,957,959],{"class":37,"line":958},72,[35,960,961],{"class":234},"        \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n",[35,963,965],{"class":37,"line":964},73,[35,966,932],{"class":113},[35,968,970,972],{"class":37,"line":969},74,[35,971,900],{"class":271},[35,973,903],{"class":113},[35,975,977,980],{"class":37,"line":976},75,[35,978,979],{"class":234},"        \"Fault-tolerant (logical)\"",[35,981,912],{"class":113},[35,983,985,988],{"class":37,"line":984},76,[35,986,987],{"class":245},"        0.90",[35,989,912],{"class":113},[35,991,993],{"class":37,"line":992},77,[35,994,995],{"class":234},"        \"Error-corrected logical qubit; high single-shot success probability.\"\n",[35,997,999],{"class":37,"line":998},78,[35,1000,932],{"class":113},[35,1002,1004,1006],{"class":37,"line":1003},79,[35,1005,900],{"class":271},[35,1007,903],{"class":113},[35,1009,1011,1014],{"class":37,"line":1010},80,[35,1012,1013],{"class":234},"        \"Topologically protected\"",[35,1015,912],{"class":113},[35,1017,1019,1022],{"class":37,"line":1018},81,[35,1020,1021],{"class":245},"        0.98",[35,1023,912],{"class":113},[35,1025,1027],{"class":37,"line":1026},82,[35,1028,1029],{"class":234},"        \"Intrinsic protection; very high single-shot success probability.\"\n",[35,1031,1033],{"class":37,"line":1032},83,[35,1034,932],{"class":113},[35,1036,1038,1040],{"class":37,"line":1037},84,[35,1039,900],{"class":271},[35,1041,903],{"class":113},[35,1043,1045,1048],{"class":37,"line":1044},85,[35,1046,1047],{"class":234},"        \"Ideal qubit\"",[35,1049,912],{"class":113},[35,1051,1053,1056],{"class":37,"line":1052},86,[35,1054,1055],{"class":245},"        1.00",[35,1057,912],{"class":113},[35,1059,1061],{"class":37,"line":1060},87,[35,1062,1063],{"class":234},"        \"Theoretical perfect qubit; always successful in a single shot.\"\n",[35,1065,1067],{"class":37,"line":1066},88,[35,1068,932],{"class":113},[35,1070,1072],{"class":37,"line":1071},89,[35,1073,1074],{"class":113},"]\n",[35,1076,1078],{"class":37,"line":1077},90,[35,1079,103],{"emptyLinePlaceholder":102},[35,1081,1083],{"class":37,"line":1082},91,[35,1084,1085],{"class":41},"# ---------- Simulation configuration ----------\n",[35,1087,1089],{"class":37,"line":1088},92,[35,1090,103],{"emptyLinePlaceholder":102},[35,1092,1094,1097,1100,1102],{"class":37,"line":1093},93,[35,1095,1096],{"class":245},"CONFIG",[35,1098,1099],{"class":113},": Dict ",[35,1101,242],{"class":241},[35,1103,1104],{"class":113}," {\n",[35,1106,1108,1111,1113,1116],{"class":37,"line":1107},94,[35,1109,1110],{"class":234},"    \"random_seed\"",[35,1112,282],{"class":113},[35,1114,1115],{"class":245},"123456",[35,1117,912],{"class":113},[35,1119,1121,1124,1126,1129],{"class":37,"line":1120},95,[35,1122,1123],{"class":234},"    \"shots_per_model\"",[35,1125,282],{"class":113},[35,1127,1128],{"class":245},"10000",[35,1130,912],{"class":113},[35,1132,1134,1137,1140,1143,1145,1148,1150,1153,1156],{"class":37,"line":1133},96,[35,1135,1136],{"class":234},"    \"epsilons\"",[35,1138,1139],{"class":113},": [",[35,1141,1142],{"class":245},"0.02",[35,1144,288],{"class":113},[35,1146,1147],{"class":245},"0.01",[35,1149,288],{"class":113},[35,1151,1152],{"class":245},"0.005",[35,1154,1155],{"class":113},"],  ",[35,1157,1158],{"class":41},"# ±2%, ±1%, ±0.5% targets\n",[35,1160,1162,1165],{"class":37,"line":1161},97,[35,1163,1164],{"class":234},"    \"cost_model\"",[35,1166,1167],{"class":113},": {\n",[35,1169,1171,1174,1176,1179],{"class":37,"line":1170},98,[35,1172,1173],{"class":234},"        \"default_cost_per_shot_usd\"",[35,1175,282],{"class":113},[35,1177,1178],{"class":245},"0.00005",[35,1180,912],{"class":113},[35,1182,1184,1187,1189],{"class":37,"line":1183},99,[35,1185,1186],{"class":234},"        \"fixed_overhead_usd\"",[35,1188,282],{"class":113},[35,1190,1191],{"class":245},"0.10\n",[35,1193,1195],{"class":37,"line":1194},100,[35,1196,1197],{"class":113},"    },\n",[35,1199,1201,1204],{"class":37,"line":1200},101,[35,1202,1203],{"class":234},"    \"timing\"",[35,1205,1167],{"class":113},[35,1207,1209,1212,1214],{"class":37,"line":1208},102,[35,1210,1211],{"class":234},"        \"time_per_shot_seconds\"",[35,1213,282],{"class":113},[35,1215,1216],{"class":245},"2e-6\n",[35,1218,1220],{"class":37,"line":1219},103,[35,1221,1222],{"class":113},"    }\n",[35,1224,1226],{"class":37,"line":1225},104,[35,1227,1228],{"class":113},"}\n",[35,1230,1232],{"class":37,"line":1231},105,[35,1233,103],{"emptyLinePlaceholder":102},[35,1235,1237,1240,1242,1245,1248,1250,1252,1255,1258],{"class":37,"line":1236},106,[35,1238,1239],{"class":113},"rng ",[35,1241,242],{"class":241},[35,1243,1244],{"class":113}," np.random.",[35,1246,1247],{"class":271},"default_rng",[35,1249,275],{"class":113},[35,1251,1096],{"class":245},[35,1253,1254],{"class":113},"[",[35,1256,1257],{"class":234},"\"random_seed\"",[35,1259,1260],{"class":113},"])\n",[35,1262,1264],{"class":37,"line":1263},107,[35,1265,103],{"emptyLinePlaceholder":102},[35,1267,1269],{"class":37,"line":1268},108,[35,1270,103],{"emptyLinePlaceholder":102},[35,1272,1274],{"class":37,"line":1273},109,[35,1275,1276],{"class":41},"# ---------- Core simulation ----------\n",[35,1278,1280],{"class":37,"line":1279},110,[35,1281,103],{"emptyLinePlaceholder":102},[35,1283,1285,1287,1290,1292,1294,1296,1298,1300,1303,1305,1307,1309,1312],{"class":37,"line":1284},111,[35,1286,268],{"class":109},[35,1288,1289],{"class":271}," simulate_bitstring",[35,1291,275],{"class":113},[35,1293,12],{"class":278},[35,1295,282],{"class":113},[35,1297,305],{"class":241},[35,1299,288],{"class":113},[35,1301,1302],{"class":278},"shots",[35,1304,282],{"class":113},[35,1306,285],{"class":241},[35,1308,288],{"class":113},[35,1310,1311],{"class":278},"rng_",[35,1313,1314],{"class":113},": np.random.Generator) -> np.ndarray:\n",[35,1316,1318],{"class":37,"line":1317},112,[35,1319,329],{"class":234},[35,1321,1323],{"class":37,"line":1322},113,[35,1324,1325],{"class":234},"    Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n",[35,1327,1329],{"class":37,"line":1328},114,[35,1330,329],{"class":234},[35,1332,1334,1336,1339,1342,1344,1346,1349,1352,1354,1357,1360,1362,1364],{"class":37,"line":1333},115,[35,1335,549],{"class":109},[35,1337,1338],{"class":113}," rng_.",[35,1340,1341],{"class":271},"binomial",[35,1343,275],{"class":113},[35,1345,500],{"class":245},[35,1347,1348],{"class":113},", p, ",[35,1350,1351],{"class":822},"size",[35,1353,242],{"class":241},[35,1355,1356],{"class":113},"shots).",[35,1358,1359],{"class":271},"astype",[35,1361,275],{"class":113},[35,1363,285],{"class":241},[35,1365,384],{"class":113},[35,1367,1369],{"class":37,"line":1368},116,[35,1370,103],{"emptyLinePlaceholder":102},[35,1372,1374],{"class":37,"line":1373},117,[35,1375,103],{"emptyLinePlaceholder":102},[35,1377,1379,1381,1384,1386,1389,1392,1395,1398,1401,1404,1406],{"class":37,"line":1378},118,[35,1380,268],{"class":109},[35,1382,1383],{"class":271}," summarize_model",[35,1385,275],{"class":113},[35,1387,1388],{"class":278},"model",[35,1390,1391],{"class":113},": QubitModel, ",[35,1393,1394],{"class":278},"bits",[35,1396,1397],{"class":113},": np.ndarray, ",[35,1399,1400],{"class":278},"epsilons",[35,1402,1403],{"class":113},": List[",[35,1405,305],{"class":241},[35,1407,1408],{"class":113},"]) -> Dict:\n",[35,1410,1412],{"class":37,"line":1411},119,[35,1413,329],{"class":234},[35,1415,1417],{"class":37,"line":1416},120,[35,1418,1419],{"class":234},"    Calculate summary statistics for a given model and its simulated bitstring.\n",[35,1421,1423],{"class":37,"line":1422},121,[35,1424,329],{"class":234},[35,1426,1428,1430,1432,1435],{"class":37,"line":1427},122,[35,1429,740],{"class":113},[35,1431,242],{"class":241},[35,1433,1434],{"class":241}," int",[35,1436,1437],{"class":113},"(bits.size)\n",[35,1439,1441,1444,1446,1448,1451,1454],{"class":37,"line":1440},123,[35,1442,1443],{"class":113},"    ones ",[35,1445,242],{"class":241},[35,1447,1434],{"class":241},[35,1449,1450],{"class":113},"(bits.",[35,1452,1453],{"class":271},"sum",[35,1455,1456],{"class":113},"())\n",[35,1458,1460,1463,1465,1467,1470,1472],{"class":37,"line":1459},124,[35,1461,1462],{"class":113},"    zeros ",[35,1464,242],{"class":241},[35,1466,1434],{"class":241},[35,1468,1469],{"class":113},"(n ",[35,1471,564],{"class":241},[35,1473,1474],{"class":113}," ones)\n",[35,1476,1478,1481,1483,1486,1488,1490,1493,1495,1498,1500,1503,1506,1508,1511],{"class":37,"line":1477},125,[35,1479,1480],{"class":113},"    frac ",[35,1482,242],{"class":241},[35,1484,1485],{"class":113}," ones ",[35,1487,398],{"class":241},[35,1489,355],{"class":113},[35,1491,1492],{"class":109},"if",[35,1494,355],{"class":113},[35,1496,1497],{"class":241},">",[35,1499,361],{"class":245},[35,1501,1502],{"class":109}," else",[35,1504,1505],{"class":241}," float",[35,1507,275],{"class":113},[35,1509,1510],{"class":234},"\"nan\"",[35,1512,384],{"class":113},[35,1514,1516,1519,1521,1523],{"class":37,"line":1515},126,[35,1517,1518],{"class":113},"    lo, hi ",[35,1520,242],{"class":241},[35,1522,272],{"class":271},[35,1524,1525],{"class":113},"(ones, n)\n",[35,1527,1529],{"class":37,"line":1528},127,[35,1530,103],{"emptyLinePlaceholder":102},[35,1532,1534,1537,1539],{"class":37,"line":1533},128,[35,1535,1536],{"class":113},"    row ",[35,1538,242],{"class":241},[35,1540,1104],{"class":113},[35,1542,1544,1547],{"class":37,"line":1543},129,[35,1545,1546],{"class":234},"        \"Model\"",[35,1548,1549],{"class":113},": model.name,\n",[35,1551,1553,1556],{"class":37,"line":1552},130,[35,1554,1555],{"class":234},"        \"Description\"",[35,1557,1558],{"class":113},": model.description,\n",[35,1560,1562,1565],{"class":37,"line":1561},131,[35,1563,1564],{"class":234},"        \"Configured p (target)\"",[35,1566,1567],{"class":113},": model.satisfaction_prob,\n",[35,1569,1571,1574],{"class":37,"line":1570},132,[35,1572,1573],{"class":234},"        \"Shots\"",[35,1575,1576],{"class":113},": n,\n",[35,1578,1580,1583],{"class":37,"line":1579},133,[35,1581,1582],{"class":234},"        \"Ones (successes)\"",[35,1584,1585],{"class":113},": ones,\n",[35,1587,1589,1592],{"class":37,"line":1588},134,[35,1590,1591],{"class":234},"        \"Zeros (failures)\"",[35,1593,1594],{"class":113},": zeros,\n",[35,1596,1598,1601],{"class":37,"line":1597},135,[35,1599,1600],{"class":234},"        \"Ones fraction (observed)\"",[35,1602,1603],{"class":113},": frac,\n",[35,1605,1607,1610,1613,1616],{"class":37,"line":1606},136,[35,1608,1609],{"class":234},"        \"Wilson CI 95",[35,1611,1612],{"class":245},"% lo",[35,1614,1615],{"class":234},"w\"",[35,1617,1618],{"class":113},": lo,\n",[35,1620,1622,1624,1627,1630],{"class":37,"line":1621},137,[35,1623,1609],{"class":234},[35,1625,1626],{"class":245},"% hi",[35,1628,1629],{"class":234},"gh\"",[35,1631,1632],{"class":113},": hi,\n",[35,1634,1636,1639,1642,1644],{"class":37,"line":1635},138,[35,1637,1638],{"class":234},"        \"CI width (95%)\"",[35,1640,1641],{"class":113},": hi ",[35,1643,564],{"class":241},[35,1645,1646],{"class":113}," lo,\n",[35,1648,1650],{"class":37,"line":1649},139,[35,1651,1222],{"class":113},[35,1653,1655,1658,1661,1664],{"class":37,"line":1654},140,[35,1656,1657],{"class":109},"    for",[35,1659,1660],{"class":113}," eps ",[35,1662,1663],{"class":109},"in",[35,1665,1666],{"class":113}," epsilons:\n",[35,1668,1670,1673,1676,1679,1682,1684,1687,1689,1692,1695,1698,1701,1703,1705,1707],{"class":37,"line":1669},141,[35,1671,1672],{"class":113},"        row[",[35,1674,1675],{"class":109},"f",[35,1677,1678],{"class":234},"\"Required shots for ±",[35,1680,1681],{"class":245},"{",[35,1683,285],{"class":241},[35,1685,1686],{"class":113},"(eps",[35,1688,484],{"class":241},[35,1690,1691],{"class":245},"100",[35,1693,1694],{"class":113},")",[35,1696,1697],{"class":245},"}",[35,1699,1700],{"class":234},"%\"",[35,1702,238],{"class":113},[35,1704,242],{"class":241},[35,1706,599],{"class":271},[35,1708,1709],{"class":113},"(model.satisfaction_prob, eps)\n",[35,1711,1713,1715],{"class":37,"line":1712},142,[35,1714,549],{"class":109},[35,1716,1717],{"class":113}," row\n",[35,1719,1721],{"class":37,"line":1720},143,[35,1722,103],{"emptyLinePlaceholder":102},[35,1724,1726],{"class":37,"line":1725},144,[35,1727,103],{"emptyLinePlaceholder":102},[35,1729,1731,1733,1736,1738,1740,1742,1744,1746,1749,1751,1753,1755,1758,1760,1762,1764,1766],{"class":37,"line":1730},145,[35,1732,268],{"class":109},[35,1734,1735],{"class":271}," apply_costs",[35,1737,275],{"class":113},[35,1739,1302],{"class":278},[35,1741,282],{"class":113},[35,1743,285],{"class":241},[35,1745,288],{"class":113},[35,1747,1748],{"class":278},"cost_per_shot",[35,1750,282],{"class":113},[35,1752,305],{"class":241},[35,1754,288],{"class":113},[35,1756,1757],{"class":278},"fixed_overhead",[35,1759,282],{"class":113},[35,1761,305],{"class":241},[35,1763,636],{"class":113},[35,1765,305],{"class":241},[35,1767,364],{"class":113},[35,1769,1771],{"class":37,"line":1770},146,[35,1772,329],{"class":234},[35,1774,1776],{"class":37,"line":1775},147,[35,1777,1778],{"class":234},"    Calculate illustrative cost based on shots and cost model.\n",[35,1780,1782],{"class":37,"line":1781},148,[35,1783,329],{"class":234},[35,1785,1787,1789,1792,1794,1797,1799],{"class":37,"line":1786},149,[35,1788,549],{"class":109},[35,1790,1791],{"class":113}," fixed_overhead ",[35,1793,445],{"class":241},[35,1795,1796],{"class":113}," shots ",[35,1798,484],{"class":241},[35,1800,1801],{"class":113}," cost_per_shot\n",[35,1803,1805],{"class":37,"line":1804},150,[35,1806,103],{"emptyLinePlaceholder":102},[35,1808,1810],{"class":37,"line":1809},151,[35,1811,103],{"emptyLinePlaceholder":102},[35,1813,1815],{"class":37,"line":1814},152,[35,1816,1817],{"class":41},"# ---------- Run the simulation ----------\n",[35,1819,1821],{"class":37,"line":1820},153,[35,1822,103],{"emptyLinePlaceholder":102},[35,1824,1826,1829,1832,1835,1837],{"class":37,"line":1825},154,[35,1827,1828],{"class":113},"all_bitstrings: Dict[",[35,1830,1831],{"class":241},"str",[35,1833,1834],{"class":113},", np.ndarray] ",[35,1836,242],{"class":241},[35,1838,1839],{"class":113}," {}\n",[35,1841,1843,1846,1848],{"class":37,"line":1842},155,[35,1844,1845],{"class":113},"summary_rows: List[Dict] ",[35,1847,242],{"class":241},[35,1849,1850],{"class":113}," []\n",[35,1852,1854],{"class":37,"line":1853},156,[35,1855,103],{"emptyLinePlaceholder":102},[35,1857,1859,1862,1864,1867],{"class":37,"line":1858},157,[35,1860,1861],{"class":241},"print",[35,1863,275],{"class":113},[35,1865,1866],{"class":234},"\"Running simulation...\"",[35,1868,384],{"class":113},[35,1870,1872,1875,1878,1880,1883],{"class":37,"line":1871},158,[35,1873,1874],{"class":109},"for",[35,1876,1877],{"class":113}," m ",[35,1879,1663],{"class":109},[35,1881,1882],{"class":245}," DEFAULT_MODELS",[35,1884,364],{"class":113},[35,1886,1888,1891,1893,1895,1898,1900,1902,1905],{"class":37,"line":1887},159,[35,1889,1890],{"class":113},"    bits ",[35,1892,242],{"class":241},[35,1894,1289],{"class":271},[35,1896,1897],{"class":113},"(m.satisfaction_prob, ",[35,1899,1096],{"class":245},[35,1901,1254],{"class":113},[35,1903,1904],{"class":234},"\"shots_per_model\"",[35,1906,1907],{"class":113},"], rng)\n",[35,1909,1911,1914,1916],{"class":37,"line":1910},160,[35,1912,1913],{"class":113},"    all_bitstrings[m.name] ",[35,1915,242],{"class":241},[35,1917,1918],{"class":113}," bits\n",[35,1920,1922,1925,1928,1930,1933,1936,1938,1940,1943],{"class":37,"line":1921},161,[35,1923,1924],{"class":113},"    summary_rows.",[35,1926,1927],{"class":271},"append",[35,1929,275],{"class":113},[35,1931,1932],{"class":271},"summarize_model",[35,1934,1935],{"class":113},"(m, bits, ",[35,1937,1096],{"class":245},[35,1939,1254],{"class":113},[35,1941,1942],{"class":234},"\"epsilons\"",[35,1944,1945],{"class":113},"]))\n",[35,1947,1949],{"class":37,"line":1948},162,[35,1950,103],{"emptyLinePlaceholder":102},[35,1952,1954,1957,1959,1962,1965],{"class":37,"line":1953},163,[35,1955,1956],{"class":113},"summary_df ",[35,1958,242],{"class":241},[35,1960,1961],{"class":113}," pd.",[35,1963,1964],{"class":271},"DataFrame",[35,1966,1967],{"class":113},"(summary_rows)\n",[35,1969,1971],{"class":37,"line":1970},164,[35,1972,103],{"emptyLinePlaceholder":102},[35,1974,1976],{"class":37,"line":1975},165,[35,1977,1978],{"class":41},"# Add cost estimates\n",[35,1980,1982,1985,1987,1990,1992,1995,1998,2001],{"class":37,"line":1981},166,[35,1983,1984],{"class":113},"cps ",[35,1986,242],{"class":241},[35,1988,1989],{"class":245}," CONFIG",[35,1991,1254],{"class":113},[35,1993,1994],{"class":234},"\"cost_model\"",[35,1996,1997],{"class":113},"][",[35,1999,2000],{"class":234},"\"default_cost_per_shot_usd\"",[35,2002,1074],{"class":113},[35,2004,2006,2009,2011,2013,2015,2017,2019,2022],{"class":37,"line":2005},167,[35,2007,2008],{"class":113},"over ",[35,2010,242],{"class":241},[35,2012,1989],{"class":245},[35,2014,1254],{"class":113},[35,2016,1994],{"class":234},[35,2018,1997],{"class":113},[35,2020,2021],{"class":234},"\"fixed_overhead_usd\"",[35,2023,1074],{"class":113},[35,2025,2027,2029,2032,2034,2036,2039,2042,2044,2046],{"class":37,"line":2026},168,[35,2028,1492],{"class":109},[35,2030,2031],{"class":113}," cps ",[35,2033,1497],{"class":241},[35,2035,361],{"class":245},[35,2037,2038],{"class":109}," or",[35,2040,2041],{"class":113}," over ",[35,2043,1497],{"class":241},[35,2045,361],{"class":245},[35,2047,364],{"class":113},[35,2049,2051,2054,2057,2059,2061],{"class":37,"line":2050},169,[35,2052,2053],{"class":113},"    summary_df[",[35,2055,2056],{"class":234},"\"Illustrative cost (USD)\"",[35,2058,238],{"class":113},[35,2060,242],{"class":241},[35,2062,894],{"class":113},[35,2064,2066,2069,2071,2073,2076,2079],{"class":37,"line":2065},170,[35,2067,2068],{"class":271},"        apply_costs",[35,2070,275],{"class":113},[35,2072,285],{"class":241},[35,2074,2075],{"class":113},"(row[",[35,2077,2078],{"class":234},"\"Shots\"",[35,2080,2081],{"class":113},"]), cps, over)\n",[35,2083,2085,2088,2091,2093,2096,2099],{"class":37,"line":2084},171,[35,2086,2087],{"class":109},"        for",[35,2089,2090],{"class":113}," _, row ",[35,2092,1663],{"class":109},[35,2094,2095],{"class":113}," summary_df.",[35,2097,2098],{"class":271},"iterrows",[35,2100,2101],{"class":113},"()\n",[35,2103,2105],{"class":37,"line":2104},172,[35,2106,2107],{"class":113},"    ]\n",[35,2109,2111],{"class":37,"line":2110},173,[35,2112,103],{"emptyLinePlaceholder":102},[35,2114,2116],{"class":37,"line":2115},174,[35,2117,2118],{"class":41},"# ---------- Save artifacts ----------\n",[35,2120,2122],{"class":37,"line":2121},175,[35,2123,103],{"emptyLinePlaceholder":102},[35,2125,2127],{"class":37,"line":2126},176,[35,2128,2129],{"class":41},"# Save raw bitstrings and summary CSV\n",[35,2131,2133,2135,2138,2140,2143,2146],{"class":37,"line":2132},177,[35,2134,1874],{"class":109},[35,2136,2137],{"class":113}," name, arr ",[35,2139,1663],{"class":109},[35,2141,2142],{"class":113}," all_bitstrings.",[35,2144,2145],{"class":271},"items",[35,2147,2148],{"class":113},"():\n",[35,2150,2152,2155,2157,2160,2163,2165,2168,2171,2173,2176,2178,2181,2183,2185],{"class":37,"line":2151},178,[35,2153,2154],{"class":113},"    out_path ",[35,2156,242],{"class":241},[35,2158,2159],{"class":109}," f",[35,2161,2162],{"class":234},"\"bitstrings_",[35,2164,1681],{"class":245},[35,2166,2167],{"class":113},"name.",[35,2169,2170],{"class":271},"replace",[35,2172,275],{"class":113},[35,2174,2175],{"class":234},"' '",[35,2177,288],{"class":113},[35,2179,2180],{"class":234},"'_'",[35,2182,1694],{"class":113},[35,2184,1697],{"class":245},[35,2186,2187],{"class":234},".csv\"\n",[35,2189,2191,2194,2196,2199,2202,2205,2208,2211,2214,2216,2219],{"class":37,"line":2190},179,[35,2192,2193],{"class":113},"    pd.",[35,2195,1964],{"class":271},[35,2197,2198],{"class":113},"({",[35,2200,2201],{"class":234},"\"bit\"",[35,2203,2204],{"class":113},": arr}).",[35,2206,2207],{"class":271},"to_csv",[35,2209,2210],{"class":113},"(out_path, ",[35,2212,2213],{"class":822},"index",[35,2215,242],{"class":241},[35,2217,2218],{"class":245},"False",[35,2220,384],{"class":113},[35,2222,2224],{"class":37,"line":2223},180,[35,2225,103],{"emptyLinePlaceholder":102},[35,2227,2229,2232,2234],{"class":37,"line":2228},181,[35,2230,2231],{"class":113},"summary_csv_path ",[35,2233,242],{"class":241},[35,2235,2236],{"class":234}," \"summary.csv\"\n",[35,2238,2240,2243,2245,2248,2250,2252,2254],{"class":37,"line":2239},182,[35,2241,2242],{"class":113},"summary_df.",[35,2244,2207],{"class":271},[35,2246,2247],{"class":113},"(summary_csv_path, ",[35,2249,2213],{"class":822},[35,2251,242],{"class":241},[35,2253,2218],{"class":245},[35,2255,384],{"class":113},[35,2257,2259],{"class":37,"line":2258},183,[35,2260,103],{"emptyLinePlaceholder":102},[35,2262,2264],{"class":37,"line":2263},184,[35,2265,2266],{"class":41},"# Save config JSON\n",[35,2268,2270,2273,2276,2278,2281,2283,2286,2288,2290],{"class":37,"line":2269},185,[35,2271,2272],{"class":109},"with",[35,2274,2275],{"class":241}," open",[35,2277,275],{"class":113},[35,2279,2280],{"class":234},"\"simulation_config.json\"",[35,2282,288],{"class":113},[35,2284,2285],{"class":234},"\"w\"",[35,2287,427],{"class":113},[35,2289,165],{"class":109},[35,2291,2292],{"class":113}," f:\n",[35,2294,2296,2299,2302,2304,2306,2309,2312,2314,2316],{"class":37,"line":2295},186,[35,2297,2298],{"class":113},"    json.",[35,2300,2301],{"class":271},"dump",[35,2303,275],{"class":113},[35,2305,1096],{"class":245},[35,2307,2308],{"class":113},", f, ",[35,2310,2311],{"class":822},"indent",[35,2313,242],{"class":241},[35,2315,460],{"class":245},[35,2317,384],{"class":113},[35,2319,2321],{"class":37,"line":2320},187,[35,2322,103],{"emptyLinePlaceholder":102},[35,2324,2326,2328,2330,2333],{"class":37,"line":2325},188,[35,2327,1861],{"class":241},[35,2329,275],{"class":113},[35,2331,2332],{"class":234},"\"Saved data to CSV and JSON files.\"",[35,2334,384],{"class":113},[35,2336,2338],{"class":37,"line":2337},189,[35,2339,103],{"emptyLinePlaceholder":102},[35,2341,2343],{"class":37,"line":2342},190,[35,2344,2345],{"class":41},"# ---------- Visualizations ----------\n",[35,2347,2349],{"class":37,"line":2348},191,[35,2350,103],{"emptyLinePlaceholder":102},[35,2352,2354],{"class":37,"line":2353},192,[35,2355,2356],{"class":41},"# Figure 1: Observed ones fraction with Wilson CI\n",[35,2358,2360,2363,2366,2368,2371],{"class":37,"line":2359},193,[35,2361,2362],{"class":113},"plt.style.",[35,2364,2365],{"class":271},"use",[35,2367,275],{"class":113},[35,2369,2370],{"class":234},"'seaborn-v0_8-whitegrid'",[35,2372,384],{"class":113},[35,2374,2376,2379,2382,2384,2387,2389,2391,2394,2396,2399],{"class":37,"line":2375},194,[35,2377,2378],{"class":113},"plt.",[35,2380,2381],{"class":271},"figure",[35,2383,275],{"class":113},[35,2385,2386],{"class":822},"figsize",[35,2388,242],{"class":241},[35,2390,275],{"class":113},[35,2392,2393],{"class":245},"10",[35,2395,288],{"class":113},[35,2397,2398],{"class":245},"6",[35,2400,2401],{"class":113},"))\n",[35,2403,2405,2408,2410,2413,2416,2418,2421],{"class":37,"line":2404},195,[35,2406,2407],{"class":113},"x ",[35,2409,242],{"class":241},[35,2411,2412],{"class":113}," np.",[35,2414,2415],{"class":271},"arange",[35,2417,275],{"class":113},[35,2419,2420],{"class":241},"len",[35,2422,2423],{"class":113},"(summary_df))\n",[35,2425,2427,2430,2432,2435,2438,2441,2444],{"class":37,"line":2426},196,[35,2428,2429],{"class":113},"y ",[35,2431,242],{"class":241},[35,2433,2434],{"class":113}," summary_df[",[35,2436,2437],{"class":234},"\"Ones fraction (observed)\"",[35,2439,2440],{"class":113},"].",[35,2442,2443],{"class":271},"to_numpy",[35,2445,2101],{"class":113},[35,2447,2449],{"class":37,"line":2448},197,[35,2450,103],{"emptyLinePlaceholder":102},[35,2452,2454],{"class":37,"line":2453},198,[35,2455,2456],{"class":41},"# --- FIX APPLIED HERE ---\n",[35,2458,2460],{"class":37,"line":2459},199,[35,2461,2462],{"class":41},"# Take the absolute value to prevent tiny negative numbers from floating-point\n",[35,2464,2466],{"class":37,"line":2465},200,[35,2467,2468],{"class":41},"# inaccuracies from causing a ValueError in the errorbar plot.\n",[35,2470,2472,2475,2477,2480,2482,2484,2487,2489,2491,2493,2495],{"class":37,"line":2471},201,[35,2473,2474],{"class":113},"lower_error ",[35,2476,242],{"class":241},[35,2478,2479],{"class":113}," y ",[35,2481,564],{"class":241},[35,2483,2434],{"class":113},[35,2485,2486],{"class":234},"\"Wilson CI 95",[35,2488,1612],{"class":245},[35,2490,1615],{"class":234},[35,2492,2440],{"class":113},[35,2494,2443],{"class":271},[35,2496,2101],{"class":113},[35,2498,2500,2503,2505,2507,2509,2511,2513,2515,2517,2520,2522],{"class":37,"line":2499},202,[35,2501,2502],{"class":113},"upper_error ",[35,2504,242],{"class":241},[35,2506,2434],{"class":113},[35,2508,2486],{"class":234},[35,2510,1626],{"class":245},[35,2512,1629],{"class":234},[35,2514,2440],{"class":113},[35,2516,2443],{"class":271},[35,2518,2519],{"class":113},"() ",[35,2521,564],{"class":241},[35,2523,2524],{"class":113}," y\n",[35,2526,2528,2531,2533,2535,2538,2541,2544,2547,2549],{"class":37,"line":2527},203,[35,2529,2530],{"class":113},"yerr ",[35,2532,242],{"class":241},[35,2534,2412],{"class":113},[35,2536,2537],{"class":271},"vstack",[35,2539,2540],{"class":113},"([np.",[35,2542,2543],{"class":271},"abs",[35,2545,2546],{"class":113},"(lower_error), np.",[35,2548,2543],{"class":271},[35,2550,2551],{"class":113},"(upper_error)])\n",[35,2553,2555],{"class":37,"line":2554},204,[35,2556,2557],{"class":41},"# --- END OF FIX ---\n",[35,2559,2561],{"class":37,"line":2560},205,[35,2562,103],{"emptyLinePlaceholder":102},[35,2564,2566,2568,2571,2574,2577,2579,2582,2584,2587,2589,2592],{"class":37,"line":2565},206,[35,2567,2378],{"class":113},[35,2569,2570],{"class":271},"bar",[35,2572,2573],{"class":113},"(x, y, ",[35,2575,2576],{"class":822},"color",[35,2578,242],{"class":241},[35,2580,2581],{"class":234},"'skyblue'",[35,2583,288],{"class":113},[35,2585,2586],{"class":822},"edgecolor",[35,2588,242],{"class":241},[35,2590,2591],{"class":234},"'black'",[35,2593,384],{"class":113},[35,2595,2597,2599,2602,2604,2607,2609,2612,2615,2617,2620,2622,2625,2627,2630,2632,2634,2636,2639,2641,2644,2646,2649],{"class":37,"line":2596},207,[35,2598,2378],{"class":113},[35,2600,2601],{"class":271},"errorbar",[35,2603,2573],{"class":113},[35,2605,2606],{"class":822},"yerr",[35,2608,242],{"class":241},[35,2610,2611],{"class":113},"yerr, ",[35,2613,2614],{"class":822},"fmt",[35,2616,242],{"class":241},[35,2618,2619],{"class":234},"\"none\"",[35,2621,288],{"class":113},[35,2623,2624],{"class":822},"capsize",[35,2626,242],{"class":241},[35,2628,2629],{"class":245},"5",[35,2631,288],{"class":113},[35,2633,2576],{"class":822},[35,2635,242],{"class":241},[35,2637,2638],{"class":234},"\"black\"",[35,2640,288],{"class":113},[35,2642,2643],{"class":822},"elinewidth",[35,2645,242],{"class":241},[35,2647,2648],{"class":245},"1.5",[35,2650,384],{"class":113},[35,2652,2654,2656,2659,2662,2665,2667,2670,2673,2676,2678,2681,2683,2686,2688,2691],{"class":37,"line":2653},208,[35,2655,2378],{"class":113},[35,2657,2658],{"class":271},"xticks",[35,2660,2661],{"class":113},"(x, summary_df[",[35,2663,2664],{"class":234},"\"Model\"",[35,2666,2440],{"class":113},[35,2668,2669],{"class":271},"tolist",[35,2671,2672],{"class":113},"(), ",[35,2674,2675],{"class":822},"rotation",[35,2677,242],{"class":241},[35,2679,2680],{"class":245},"25",[35,2682,288],{"class":113},[35,2684,2685],{"class":822},"ha",[35,2687,242],{"class":241},[35,2689,2690],{"class":234},"\"right\"",[35,2692,384],{"class":113},[35,2694,2696,2698,2701,2703,2706],{"class":37,"line":2695},209,[35,2697,2378],{"class":113},[35,2699,2700],{"class":271},"ylabel",[35,2702,275],{"class":113},[35,2704,2705],{"class":234},"\"Observed Ones Fraction\"",[35,2707,384],{"class":113},[35,2709,2711,2713,2716,2718,2721,2723,2726,2728,2731],{"class":37,"line":2710},210,[35,2712,2378],{"class":113},[35,2714,2715],{"class":271},"title",[35,2717,275],{"class":113},[35,2719,2720],{"class":234},"\"Observed Single-Shot Success Rates with 95% Wilson CI\"",[35,2722,288],{"class":113},[35,2724,2725],{"class":822},"fontsize",[35,2727,242],{"class":241},[35,2729,2730],{"class":245},"14",[35,2732,384],{"class":113},[35,2734,2736,2738,2741],{"class":37,"line":2735},211,[35,2737,2378],{"class":113},[35,2739,2740],{"class":271},"tight_layout",[35,2742,2101],{"class":113},[35,2744,2746,2749,2751],{"class":37,"line":2745},212,[35,2747,2748],{"class":113},"fig1_path ",[35,2750,242],{"class":241},[35,2752,2753],{"class":234}," \"fig_ones_fraction.png\"\n",[35,2755,2757,2759,2762,2765,2768,2770,2773,2775,2778,2780,2783],{"class":37,"line":2756},213,[35,2758,2378],{"class":113},[35,2760,2761],{"class":271},"savefig",[35,2763,2764],{"class":113},"(fig1_path, ",[35,2766,2767],{"class":822},"dpi",[35,2769,242],{"class":241},[35,2771,2772],{"class":245},"200",[35,2774,288],{"class":113},[35,2776,2777],{"class":822},"bbox_inches",[35,2779,242],{"class":241},[35,2781,2782],{"class":234},"'tight'",[35,2784,384],{"class":113},[35,2786,2788,2790,2793],{"class":37,"line":2787},214,[35,2789,2378],{"class":113},[35,2791,2792],{"class":271},"show",[35,2794,2101],{"class":113},[35,2796,2798],{"class":37,"line":2797},215,[35,2799,103],{"emptyLinePlaceholder":102},[35,2801,2803],{"class":37,"line":2802},216,[35,2804,2805],{"class":41},"# Figure 2: Required shots at several ±epsilon targets\n",[35,2807,2809,2811,2813,2815,2817,2819,2821,2823,2825,2827],{"class":37,"line":2808},217,[35,2810,2378],{"class":113},[35,2812,2381],{"class":271},[35,2814,275],{"class":113},[35,2816,2386],{"class":822},[35,2818,242],{"class":241},[35,2820,275],{"class":113},[35,2822,2393],{"class":245},[35,2824,288],{"class":113},[35,2826,2398],{"class":245},[35,2828,2401],{"class":113},[35,2830,2832,2835,2837],{"class":37,"line":2831},218,[35,2833,2834],{"class":113},"width ",[35,2836,242],{"class":241},[35,2838,2839],{"class":245}," 0.2\n",[35,2841,2843,2845,2847,2849,2851,2853,2855,2857,2859],{"class":37,"line":2842},219,[35,2844,2407],{"class":113},[35,2846,242],{"class":241},[35,2848,2412],{"class":113},[35,2850,2415],{"class":271},[35,2852,275],{"class":113},[35,2854,2420],{"class":241},[35,2856,275],{"class":113},[35,2858,886],{"class":245},[35,2860,2401],{"class":113},[35,2862,2864,2867,2869,2872,2875,2878,2881,2883,2886],{"class":37,"line":2863},220,[35,2865,2866],{"class":113},"colors ",[35,2868,242],{"class":241},[35,2870,2871],{"class":113}," [",[35,2873,2874],{"class":234},"'#ff9999'",[35,2876,2877],{"class":113},",",[35,2879,2880],{"class":234},"'#66b3ff'",[35,2882,2877],{"class":113},[35,2884,2885],{"class":234},"'#99ff99'",[35,2887,1074],{"class":113},[35,2889,2891],{"class":37,"line":2890},221,[35,2892,103],{"emptyLinePlaceholder":102},[35,2894,2896,2898,2901,2903,2906,2908,2910,2912,2914],{"class":37,"line":2895},222,[35,2897,1874],{"class":109},[35,2899,2900],{"class":113}," i, eps ",[35,2902,1663],{"class":109},[35,2904,2905],{"class":241}," enumerate",[35,2907,275],{"class":113},[35,2909,1096],{"class":245},[35,2911,1254],{"class":113},[35,2913,1942],{"class":234},[35,2915,2916],{"class":113},"]):\n",[35,2918,2920,2923,2925,2927,2930,2933,2935,2937,2939,2941],{"class":37,"line":2919},223,[35,2921,2922],{"class":113},"    req ",[35,2924,242],{"class":241},[35,2926,2871],{"class":113},[35,2928,2929],{"class":271},"shots_for_epsilon",[35,2931,2932],{"class":113},"(m.satisfaction_prob, eps) ",[35,2934,1874],{"class":109},[35,2936,1877],{"class":113},[35,2938,1663],{"class":109},[35,2940,1882],{"class":245},[35,2942,1074],{"class":113},[35,2944,2946,2949,2951,2954,2956,2959,2961,2964,2967,2969,2971,2974,2976,2979,2982,2984,2987,2989,2991,2993,2996,2998,3000,3002],{"class":37,"line":2945},224,[35,2947,2948],{"class":113},"    plt.",[35,2950,2570],{"class":271},[35,2952,2953],{"class":113},"(x ",[35,2955,445],{"class":241},[35,2957,2958],{"class":113}," i ",[35,2960,484],{"class":241},[35,2962,2963],{"class":113}," width, req, width, ",[35,2965,2966],{"class":822},"label",[35,2968,242],{"class":241},[35,2970,1675],{"class":109},[35,2972,2973],{"class":234},"\"±",[35,2975,1681],{"class":245},[35,2977,2978],{"class":113},"eps",[35,2980,2981],{"class":109},":.1%",[35,2983,1697],{"class":245},[35,2985,2986],{"class":234},"\"",[35,2988,288],{"class":113},[35,2990,2576],{"class":822},[35,2992,242],{"class":241},[35,2994,2995],{"class":113},"colors[i], ",[35,2997,2586],{"class":822},[35,2999,242],{"class":241},[35,3001,2591],{"class":234},[35,3003,384],{"class":113},[35,3005,3007],{"class":37,"line":3006},225,[35,3008,103],{"emptyLinePlaceholder":102},[35,3010,3012,3014,3016,3018,3020,3023,3025,3027,3029,3031,3034,3036,3038,3040,3042,3044,3046,3048],{"class":37,"line":3011},226,[35,3013,2378],{"class":113},[35,3015,2658],{"class":271},[35,3017,2953],{"class":113},[35,3019,445],{"class":241},[35,3021,3022],{"class":113}," width, [m.name ",[35,3024,1874],{"class":109},[35,3026,1877],{"class":113},[35,3028,1663],{"class":109},[35,3030,1882],{"class":245},[35,3032,3033],{"class":113},"], ",[35,3035,2675],{"class":822},[35,3037,242],{"class":241},[35,3039,2680],{"class":245},[35,3041,288],{"class":113},[35,3043,2685],{"class":822},[35,3045,242],{"class":241},[35,3047,2690],{"class":234},[35,3049,384],{"class":113},[35,3051,3053,3055,3057,3059,3062],{"class":37,"line":3052},227,[35,3054,2378],{"class":113},[35,3056,2700],{"class":271},[35,3058,275],{"class":113},[35,3060,3061],{"class":234},"\"Required Shots (Log Scale)\"",[35,3063,384],{"class":113},[35,3065,3067,3069,3071,3073,3076,3078,3080,3082,3084],{"class":37,"line":3066},228,[35,3068,2378],{"class":113},[35,3070,2715],{"class":271},[35,3072,275],{"class":113},[35,3074,3075],{"class":234},"\"Approximate Shots to Estimate Success Rate within ±ε\"",[35,3077,288],{"class":113},[35,3079,2725],{"class":822},[35,3081,242],{"class":241},[35,3083,2730],{"class":245},[35,3085,384],{"class":113},[35,3087,3089,3091,3094,3096,3099],{"class":37,"line":3088},229,[35,3090,2378],{"class":113},[35,3092,3093],{"class":271},"yscale",[35,3095,275],{"class":113},[35,3097,3098],{"class":234},"'log'",[35,3100,384],{"class":113},[35,3102,3104,3106,3109,3111,3113,3115,3118],{"class":37,"line":3103},230,[35,3105,2378],{"class":113},[35,3107,3108],{"class":271},"legend",[35,3110,275],{"class":113},[35,3112,2715],{"class":822},[35,3114,242],{"class":241},[35,3116,3117],{"class":234},"\"Error Tolerance\"",[35,3119,384],{"class":113},[35,3121,3123,3125,3127],{"class":37,"line":3122},231,[35,3124,2378],{"class":113},[35,3126,2740],{"class":271},[35,3128,2101],{"class":113},[35,3130,3132,3135,3137],{"class":37,"line":3131},232,[35,3133,3134],{"class":113},"fig2_path ",[35,3136,242],{"class":241},[35,3138,3139],{"class":234}," \"fig_required_shots.png\"\n",[35,3141,3143,3145,3147,3150,3152,3154,3156,3158,3160,3162,3164],{"class":37,"line":3142},233,[35,3144,2378],{"class":113},[35,3146,2761],{"class":271},[35,3148,3149],{"class":113},"(fig2_path, ",[35,3151,2767],{"class":822},[35,3153,242],{"class":241},[35,3155,2772],{"class":245},[35,3157,288],{"class":113},[35,3159,2777],{"class":822},[35,3161,242],{"class":241},[35,3163,2782],{"class":234},[35,3165,384],{"class":113},[35,3167,3169,3171,3173],{"class":37,"line":3168},234,[35,3170,2378],{"class":113},[35,3172,2792],{"class":271},[35,3174,2101],{"class":113},[35,3176,3178],{"class":37,"line":3177},235,[35,3179,103],{"emptyLinePlaceholder":102},[35,3181,3183,3185,3187,3190],{"class":37,"line":3182},236,[35,3184,1861],{"class":241},[35,3186,275],{"class":113},[35,3188,3189],{"class":234},"\"Generated and saved plots.\"",[35,3191,384],{"class":113},[35,3193,3195],{"class":37,"line":3194},237,[35,3196,103],{"emptyLinePlaceholder":102},[35,3198,3200],{"class":37,"line":3199},238,[35,3201,3202],{"class":41},"# ---------- Display interactive table ----------\n",[35,3204,3206],{"class":37,"line":3205},239,[35,3207,103],{"emptyLinePlaceholder":102},[35,3209,3211,3214,3216,3219,3221,3224],{"class":37,"line":3210},240,[35,3212,3213],{"class":271},"display",[35,3215,275],{"class":113},[35,3217,3218],{"class":271},"Markdown",[35,3220,275],{"class":113},[35,3222,3223],{"class":234},"\"##  Qubit Sampling Budget - Summary\"",[35,3225,2401],{"class":113},[35,3227,3229,3231,3234,3237],{"class":37,"line":3228},241,[35,3230,3213],{"class":271},[35,3232,3233],{"class":113},"(summary_df.style.",[35,3235,3236],{"class":271},"format",[35,3238,3239],{"class":113},"({\n",[35,3241,3243,3246,3248,3250,3252,3255,3257,3259],{"class":37,"line":3242},242,[35,3244,3245],{"class":234},"    \"Configured p (target)\"",[35,3247,282],{"class":113},[35,3249,2986],{"class":234},[35,3251,1681],{"class":245},[35,3253,3254],{"class":109},":.2f",[35,3256,1697],{"class":245},[35,3258,2986],{"class":234},[35,3260,912],{"class":113},[35,3262,3264,3267,3269,3271,3273,3276,3278,3280],{"class":37,"line":3263},243,[35,3265,3266],{"class":234},"    \"Ones fraction (observed)\"",[35,3268,282],{"class":113},[35,3270,2986],{"class":234},[35,3272,1681],{"class":245},[35,3274,3275],{"class":109},":.4f",[35,3277,1697],{"class":245},[35,3279,2986],{"class":234},[35,3281,912],{"class":113},[35,3283,3285,3288,3290,3292,3294,3296,3298,3300,3302,3304],{"class":37,"line":3284},244,[35,3286,3287],{"class":234},"    \"Wilson CI 95",[35,3289,1612],{"class":245},[35,3291,1615],{"class":234},[35,3293,282],{"class":113},[35,3295,2986],{"class":234},[35,3297,1681],{"class":245},[35,3299,3275],{"class":109},[35,3301,1697],{"class":245},[35,3303,2986],{"class":234},[35,3305,912],{"class":113},[35,3307,3309,3311,3313,3315,3317,3319,3321,3323,3325,3327],{"class":37,"line":3308},245,[35,3310,3287],{"class":234},[35,3312,1626],{"class":245},[35,3314,1629],{"class":234},[35,3316,282],{"class":113},[35,3318,2986],{"class":234},[35,3320,1681],{"class":245},[35,3322,3275],{"class":109},[35,3324,1697],{"class":245},[35,3326,2986],{"class":234},[35,3328,912],{"class":113},[35,3330,3332,3335,3337,3339,3341,3343,3345,3347],{"class":37,"line":3331},246,[35,3333,3334],{"class":234},"    \"CI width (95%)\"",[35,3336,282],{"class":113},[35,3338,2986],{"class":234},[35,3340,1681],{"class":245},[35,3342,3275],{"class":109},[35,3344,1697],{"class":245},[35,3346,2986],{"class":234},[35,3348,912],{"class":113},[35,3350,3352,3355,3357,3360,3362,3364,3366],{"class":37,"line":3351},247,[35,3353,3354],{"class":234},"    \"Illustrative cost (USD)\"",[35,3356,282],{"class":113},[35,3358,3359],{"class":234},"\"$",[35,3361,1681],{"class":245},[35,3363,3254],{"class":109},[35,3365,1697],{"class":245},[35,3367,3368],{"class":234},"\"\n",[35,3370,3372],{"class":37,"line":3371},248,[35,3373,3374],{"class":113},"}))\n",[35,3376,3378],{"class":37,"line":3377},249,[35,3379,103],{"emptyLinePlaceholder":102},[35,3381,3383],{"class":37,"line":3382},250,[35,3384,103],{"emptyLinePlaceholder":102},[35,3386,3388],{"class":37,"line":3387},251,[35,3389,3390],{"class":41},"# ---------- Build a ready-to-run Colab notebook ----------\n",[35,3392,3394],{"class":37,"line":3393},252,[35,3395,103],{"emptyLinePlaceholder":102},[35,3397,3399,3402,3404,3407],{"class":37,"line":3398},253,[35,3400,3401],{"class":113},"md_intro ",[35,3403,242],{"class":241},[35,3405,3406],{"class":109}," r",[35,3408,3410],{"class":3409},"sVyAn","\"\"\"\n",[35,3412,3414],{"class":37,"line":3413},254,[35,3415,3416],{"class":41},"# Sampling Budget & Bit-String Analogy for Qubit Classes\n",[35,3418,3420],{"class":37,"line":3419},255,[35,3421,103],{"emptyLinePlaceholder":102},[35,3423,3425,3427,3430,3432,3435,3437,3440,3442,3445,3447,3450,3452],{"class":37,"line":3424},256,[35,3426,421],{"class":245},[35,3428,3429],{"class":3409},"High school level ",[35,3431,275],{"class":245},[35,3433,3434],{"class":3409},"concise",[35,3436,1694],{"class":245},[35,3438,3439],{"class":3409},":",[35,3441,421],{"class":245},[35,3443,3444],{"class":3409}," We treat each measurement ",[35,3446,275],{"class":245},[35,3448,3449],{"class":3409},"“single shot”",[35,3451,1694],{"class":245},[35,3453,3454],{"class":3409}," as a bit: `1` if the outcome matches the analytical prediction, `0` if it does not. Different qubit types have different chances of landing a `1`. We simulate many shots, show how often we get `1`s, and estimate how many shots we need for a target accuracy.\n",[35,3456,3458],{"class":37,"line":3457},257,[35,3459,103],{"emptyLinePlaceholder":102},[35,3461,3463,3465,3468,3470,3472,3474,3476,3478,3481,3483,3486,3488,3491,3493,3496,3498,3501,3503,3506,3508,3511,3514,3517,3520,3522,3524,3527,3529,3531,3534,3537,3539],{"class":37,"line":3462},258,[35,3464,421],{"class":245},[35,3466,3467],{"class":3409},"Graduate level ",[35,3469,275],{"class":245},[35,3471,3434],{"class":3409},[35,3473,1694],{"class":245},[35,3475,3439],{"class":3409},[35,3477,421],{"class":245},[35,3479,3480],{"class":3409}," We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ ",[35,3482,275],{"class":245},[35,3484,3485],{"class":3409},"noisy intermediate-scale quantum",[35,3487,1694],{"class":245},[35,3489,3490],{"class":3409},", fault-tolerant ",[35,3492,275],{"class":245},[35,3494,3495],{"class":3409},"logical",[35,3497,1694],{"class":245},[35,3499,3500],{"class":3409},", topologically protected, and ideal. We report Wilson score intervals ",[35,3502,275],{"class":245},[35,3504,3505],{"class":3409},"95%",[35,3507,1694],{"class":245},[35,3509,3510],{"class":3409}," for observed proportions and estimate the required shot budget $n ",[35,3512,3513],{"class":241},"\\a",[35,3515,3516],{"class":3409},"pprox z^2",[35,3518,3519],{"class":241},"\\,",[35,3521,12],{"class":3409},[35,3523,275],{"class":245},[35,3525,3526],{"class":3409},"1-p",[35,3528,1694],{"class":245},[35,3530,398],{"class":3409},[35,3532,3533],{"class":241},"\\e",[35,3535,3536],{"class":3409},"psilon^2$ to bound absolute error by ±$",[35,3538,3533],{"class":241},[35,3540,3541],{"class":3409},"psilon$ at ~95% confidence.\n",[35,3543,3545],{"class":37,"line":3544},259,[35,3546,3410],{"class":3409},[35,3548,3550],{"class":37,"line":3549},260,[35,3551,103],{"emptyLinePlaceholder":102},[35,3553,3555,3558,3560,3562],{"class":37,"line":3554},261,[35,3556,3557],{"class":113},"code_config ",[35,3559,242],{"class":241},[35,3561,3406],{"class":109},[35,3563,3410],{"class":3409},[35,3565,3567],{"class":37,"line":3566},262,[35,3568,3569],{"class":41},"# @title Configuration\n",[35,3571,3573],{"class":37,"line":3572},263,[35,3574,3575],{"class":3409},"from dataclasses import dataclass\n",[35,3577,3579],{"class":37,"line":3578},264,[35,3580,3581],{"class":3409},"from typing import List, Dict\n",[35,3583,3585],{"class":37,"line":3584},265,[35,3586,3587],{"class":3409},"import numpy as np\n",[35,3589,3591],{"class":37,"line":3590},266,[35,3592,103],{"emptyLinePlaceholder":102},[35,3594,3596,3599],{"class":37,"line":3595},267,[35,3597,3598],{"class":3409},"RANDOM_SEED = 579345  ",[35,3600,3601],{"class":41},"# @param {type:\"number\"}\n",[35,3603,3605,3608],{"class":37,"line":3604},268,[35,3606,3607],{"class":3409},"SHOTS_PER_MODEL = 10000  ",[35,3609,3601],{"class":41},[35,3611,3613,3616,3619],{"class":37,"line":3612},269,[35,3614,3615],{"class":3409},"EPSILONS = ",[35,3617,3618],{"class":245},"[0.02, 0.01, 0.005]",[35,3620,3621],{"class":41},"  # @param\n",[35,3623,3625,3628],{"class":37,"line":3624},270,[35,3626,3627],{"class":3409},"COST_PER_SHOT_USD = 0.00005  ",[35,3629,3601],{"class":41},[35,3631,3633,3636],{"class":37,"line":3632},271,[35,3634,3635],{"class":3409},"FIXED_OVERHEAD_USD = 0.10    ",[35,3637,3601],{"class":41},[35,3639,3641],{"class":37,"line":3640},272,[35,3642,103],{"emptyLinePlaceholder":102},[35,3644,3646,3648,3650,3653],{"class":37,"line":3645},273,[35,3647,817],{"class":3409},[35,3649,275],{"class":245},[35,3651,3652],{"class":3409},"frozen=True",[35,3654,384],{"class":245},[35,3656,3658],{"class":37,"line":3657},274,[35,3659,3660],{"class":3409},"class QubitModel:\n",[35,3662,3664],{"class":37,"line":3663},275,[35,3665,3666],{"class":3409},"    name: str\n",[35,3668,3670],{"class":37,"line":3669},276,[35,3671,3672],{"class":3409},"    satisfaction_prob: float\n",[35,3674,3676],{"class":37,"line":3675},277,[35,3677,3678],{"class":3409},"    description: str\n",[35,3680,3682],{"class":37,"line":3681},278,[35,3683,103],{"emptyLinePlaceholder":102},[35,3685,3687,3690,3693,3696],{"class":37,"line":3686},279,[35,3688,3689],{"class":3409},"MODELS: List",[35,3691,3692],{"class":245},"[QubitModel]",[35,3694,3695],{"class":3409}," = ",[35,3697,3698],{"class":245},"[\n",[35,3700,3702],{"class":37,"line":3701},280,[35,3703,3704],{"class":245},"    QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n",[35,3706,3708],{"class":37,"line":3707},281,[35,3709,3710],{"class":245},"    QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n",[35,3712,3714],{"class":37,"line":3713},282,[35,3715,3716],{"class":245},"    QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n",[35,3718,3720],{"class":37,"line":3719},283,[35,3721,3722],{"class":245},"    QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n",[35,3724,3726],{"class":37,"line":3725},284,[35,3727,3728],{"class":245},"    QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n",[35,3730,3732],{"class":37,"line":3731},285,[35,3733,1074],{"class":245},[35,3735,3737],{"class":37,"line":3736},286,[35,3738,103],{"emptyLinePlaceholder":102},[35,3740,3742,3745,3747,3750],{"class":37,"line":3741},287,[35,3743,3744],{"class":3409},"rng = np.random.default_rng",[35,3746,275],{"class":245},[35,3748,3749],{"class":3409},"RANDOM_SEED",[35,3751,384],{"class":245},[35,3753,3755],{"class":37,"line":3754},288,[35,3756,3410],{"class":3409},[35,3758,3760],{"class":37,"line":3759},289,[35,3761,103],{"emptyLinePlaceholder":102},[35,3763,3765,3768,3770,3772],{"class":37,"line":3764},290,[35,3766,3767],{"class":113},"code_lib_and_run ",[35,3769,242],{"class":241},[35,3771,3406],{"class":109},[35,3773,3410],{"class":3409},[35,3775,3777],{"class":37,"line":3776},291,[35,3778,3779],{"class":41},"# @title Run Simulation & Display Results\n",[35,3781,3783],{"class":37,"line":3782},292,[35,3784,3785],{"class":3409},"import math\n",[35,3787,3789],{"class":37,"line":3788},293,[35,3790,3791],{"class":3409},"from typing import Tuple, Dict\n",[35,3793,3795],{"class":37,"line":3794},294,[35,3796,3797],{"class":3409},"import pandas as pd\n",[35,3799,3801],{"class":37,"line":3800},295,[35,3802,3587],{"class":3409},[35,3804,3806],{"class":37,"line":3805},296,[35,3807,3808],{"class":3409},"import matplotlib.pyplot as plt\n",[35,3810,3812],{"class":37,"line":3811},297,[35,3813,3814],{"class":3409},"from IPython.display import display, Markdown\n",[35,3816,3818],{"class":37,"line":3817},298,[35,3819,103],{"emptyLinePlaceholder":102},[35,3821,3823,3826,3828,3831,3833,3836,3839],{"class":37,"line":3822},299,[35,3824,3825],{"class":3409},"def wilson_ci",[35,3827,275],{"class":245},[35,3829,3830],{"class":3409},"k: int, n: int, z: float = 1.96",[35,3832,1694],{"class":245},[35,3834,3835],{"class":3409}," -> Tuple",[35,3837,3838],{"class":245},"[float, float]",[35,3840,364],{"class":3409},[35,3842,3844,3847,3849,3852],{"class":37,"line":3843},300,[35,3845,3846],{"class":3409},"    if n == 0: return ",[35,3848,275],{"class":245},[35,3850,3851],{"class":3409},"0.0, 1.0",[35,3853,384],{"class":245},[35,3855,3857,3860,3862],{"class":37,"line":3856},301,[35,3858,3859],{"class":3409},"    p_hat, z2_n = k \u002F n, z",[35,3861,421],{"class":245},[35,3863,3864],{"class":3409},"2 \u002F n\n",[35,3866,3868,3871,3873],{"class":37,"line":3867},302,[35,3869,3870],{"class":3409},"    denom = 1.0 ",[35,3872,445],{"class":245},[35,3874,3875],{"class":3409}," z2_n\n",[35,3877,3879,3882,3884,3887,3889,3892,3894],{"class":37,"line":3878},303,[35,3880,3881],{"class":3409},"    center = ",[35,3883,275],{"class":245},[35,3885,3886],{"class":3409},"p_hat ",[35,3888,445],{"class":245},[35,3890,3891],{"class":3409}," z2_n \u002F 2",[35,3893,1694],{"class":245},[35,3895,3896],{"class":3409}," \u002F denom\n",[35,3898,3900,3903,3905,3908,3910,3913,3915,3917,3919,3921,3924,3926,3929,3931,3934,3936,3939,3941,3944,3946,3949,3951,3953,3956],{"class":37,"line":3899},304,[35,3901,3902],{"class":3409},"    span = ",[35,3904,275],{"class":245},[35,3906,3907],{"class":3409},"z ",[35,3909,484],{"class":245},[35,3911,3912],{"class":3409}," math.sqrt",[35,3914,275],{"class":245},[35,3916,3886],{"class":3409},[35,3918,484],{"class":245},[35,3920,373],{"class":245},[35,3922,3923],{"class":3409},"1 - p_hat",[35,3925,1694],{"class":245},[35,3927,3928],{"class":3409}," \u002F n ",[35,3930,445],{"class":245},[35,3932,3933],{"class":3409}," z",[35,3935,421],{"class":245},[35,3937,3938],{"class":3409},"2 \u002F ",[35,3940,275],{"class":245},[35,3942,3943],{"class":3409},"4 ",[35,3945,484],{"class":245},[35,3947,3948],{"class":3409}," n",[35,3950,421],{"class":245},[35,3952,460],{"class":3409},[35,3954,3955],{"class":245},")))",[35,3957,3896],{"class":3409},[35,3959,3961,3964,3966,3968,3970,3973,3975,3978,3980,3983,3985,3988],{"class":37,"line":3960},305,[35,3962,3963],{"class":3409},"    return ",[35,3965,275],{"class":245},[35,3967,554],{"class":3409},[35,3969,275],{"class":245},[35,3971,3972],{"class":3409},"0.0, center - span",[35,3974,1694],{"class":245},[35,3976,3977],{"class":3409},", min",[35,3979,275],{"class":245},[35,3981,3982],{"class":3409},"1.0, center ",[35,3984,445],{"class":245},[35,3986,3987],{"class":3409}," span",[35,3989,2401],{"class":245},[35,3991,3993],{"class":37,"line":3992},306,[35,3994,103],{"emptyLinePlaceholder":102},[35,3996,3998,4001,4003,4006,4008],{"class":37,"line":3997},307,[35,3999,4000],{"class":3409},"def shots_for_epsilon",[35,4002,275],{"class":245},[35,4004,4005],{"class":3409},"p: float, epsilon: float = 0.01, z: float = 1.96",[35,4007,1694],{"class":245},[35,4009,4010],{"class":3409}," -> int:\n",[35,4012,4014,4017,4019,4021,4023,4025],{"class":37,"line":4013},308,[35,4015,4016],{"class":3409},"    if p ",[35,4018,484],{"class":245},[35,4020,373],{"class":245},[35,4022,3526],{"class":3409},[35,4024,1694],{"class":245},[35,4026,4027],{"class":3409}," == 0: return 1\n",[35,4029,4031,4034,4036,4039,4041,4044,4047,4049,4051,4054,4056,4058,4060,4062,4064,4067,4070,4072,4074,4076,4078],{"class":37,"line":4030},309,[35,4032,4033],{"class":3409},"    return max",[35,4035,275],{"class":245},[35,4037,4038],{"class":3409},"1, int",[35,4040,275],{"class":245},[35,4042,4043],{"class":3409},"math.ceil",[35,4045,4046],{"class":245},"((",[35,4048,300],{"class":3409},[35,4050,421],{"class":245},[35,4052,4053],{"class":3409},"2 ",[35,4055,484],{"class":245},[35,4057,700],{"class":3409},[35,4059,484],{"class":245},[35,4061,373],{"class":245},[35,4063,3526],{"class":3409},[35,4065,4066],{"class":245},"))",[35,4068,4069],{"class":3409}," \u002F ",[35,4071,275],{"class":245},[35,4073,612],{"class":3409},[35,4075,421],{"class":245},[35,4077,460],{"class":3409},[35,4079,4080],{"class":245},"))))\n",[35,4082,4084],{"class":37,"line":4083},310,[35,4085,103],{"emptyLinePlaceholder":102},[35,4087,4089,4092,4094,4097,4099],{"class":37,"line":4088},311,[35,4090,4091],{"class":3409},"def summarize_model",[35,4093,275],{"class":245},[35,4095,4096],{"class":3409},"model, bits: np.ndarray, epsilons",[35,4098,1694],{"class":245},[35,4100,4101],{"class":3409}," -> Dict:\n",[35,4103,4105,4108,4110,4112],{"class":37,"line":4104},312,[35,4106,4107],{"class":3409},"    n = len",[35,4109,275],{"class":245},[35,4111,1394],{"class":3409},[35,4113,384],{"class":245},[35,4115,4117,4120,4122,4124],{"class":37,"line":4116},313,[35,4118,4119],{"class":3409},"    ones = sum",[35,4121,275],{"class":245},[35,4123,1394],{"class":3409},[35,4125,384],{"class":245},[35,4127,4129,4132,4134,4137],{"class":37,"line":4128},314,[35,4130,4131],{"class":3409},"    lo, hi = wilson_ci",[35,4133,275],{"class":245},[35,4135,4136],{"class":3409},"ones, n",[35,4138,384],{"class":245},[35,4140,4142],{"class":37,"line":4141},315,[35,4143,4144],{"class":3409},"    row = {\n",[35,4146,4148,4151,4153,4156,4158],{"class":37,"line":4147},316,[35,4149,4150],{"class":3409},"        \"Model\": model.name, \"p ",[35,4152,275],{"class":245},[35,4154,4155],{"class":3409},"target",[35,4157,1694],{"class":245},[35,4159,4160],{"class":3409},"\": model.satisfaction_prob, \"Shots\": n,\n",[35,4162,4164,4167,4169,4172,4174],{"class":37,"line":4163},317,[35,4165,4166],{"class":3409},"        \"Ones\": ones, \"p ",[35,4168,275],{"class":245},[35,4170,4171],{"class":3409},"observed",[35,4173,1694],{"class":245},[35,4175,4176],{"class":3409},"\": ones \u002F n if n > 0 else 0,\n",[35,4178,4180],{"class":37,"line":4179},318,[35,4181,4182],{"class":3409},"        \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n",[35,4184,4186],{"class":37,"line":4185},319,[35,4187,1222],{"class":3409},[35,4189,4191],{"class":37,"line":4190},320,[35,4192,4193],{"class":3409},"    for eps in epsilons:\n",[35,4195,4197,4200,4203,4206,4208,4211],{"class":37,"line":4196},321,[35,4198,4199],{"class":3409},"        row",[35,4201,4202],{"class":245},"[f\"Shots for ±{eps:.1%}\"]",[35,4204,4205],{"class":3409}," = shots_for_epsilon",[35,4207,275],{"class":245},[35,4209,4210],{"class":3409},"model.satisfaction_prob, eps",[35,4212,384],{"class":245},[35,4214,4216],{"class":37,"line":4215},322,[35,4217,4218],{"class":3409},"    return row\n",[35,4220,4222],{"class":37,"line":4221},323,[35,4223,103],{"emptyLinePlaceholder":102},[35,4225,4227],{"class":37,"line":4226},324,[35,4228,4229],{"class":41},"# --- Run Simulation ---\n",[35,4231,4233,4236],{"class":37,"line":4232},325,[35,4234,4235],{"class":3409},"summary_rows = ",[35,4237,4238],{"class":245},"[summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\n",[35,4240,4242,4245,4247,4250],{"class":37,"line":4241},326,[35,4243,4244],{"class":3409},"summary_df = pd.DataFrame",[35,4246,275],{"class":245},[35,4248,4249],{"class":3409},"summary_rows",[35,4251,384],{"class":245},[35,4253,4255],{"class":37,"line":4254},327,[35,4256,4257],{"class":3409},"if COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n",[35,4259,4261,4264,4267,4270,4272,4275,4278,4280],{"class":37,"line":4260},328,[35,4262,4263],{"class":3409},"    summary_df",[35,4265,4266],{"class":245},"[\"Cost (USD)\"]",[35,4268,4269],{"class":3409}," = FIXED_OVERHEAD_USD ",[35,4271,445],{"class":245},[35,4273,4274],{"class":3409}," summary_df",[35,4276,4277],{"class":245},"[\"Shots\"]",[35,4279,463],{"class":245},[35,4281,4282],{"class":3409}," COST_PER_SHOT_USD\n",[35,4284,4286],{"class":37,"line":4285},329,[35,4287,103],{"emptyLinePlaceholder":102},[35,4289,4291,4293,4295,4297,4299,4301],{"class":37,"line":4290},330,[35,4292,3213],{"class":3409},[35,4294,275],{"class":245},[35,4296,3218],{"class":3409},[35,4298,275],{"class":245},[35,4300,2986],{"class":3409},[35,4302,4303],{"class":41},"### Simulation Summary\"))\n",[35,4305,4307,4309,4311,4314,4316,4319,4321,4324,4326,4329],{"class":37,"line":4306},331,[35,4308,3213],{"class":3409},[35,4310,275],{"class":245},[35,4312,4313],{"class":3409},"summary_df.style.format",[35,4315,275],{"class":245},[35,4317,4318],{"class":3409},"precision=4",[35,4320,1694],{"class":245},[35,4322,4323],{"class":3409},".background_gradient",[35,4325,275],{"class":245},[35,4327,4328],{"class":3409},"cmap='viridis', subset=",[35,4330,4331],{"class":245},"['p (observed)', 'CI width']))\n",[35,4333,4335],{"class":37,"line":4334},332,[35,4336,103],{"emptyLinePlaceholder":102},[35,4338,4340],{"class":37,"line":4339},333,[35,4341,4342],{"class":41},"# --- Plotting ---\n",[35,4344,4346,4349,4351,4353],{"class":37,"line":4345},334,[35,4347,4348],{"class":3409},"plt.style.use",[35,4350,275],{"class":245},[35,4352,2370],{"class":3409},[35,4354,384],{"class":245},[35,4356,4358,4361,4363,4366,4368,4371,4373,4376,4378,4381],{"class":37,"line":4357},335,[35,4359,4360],{"class":3409},"fig, ",[35,4362,275],{"class":245},[35,4364,4365],{"class":3409},"ax1, ax2",[35,4367,1694],{"class":245},[35,4369,4370],{"class":3409}," = plt.subplots",[35,4372,275],{"class":245},[35,4374,4375],{"class":3409},"1, 2, figsize=",[35,4377,275],{"class":245},[35,4379,4380],{"class":3409},"16, 6",[35,4382,2401],{"class":245},[35,4384,4386],{"class":37,"line":4385},336,[35,4387,103],{"emptyLinePlaceholder":102},[35,4389,4391],{"class":37,"line":4390},337,[35,4392,4393],{"class":41},"# Plot 1\n",[35,4395,4397,4400,4403,4406],{"class":37,"line":4396},338,[35,4398,4399],{"class":3409},"y = summary_df",[35,4401,4402],{"class":245},"[\"p (observed)\"]",[35,4404,4405],{"class":3409},".to_numpy",[35,4407,2101],{"class":245},[35,4409,4411,4414],{"class":37,"line":4410},339,[35,4412,4413],{"class":3409},"lower_error = y - summary_df",[35,4415,4416],{"class":245},"[\"CI 95% low\"]\n",[35,4418,4420,4423,4426],{"class":37,"line":4419},340,[35,4421,4422],{"class":3409},"upper_error = summary_df",[35,4424,4425],{"class":245},"[\"CI 95% high\"]",[35,4427,4428],{"class":3409}," - y\n",[35,4430,4432,4435],{"class":37,"line":4431},341,[35,4433,4434],{"class":3409},"yerr = np.vstack",[35,4436,4437],{"class":245},"([np.abs(lower_error), np.abs(upper_error)])\n",[35,4439,4441],{"class":37,"line":4440},342,[35,4442,103],{"emptyLinePlaceholder":102},[35,4444,4446,4449,4451,4454,4457,4460],{"class":37,"line":4445},343,[35,4447,4448],{"class":3409},"ax1.bar",[35,4450,275],{"class":245},[35,4452,4453],{"class":3409},"summary_df",[35,4455,4456],{"class":245},"[\"Model\"]",[35,4458,4459],{"class":3409},", y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black'",[35,4461,384],{"class":245},[35,4463,4465,4468,4470,4473,4475,4478,4480,4483],{"class":37,"line":4464},344,[35,4466,4467],{"class":3409},"ax1.set_title",[35,4469,275],{"class":245},[35,4471,4472],{"class":3409},"\"Observed Success Rate ",[35,4474,275],{"class":245},[35,4476,4477],{"class":3409},"95% Wilson CI",[35,4479,1694],{"class":245},[35,4481,4482],{"class":3409},"\", fontsize=14",[35,4484,384],{"class":245},[35,4486,4488,4491,4493,4496],{"class":37,"line":4487},345,[35,4489,4490],{"class":3409},"ax1.tick_params",[35,4492,275],{"class":245},[35,4494,4495],{"class":3409},"axis='x', rotation=30, labelsize=10",[35,4497,384],{"class":245},[35,4499,4501],{"class":37,"line":4500},346,[35,4502,103],{"emptyLinePlaceholder":102},[35,4504,4506],{"class":37,"line":4505},347,[35,4507,4508],{"class":41},"# Plot 2\n",[35,4510,4512],{"class":37,"line":4511},348,[35,4513,4514],{"class":3409},"width = 0.2\n",[35,4516,4518,4521,4523,4525,4527,4530],{"class":37,"line":4517},349,[35,4519,4520],{"class":3409},"x = np.arange",[35,4522,275],{"class":245},[35,4524,2420],{"class":3409},[35,4526,275],{"class":245},[35,4528,4529],{"class":3409},"MODELS",[35,4531,2401],{"class":245},[35,4533,4535,4538],{"class":37,"line":4534},350,[35,4536,4537],{"class":3409},"colors = ",[35,4539,4540],{"class":245},"['#ff9999','#66b3ff','#99ff99']\n",[35,4542,4544,4547,4549,4552,4554],{"class":37,"line":4543},351,[35,4545,4546],{"class":3409},"for i, eps in enumerate",[35,4548,275],{"class":245},[35,4550,4551],{"class":3409},"EPSILONS",[35,4553,1694],{"class":245},[35,4555,364],{"class":3409},[35,4557,4559,4562],{"class":37,"line":4558},352,[35,4560,4561],{"class":3409},"    req = summary_df",[35,4563,4564],{"class":245},"[f\"Shots for ±{eps:.1%}\"]\n",[35,4566,4568,4571,4573,4575,4577,4579,4581,4584,4587,4590],{"class":37,"line":4567},353,[35,4569,4570],{"class":3409},"    ax2.bar",[35,4572,275],{"class":245},[35,4574,2407],{"class":3409},[35,4576,445],{"class":245},[35,4578,2958],{"class":3409},[35,4580,484],{"class":245},[35,4582,4583],{"class":3409}," width, req, width, label=f\"±{eps:.1%}\", color=colors",[35,4585,4586],{"class":245},"[i]",[35,4588,4589],{"class":3409},", edgecolor='black'",[35,4591,384],{"class":245},[35,4593,4595],{"class":37,"line":4594},354,[35,4596,103],{"emptyLinePlaceholder":102},[35,4598,4600,4603,4605,4607,4609,4612],{"class":37,"line":4599},355,[35,4601,4602],{"class":3409},"ax2.set_xticks",[35,4604,275],{"class":245},[35,4606,2407],{"class":3409},[35,4608,445],{"class":245},[35,4610,4611],{"class":3409}," width, summary_df",[35,4613,4614],{"class":245},"[\"Model\"])\n",[35,4616,4618,4621,4623,4626,4628,4631,4633,4635],{"class":37,"line":4617},356,[35,4619,4620],{"class":3409},"ax2.set_ylabel",[35,4622,275],{"class":245},[35,4624,4625],{"class":3409},"\"Required Shots ",[35,4627,275],{"class":245},[35,4629,4630],{"class":3409},"Log Scale",[35,4632,1694],{"class":245},[35,4634,2986],{"class":3409},[35,4636,384],{"class":245},[35,4638,4640,4643,4645,4647],{"class":37,"line":4639},357,[35,4641,4642],{"class":3409},"ax2.set_yscale",[35,4644,275],{"class":245},[35,4646,3098],{"class":3409},[35,4648,384],{"class":245},[35,4650,4652,4655,4657,4660],{"class":37,"line":4651},358,[35,4653,4654],{"class":3409},"ax2.set_title",[35,4656,275],{"class":245},[35,4658,4659],{"class":3409},"\"Shots Needed for ±ε Tolerance\", 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nbf.v4.",[35,4756,4757],{"class":271},"new_markdown_cell",[35,4759,4760],{"class":113},"(md_intro),\n",[35,4762,4764,4766,4769],{"class":37,"line":4763},369,[35,4765,4754],{"class":113},[35,4767,4768],{"class":271},"new_code_cell",[35,4770,4771],{"class":113},"(code_config),\n",[35,4773,4775,4777,4779],{"class":37,"line":4774},370,[35,4776,4754],{"class":113},[35,4778,4768],{"class":271},[35,4780,4781],{"class":113},"(code_lib_and_run),\n",[35,4783,4785],{"class":37,"line":4784},371,[35,4786,1074],{"class":113},[35,4788,4790],{"class":37,"line":4789},372,[35,4791,103],{"emptyLinePlaceholder":102},[35,4793,4795,4798,4800],{"class":37,"line":4794},373,[35,4796,4797],{"class":113},"nb_path ",[35,4799,242],{"class":241},[35,4801,4802],{"class":234}," \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\n",[35,4804,4806,4808,4810,4813,4815,4817,4820,4822,4825,4827,4829],{"class":37,"line":4805},374,[35,4807,2272],{"class":109},[35,4809,2275],{"class":241},[35,4811,4812],{"class":113},"(nb_path, ",[35,4814,2285],{"class":234},[35,4816,288],{"class":113},[35,4818,4819],{"class":822},"encoding",[35,4821,242],{"class":241},[35,4823,4824],{"class":234},"\"utf-8\"",[35,4826,427],{"class":113},[35,4828,165],{"class":109},[35,4830,2292],{"class":113},[35,4832,4834,4837,4840],{"class":37,"line":4833},375,[35,4835,4836],{"class":113},"    nbf.",[35,4838,4839],{"class":271},"write",[35,4841,4842],{"class":113},"(nb, f)\n",[35,4844,4846],{"class":37,"line":4845},376,[35,4847,103],{"emptyLinePlaceholder":102},[35,4849,4851],{"class":37,"line":4850},377,[35,4852,4853],{"class":41},"# ---------- Final confirmation ----------\n",[35,4855,4857,4859,4861,4863,4866,4868,4870,4873,4875,4878],{"class":37,"line":4856},378,[35,4858,1861],{"class":241},[35,4860,275],{"class":113},[35,4862,2986],{"class":234},[35,4864,4865],{"class":241},"\\n",[35,4867,2986],{"class":234},[35,4869,415],{"class":241},[35,4871,4872],{"class":234}," 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",[35,4921,1681],{"class":245},[35,4923,4924],{"class":113},"nb_path",[35,4926,1697],{"class":245},[35,4928,2986],{"class":234},[35,4930,384],{"class":113},[35,4932,4934,4936,4938,4940,4943,4945,4948,4950,4952],{"class":37,"line":4933},382,[35,4935,1861],{"class":241},[35,4937,275],{"class":113},[35,4939,1675],{"class":109},[35,4941,4942],{"class":234},"\"Summary data saved to: ",[35,4944,1681],{"class":245},[35,4946,4947],{"class":113},"summary_csv_path",[35,4949,1697],{"class":245},[35,4951,2986],{"class":234},[35,4953,384],{"class":113},[35,4955,4957,4959,4961,4963,4966,4968,4971,4973,4975,4977,4980,4982,4984],{"class":37,"line":4956},383,[35,4958,1861],{"class":241},[35,4960,275],{"class":113},[35,4962,1675],{"class":109},[35,4964,4965],{"class":234},"\"Plots saved to: ",[35,4967,1681],{"class":245},[35,4969,4970],{"class":113},"fig1_path",[35,4972,1697],{"class":245},[35,4974,288],{"class":234},[35,4976,1681],{"class":245},[35,4978,4979],{"class":113},"fig2_path",[35,4981,1697],{"class":245},[35,4983,2986],{"class":234},[35,4985,384],{"class":113},[25,4987,4992],{"className":4988,"code":4990,"language":4991,"meta":30},[4989],"language-text","Running simulation...\nSaved data to CSV and JSON files.\n","text",[32,4993,4990],{"__ignoreMap":30},[4995,4996],"blog-figure",{"alt":4997,"src":4998},"Output 1 of the notebook code above","\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.webp",[4995,5000],{"alt":5001,"src":5002},"Output 2 of the notebook code above","\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-02.webp",[25,5004,5007],{"className":5005,"code":5006,"language":4991,"meta":30},[4989],"Generated and saved plots.\n",[32,5008,5006],{"__ignoreMap":30},[25,5010,5013],{"className":5011,"code":5012,"language":4991,"meta":30},[4989],"==================================================\nSCRIPT COMPLETE\n==================================================\nGoogle Colab notebook saved to: Sampling_Budget_Qubit_Bitstrings.ipynb\nSummary data saved to: summary.csv\nPlots saved to: fig_ones_fraction.png, fig_required_shots.png\n",[32,5014,5012],{"__ignoreMap":30},[5016,5017],"hr",{},[5019,5020,5022],"h2",{"id":5021},"references","References",[5024,5025,5026,5034,5041,5048,5055,5062,5069,5076,5083,5090,5097,5104,5111,5123,5130,5137],"ol",{},[5027,5028,5029,5030,5033],"li",{},"Agresti, A. and Coull, B.A. (1998) 'Approximate is better than “exact” for interval estimation of binomial proportions', ",[15,5031,5032],{},"The American Statistician",", 52(2), pp. 119-126.",[5027,5035,5036,5037,5040],{},"Brown, L.D., Cai, T.T. and DasGupta, A. (2001) 'Interval Estimation for a Binomial Proportion', ",[15,5038,5039],{},"Statistical Science",", 16(2), pp. 101-133.",[5027,5042,5043,5044,5047],{},"Cochran, W.G. (1977) ",[15,5045,5046],{},"Sampling Techniques",". 3rd edn. New York: John Wiley & Sons.",[5027,5049,5050,5051,5054],{},"Gottesman, D. (2009) 'An introduction to quantum error correction and fault-tolerant quantum computation', in ",[15,5052,5053],{},"Quantum Information Science and Its Contributions to Mathematics, Proceedings of Symposia in Applied Mathematics",", 72, pp. 13-58.",[5027,5056,5057,5058,5061],{},"Harris, C.R., Millman, K.J., van der Walt, S.J., Gommers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N.J., Kern, R., Picus, M., Hoyer, S., van Kerkwijk, M.H., Brett, M., Haldane, A., del Río, J.F., Wiebe, M., Peterson, P., Gérard-Marchant, P., Sheppard, K., Reddy, T., Weckesser, W., Abbasi, H., Gohlke, C. and Oliphant, T.E. (2020) 'Array programming with NumPy', ",[15,5059,5060],{},"Nature",", 585, pp. 357–362.",[5027,5063,5064,5065,5068],{},"Hunter, J.D. (2007) 'Matplotlib: A 2D Graphics Environment', ",[15,5066,5067],{},"Computing in Science & Engineering",", 9(3), pp. 90-95.",[5027,5070,5071,5072,5075],{},"Kitaev, A.Y. (2003) 'Fault-tolerant quantum computation by anyons', ",[15,5073,5074],{},"Annals of Physics",", 303(1), pp. 2-30.",[5027,5077,5078,5079,5082],{},"McKinney, W. (2010) 'Data Structures for Statistical Computing in Python', in ",[15,5080,5081],{},"Proceedings of the 9th Python in Science Conference",", pp. 56-61.",[5027,5084,5085,5086,5089],{},"Nayak, C., Simon, S.H., Stern, A., Freedman, M. and Das Sarma, S. (2008) 'Non-Abelian anyons and topological quantum computation', ",[15,5087,5088],{},"Reviews of Modern Physics",", 80(3), pp. 1083-1159.",[5027,5091,5092,5093,5096],{},"Newman, M. (2013) ",[15,5094,5095],{},"Computational Physics",". CreateSpace Independent Publishing Platform.",[5027,5098,5099,5100,5103],{},"Nielsen, M.A. and Chuang, I.L. (2010) ",[15,5101,5102],{},"Quantum Computation and Quantum Information: 10th Anniversary Edition",". Cambridge: Cambridge University Press.",[5027,5105,5106,5107,5110],{},"Preskill, J. (2018) 'Quantum Computing in the NISQ era and beyond', ",[15,5108,5109],{},"Quantum",", 2, p. 79.",[5027,5112,5113,5114,5117,5118,5122],{},"The pandas development team (2020) ",[15,5115,5116],{},"pandas-dev\u002Fpandas: Pandas",". Zenodo. Available at: ",[19,5119,5121],{"href":5120},"https:\u002F\u002Fwww.google.com\u002Fsearch?q=https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134","https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134",".",[5027,5124,5125,5126,5129],{},"Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Millman, K.J., Mayorov, N., Nelson, A.R.J., Jones, E., Kern, R., Larson, E., Carey, C.J., Polat, İ., Feng, Y., Moore, E.W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E.A., Harris, C.R., Archibald, A.M., Ribeiro, A.H., Pedregosa, F., van Mulbregt, P. and SciPy 1.0 Contributors (2020) 'SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python', ",[15,5127,5128],{},"Nature Methods",", 17, pp. 261–272.",[5027,5131,5132,5133,5136],{},"Wilde, M.M. (2017) ",[15,5134,5135],{},"Quantum Information Theory",". 2nd edn. Cambridge: Cambridge University Press.",[5027,5138,5139,5140,5143],{},"Wilson, E.B. (1927) 'Probable inference, the law of succession, and statistical inference', ",[15,5141,5142],{},"Journal of the American Statistical Association",", 22(158), pp. 209-212.",[5019,5145,5147],{"id":5146},"run-it-yourself","Run it yourself",[12,5149,5150,5151,5153,5154,5122],{},"Get ",[19,5152,22],{"href":21}," or ",[19,5155,5157],{"href":5156},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb","open it in Colab",[5159,5160,5161],"style",{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .sU0A5, html code.shiki .sU0A5{--shiki-default:#E5C07B}html pre.shiki code .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":30,"searchDepth":45,"depth":45,"links":5163},[5164,5165],{"id":5021,"depth":45,"text":5022},{"id":5146,"depth":45,"text":5147},[5167,5168,5169,5170],"Qollab","Blog","Expert notes","Sampling budget",[5172],{"username":7,"name":5173,"role":5174,"bio":5175,"links":5176},"Onri Jay Benally","Quantum Hardware Engineer, IBM · University of Minnesota","A quantum hardware engineer at IBM and graduate researcher at the Nano Magnetism & Quantum Spintronics Lab, University of Minnesota-Twin Cities. Onri shares runnable notebooks that make hardware-level quantum concepts visible.",[5177,5180],{"label":5178,"href":5179},"GitHub ↗","https:\u002F\u002Fgithub.com\u002FOJB-Quantum",{"label":5181,"href":21},"Original notebook ↗",{"username":7,"name":5173,"role":5183},"Quantum Hardware Engineer, IBM",null,"A Sampling-Budget & Bit-String Analogy for Qubit Classifications. A notebook by Onri Jay Benally, republished with permission.",false,"md","Deep dive · Hardware","article",{},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications","2026-07-15","4 min read",[],{"title":5,"description":5185},"blog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications",[5199,5200],"hardware","quantum","HETemA9RSbsiqyryZFFLOYbrY0iNodc-iLIt3qtwByA",[5203,5693,46138,71840],{"id":5204,"title":5205,"authors":5206,"body":5207,"breadcrumb":5651,"builders":5653,"byline":5658,"challenge":5184,"courseAuthor":5184,"courseLead":5184,"dek":5659,"description":5660,"draft":5186,"extension":5187,"eyebrow":5661,"finish":5184,"fork":5184,"hero":5184,"heroAlt":5184,"heroCta":5184,"heroImage":5184,"kind":5189,"lessonCount":5184,"meta":5662,"navigation":102,"newsItems":5184,"next":5184,"ogImage":5663,"order":5184,"outcomes":5184,"path":5664,"publishDate":5193,"readingTime":5665,"related":5666,"relatedProjects":5667,"seo":5687,"stem":5689,"tags":5690,"track":5184,"trackName":5184,"__hash__":5692},"blog\u002Fblog\u002Fexpert-notes\u002F2-qubit-state-visualization.md","2-qubit states, visualized 8 ways",[7],{"type":9,"value":5208,"toc":5639},[5209,5212,5224,5228,5231,5508,5511,5515,5518,5524,5527,5533,5537,5540,5546,5550,5553,5559,5563,5566,5572,5576,5579,5583,5586,5592,5596,5599,5605,5609,5612,5618,5624,5626,5636],[12,5210,5211],{},"A 2-qubit state is a four-dimensional complex vector. That's easy to write down and hard to picture, and entanglement makes it harder: the most interesting thing about the state is precisely the part you can't see by looking at each qubit alone. This walkthrough takes one small circuit and renders its state eight different ways, because each visualization makes a different property visible.",[12,5213,5214,5215,5219,5220,5122],{},"Everything below comes from Onri's runnable notebook, shared openly on ",[19,5216,5218],{"href":5217},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002F2_Qubit_State_Visualization_in_Python.ipynb","GitHub",". The code and figures are his, republished here with permission. You can run the whole thing end to end in ",[19,5221,5223],{"href":5222},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002F2_Qubit_State_Visualization_in_Python.ipynb","Google Colab",[5019,5225,5227],{"id":5226},"one-circuit-two-states","One circuit, two states",[12,5229,5230],{},"The experiment is a controlled comparison. First, the baseline: the product state |00⟩, where each qubit has its own well-defined state and nothing is correlated. Then the same eight visualizations run again on an entangled state, built with a Hadamard, a CNOT, and a couple of single-qubit rotations:",[5232,5233,5236],"code-block",{"name":5234,"tag":5235},"2_qubit_state_visualization.ipynb","Python",[25,5237,5239],{"className":27,"code":5238,"language":29,"meta":30,"style":30},"import numpy as np\nfrom qiskit import QuantumCircuit\nfrom qiskit.quantum_info import Statevector, DensityMatrix\n\n# Create a circuit with 2 qubits\nqc = QuantumCircuit(2)\nqc.h(0)             # Hadamard on q0\nqc.cx(0, 1)         # CNOT with q0 as control, q1 as target\nqc.ry(np.pi \u002F 4, 1) # Y-rotation of pi\u002F4 on q1\nqc.z(1)             # Phase flip (Z gate) on q1\n\nfinal_statevector = Statevector(qc)\nfinal_density_matrix = DensityMatrix(qc)\n\n# Generate all 8 plots for the final state\nplot_bloch_multivector_viz(final_density_matrix, \"Final Entangled State\")\nplot_q_sphere_viz(final_density_matrix, \"Final Entangled State\")\nplot_steering_ellipsoid(final_density_matrix, \"Steering Ellipsoid for Final State\")\nvisualize_schmidt_decomposition(final_statevector, \"Schmidt Decomposition for Final State\")\nexplain_toroidal_geometry()\nplot_correlation_invariants(final_density_matrix, \"Final Entangled State\")\nplot_wigner_function(final_density_matrix, \"Wigner Functions for Final State\")\nplot_density_matrix(final_density_matrix, \"Density Matrix for Final State\")\n",[32,5240,5241,5251,5263,5275,5279,5284,5300,5319,5340,5364,5379,5383,5396,5408,5412,5417,5430,5441,5453,5466,5473,5484,5496],{"__ignoreMap":30},[35,5242,5243,5245,5247,5249],{"class":37,"line":38},[35,5244,110],{"class":109},[35,5246,162],{"class":113},[35,5248,165],{"class":109},[35,5250,168],{"class":113},[35,5252,5253,5255,5258,5260],{"class":37,"line":45},[35,5254,128],{"class":109},[35,5256,5257],{"class":113}," qiskit ",[35,5259,110],{"class":109},[35,5261,5262],{"class":113}," QuantumCircuit\n",[35,5264,5265,5267,5270,5272],{"class":37,"line":51},[35,5266,128],{"class":109},[35,5268,5269],{"class":113}," qiskit.quantum_info ",[35,5271,110],{"class":109},[35,5273,5274],{"class":113}," Statevector, DensityMatrix\n",[35,5276,5277],{"class":37,"line":57},[35,5278,103],{"emptyLinePlaceholder":102},[35,5280,5281],{"class":37,"line":63},[35,5282,5283],{"class":41},"# Create a circuit with 2 qubits\n",[35,5285,5286,5289,5291,5294,5296,5298],{"class":37,"line":69},[35,5287,5288],{"class":113},"qc ",[35,5290,242],{"class":241},[35,5292,5293],{"class":271}," QuantumCircuit",[35,5295,275],{"class":113},[35,5297,460],{"class":245},[35,5299,384],{"class":113},[35,5301,5302,5305,5308,5310,5313,5316],{"class":37,"line":75},[35,5303,5304],{"class":113},"qc.",[35,5306,5307],{"class":271},"h",[35,5309,275],{"class":113},[35,5311,5312],{"class":245},"0",[35,5314,5315],{"class":113},")             ",[35,5317,5318],{"class":41},"# Hadamard on q0\n",[35,5320,5321,5323,5326,5328,5330,5332,5334,5337],{"class":37,"line":81},[35,5322,5304],{"class":113},[35,5324,5325],{"class":271},"cx",[35,5327,275],{"class":113},[35,5329,5312],{"class":245},[35,5331,288],{"class":113},[35,5333,500],{"class":245},[35,5335,5336],{"class":113},")         ",[35,5338,5339],{"class":41},"# CNOT with q0 as control, q1 as target\n",[35,5341,5342,5344,5347,5350,5352,5355,5357,5359,5361],{"class":37,"line":87},[35,5343,5304],{"class":113},[35,5345,5346],{"class":271},"ry",[35,5348,5349],{"class":113},"(np.pi ",[35,5351,398],{"class":241},[35,5353,5354],{"class":245}," 4",[35,5356,288],{"class":113},[35,5358,500],{"class":245},[35,5360,427],{"class":113},[35,5362,5363],{"class":41},"# Y-rotation of pi\u002F4 on q1\n",[35,5365,5366,5368,5370,5372,5374,5376],{"class":37,"line":93},[35,5367,5304],{"class":113},[35,5369,300],{"class":271},[35,5371,275],{"class":113},[35,5373,500],{"class":245},[35,5375,5315],{"class":113},[35,5377,5378],{"class":41},"# Phase flip (Z gate) on q1\n",[35,5380,5381],{"class":37,"line":99},[35,5382,103],{"emptyLinePlaceholder":102},[35,5384,5385,5388,5390,5393],{"class":37,"line":106},[35,5386,5387],{"class":113},"final_statevector ",[35,5389,242],{"class":241},[35,5391,5392],{"class":271}," Statevector",[35,5394,5395],{"class":113},"(qc)\n",[35,5397,5398,5401,5403,5406],{"class":37,"line":117},[35,5399,5400],{"class":113},"final_density_matrix ",[35,5402,242],{"class":241},[35,5404,5405],{"class":271}," DensityMatrix",[35,5407,5395],{"class":113},[35,5409,5410],{"class":37,"line":125},[35,5411,103],{"emptyLinePlaceholder":102},[35,5413,5414],{"class":37,"line":139},[35,5415,5416],{"class":41},"# Generate all 8 plots for the final state\n",[35,5418,5419,5422,5425,5428],{"class":37,"line":152},[35,5420,5421],{"class":271},"plot_bloch_multivector_viz",[35,5423,5424],{"class":113},"(final_density_matrix, ",[35,5426,5427],{"class":234},"\"Final Entangled State\"",[35,5429,384],{"class":113},[35,5431,5432,5435,5437,5439],{"class":37,"line":157},[35,5433,5434],{"class":271},"plot_q_sphere_viz",[35,5436,5424],{"class":113},[35,5438,5427],{"class":234},[35,5440,384],{"class":113},[35,5442,5443,5446,5448,5451],{"class":37,"line":171},[35,5444,5445],{"class":271},"plot_steering_ellipsoid",[35,5447,5424],{"class":113},[35,5449,5450],{"class":234},"\"Steering Ellipsoid for Final State\"",[35,5452,384],{"class":113},[35,5454,5455,5458,5461,5464],{"class":37,"line":184},[35,5456,5457],{"class":271},"visualize_schmidt_decomposition",[35,5459,5460],{"class":113},"(final_statevector, ",[35,5462,5463],{"class":234},"\"Schmidt Decomposition for Final State\"",[35,5465,384],{"class":113},[35,5467,5468,5471],{"class":37,"line":197},[35,5469,5470],{"class":271},"explain_toroidal_geometry",[35,5472,2101],{"class":113},[35,5474,5475,5478,5480,5482],{"class":37,"line":210},[35,5476,5477],{"class":271},"plot_correlation_invariants",[35,5479,5424],{"class":113},[35,5481,5427],{"class":234},[35,5483,384],{"class":113},[35,5485,5486,5489,5491,5494],{"class":37,"line":223},[35,5487,5488],{"class":271},"plot_wigner_function",[35,5490,5424],{"class":113},[35,5492,5493],{"class":234},"\"Wigner Functions for Final State\"",[35,5495,384],{"class":113},[35,5497,5498,5501,5503,5506],{"class":37,"line":228},[35,5499,5500],{"class":271},"plot_density_matrix",[35,5502,5424],{"class":113},[35,5504,5505],{"class":234},"\"Density Matrix for Final State\"",[35,5507,384],{"class":113},[12,5509,5510],{},"Every method below shows the entangled state; where the contrast with |00⟩ is the whole point, both are shown.",[5019,5512,5514],{"id":5513},"_1-two-bloch-spheres-one-giveaway","1. Two Bloch spheres, one giveaway",[12,5516,5517],{},"The Bloch multivector shows the state of each qubit on its own Bloch sphere. For |00⟩, both vectors reach the surface: each qubit is in a pure, definite state.",[4995,5519],{"alt":5520,"caption":5521,"no":5522,"src":5523},"Two Bloch spheres for the initial state |00>, each with a vector pointing to the north pole","The product state |00⟩: each qubit's vector reaches the surface of its own sphere.","Fig. 1","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fbloch-initial.webp",[12,5525,5526],{},"For the entangled state, the vectors retreat inside the sphere. Each qubit on its own looks like a mixed state. The information isn't in either qubit, it's in the correlation between them. The shrunken vector is the visual signature of entanglement.",[4995,5528],{"alt":5529,"caption":5530,"no":5531,"src":5532},"Two Bloch spheres for the entangled state, each vector noticeably inside the sphere","The entangled state: both vectors pull inside the sphere; each qubit alone is mixed.","Fig. 2","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fbloch-entangled.webp",[5019,5534,5536],{"id":5535},"_2-the-q-sphere","2. The Q-sphere",[12,5538,5539],{},"The Q-sphere maps state amplitudes and phases onto a single sphere: the size of each point is the magnitude of the amplitude for a basis state, and its color is the phase. Where the Bloch view loses the correlations, the Q-sphere shows the full 2-qubit state at once.",[4995,5541],{"alt":5542,"caption":5543,"no":5544,"src":5545},"Q-sphere of the entangled state showing amplitude points with phase coloring","The entangled state on the Q-sphere: amplitude as point size, phase as color.","Fig. 3","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fqsphere-entangled.webp",[5019,5547,5549],{"id":5548},"_3-the-steering-ellipsoid","3. The steering ellipsoid",[12,5551,5552],{},"The steering ellipsoid shows the set of states qubit B can be \"steered\" into by measuring qubit A. If it collapses to a single point, there is no steering at all; a non-trivial ellipsoid means the state is entangled. For |00⟩ the notebook can't even draw one: the ellipsoid degenerates to a point, which is its own kind of answer.",[4995,5554],{"alt":5555,"caption":5556,"no":5557,"src":5558},"Quantum steering ellipsoid for the entangled state","The entangled state's steering ellipsoid: measuring qubit A can steer qubit B anywhere on this surface.","Fig. 4","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fsteering-ellipsoid-entangled.webp",[5019,5560,5562],{"id":5561},"_4-schmidt-decomposition","4. Schmidt decomposition",[12,5564,5565],{},"Any pure 2-qubit state can be written as |ψ⟩ = λ₀|u₀⟩|v₀⟩ + λ₁|u₁⟩|v₁⟩. The Schmidt coefficients λ₀ and λ₁ quantify entanglement directly: a product state has one coefficient equal to 1 and the other 0, while an entangled state splits the weight between both terms.",[4995,5567],{"alt":5568,"caption":5569,"no":5570,"src":5571},"Bar chart of the Schmidt coefficients for the entangled state","Both Schmidt coefficients are non-zero: the weight is shared, so the state is entangled.","Fig. 5","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fschmidt-entangled.webp",[5019,5573,5575],{"id":5574},"_5-toroidal-geometry","5. Toroidal geometry",[12,5577,5578],{},"Not everything has a standard plot. The toroidal view is a highly abstract method that maps entanglement invariants onto the surface of a torus. Powerful for classifying entanglement, but beyond standard plotting libraries. The notebook explains the concept and moves on; it's included because knowing a representation exists is half of finding a use for it.",[5019,5580,5582],{"id":5581},"_6-correlation-invariants","6. Correlation invariants",[12,5584,5585],{},"This plot shows the invariants of the state under local unitary operations: the eigenvalues of TᵀT, where T is the correlation matrix. These numbers don't change when either qubit is rotated on its own, so they capture the nature and strength of the correlation itself: the part of the state that no local operation can create or destroy.",[4995,5587],{"alt":5588,"caption":5589,"no":5590,"src":5591},"Bar plot of local correlation invariants for the entangled state","Local invariants of the entangled state: what remains when you ignore everything each qubit does alone.","Fig. 6","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fcorrelation-invariants-entangled.webp",[5019,5593,5595],{"id":5594},"_7-the-wigner-function","7. The Wigner function",[12,5597,5598],{},"A phase-space representation of each qubit's reduced state. Negative values, shown in red, are a key signature of non-classicality. No classical probability distribution can go below zero.",[4995,5600],{"alt":5601,"caption":5602,"no":5603,"src":5604},"Wigner function plots for the entangled state with negative regions shown in red","Wigner functions for the entangled state: the red, negative regions have no classical counterpart.","Fig. 7","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fwigner-entangled.webp",[5019,5606,5608],{"id":5607},"_8-the-density-matrix","8. The density matrix",[12,5610,5611],{},"The workhorse. Real parts in blue, imaginary parts in red; the diagonal holds the populations and the off-diagonal elements indicate coherence and entanglement. For |00⟩, a single bar stands alone.",[4995,5613],{"alt":5614,"caption":5615,"no":5616,"src":5617},"Density matrix of the initial state |00>, a single bar at the 00 position","The density matrix of |00⟩: one population, no coherences.","Fig. 8","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fdensity-initial.webp",[4995,5619],{"alt":5620,"caption":5621,"no":5622,"src":5623},"Density matrix of the entangled state with multiple bars including off-diagonal elements","The entangled state: off-diagonal structure appears, the coherences that carry the correlations.","Fig. 9","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fdensity-entangled.webp",[5019,5625,5147],{"id":5146},[12,5627,5628,5629,5153,5632,5635],{},"Every figure above comes out of one script you can run top to bottom: ",[19,5630,5631],{"href":5217},"open the notebook on GitHub",[19,5633,5634],{"href":5222},"run it directly in Colab",". Change the circuit in Step 5, run again, and watch all eight views shift together. That's where the intuition comes from.",[5159,5637,5638],{},"html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":30,"searchDepth":45,"depth":45,"links":5640},[5641,5642,5643,5644,5645,5646,5647,5648,5649,5650],{"id":5226,"depth":45,"text":5227},{"id":5513,"depth":45,"text":5514},{"id":5535,"depth":45,"text":5536},{"id":5548,"depth":45,"text":5549},{"id":5561,"depth":45,"text":5562},{"id":5574,"depth":45,"text":5575},{"id":5581,"depth":45,"text":5582},{"id":5594,"depth":45,"text":5595},{"id":5607,"depth":45,"text":5608},{"id":5146,"depth":45,"text":5147},[5167,5168,5169,5652],"2-qubit states",[5654],{"username":7,"name":5173,"role":5174,"bio":5175,"links":5655},[5656,5657],{"label":5178,"href":5179},{"label":5181,"href":5217},{"username":7,"name":5173,"role":5183},"One entangled state, eight ways to see it. A runnable notebook that turns the same 2-qubit circuit into Bloch spheres, steering ellipsoids, Wigner functions, and more.","One entangled state, eight ways to see it. Onri Jay Benally's runnable notebook renders a 2-qubit state as Bloch spheres, Q-spheres, steering ellipsoids, Wigner functions, and more.","Deep dive · Visualization",{},"\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fbloch-entangled.png","\u002Fblog\u002Fexpert-notes\u002F2-qubit-state-visualization","6 min read",[],[5668,5675,5682],{"username":5669,"project":5670,"title":5671,"category":5672,"thumb":5673,"to":5674},"q-inho","qave","QAVE","Visualization","\u002F_content\u002Fimages\u002Fqave\u002Fthumbnail.webp","\u002Fexplore\u002Fqave",{"username":5676,"project":5677,"title":5678,"category":5679,"thumb":5680,"to":5681},"AmberPincar","quantum-garden","Quantum Garden","Art","\u002F_content\u002Fimages\u002Fquantum-garden\u002Fthumbnail.webp","\u002Fexplore\u002Fquantum-garden",{"username":5683,"project":5684,"title":5685,"category":5679,"thumb":5686},"xinyi","quantum-butterfly-field","Quantum Butterfly Field","\u002F_content\u002Fimages\u002Fquantum-butterfly-field\u002Fthumbnail.webp",{"description":5688,"title":5205},"One entangled state, eight ways to see it. Onri Jay Benally's runnable notebook renders a 2-qubit state as Bloch spheres, Q-spheres, steering ellipsoids, and more.","blog\u002Fexpert-notes\u002F2-qubit-state-visualization",[5691,5200,29],"visualization","hbpIuxS-SqH0GmOc1n1ds2B-L8OsTlp8DNHQ4vd3Ylw",{"id":5694,"title":5695,"authors":5696,"body":5698,"breadcrumb":46114,"builders":46116,"byline":46124,"challenge":5184,"courseAuthor":5184,"courseLead":5184,"dek":5184,"description":46126,"draft":5186,"extension":5187,"eyebrow":46127,"finish":5184,"fork":5184,"hero":5184,"heroAlt":5184,"heroCta":5184,"heroImage":5184,"kind":5189,"lessonCount":5184,"meta":46128,"navigation":102,"newsItems":5184,"next":5184,"ogImage":5184,"order":5184,"outcomes":5184,"path":46129,"publishDate":5193,"readingTime":46130,"related":46131,"relatedProjects":5184,"seo":46132,"stem":46133,"tags":46134,"track":5184,"trackName":5184,"__hash__":46137},"blog\u002Fblog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization.md","Krylov Quantum Diagonalization of Lattice Hamiltonians",[5697],"mirko-amico",{"type":9,"value":5699,"toc":46085},[5700,5709,5714,5729,5733,5736,5764,5768,5771,5788,5792,7042,7046,7051,7315,7623,8288,8608,8611,8615,8919,9179,9450,9654,10432,10669,10672,10910,11232,11239,12185,12190,12676,13047,13053,13057,13155,13491,13497,13500,13548,13552,13634,13700,13704,13708,13816,13940,13946,13950,13956,15226,16055,16605,16608,17594,17986,18233,18641,18647,18650,18653,18720,18726,18730,18734,18737,18743,19387,22154,22362,23581,24815,24860,24864,25412,25842,25845,25849,25939,25942,26067,26070,26128,26353,26356,26411,26417,26420,26424,26427,26533,26537,26540,26925,26928,27140,27146,27150,27616,27620,27675,28005,28011,28103,28469,28473,28498,28502,28542,28632,28663,28988,29001,29064,29403,29415,29478,29650,29733,29826,29832,29835,29852,29858,30101,30105,30108,30482,30608,31204,31236,31701,31783,31789,31855,32571,32674,32937,33191,33522,33705,34047,34787,35101,36716,36748,38244,38721,39415,40198,40532,40859,40921,40930,40934,41133,41490,41493,42021,42024,42324,42380,42644,43137,43396,44338,45342,45782,46037,46039,46048,46053,46058,46063,46067,46070,46076,46078,46082],[12,5701,5702],{},[15,5703,17,5704,5708],{},[19,5705,5707],{"href":5706},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fen\u002Ftutorials\u002Fkrylov-quantum-diagonalization","the original tutorial",". The text, code, and figures below are Mirko's, as published, lightly reformatted for this page.",[12,5710,5711],{},[15,5712,5713],{},"Usage estimate: 20 minutes on a Heron r2 (NOTE: This is an estimate only. Your runtime might vary.)",[25,5715,5717],{"className":27,"code":5716,"language":29,"meta":30,"style":30},"# This cell is hidden from users – it disables some lint rules\n# ruff: noqa: E402 E722 F601\n",[32,5718,5719,5724],{"__ignoreMap":30},[35,5720,5721],{"class":37,"line":38},[35,5722,5723],{"class":41},"# This cell is hidden from users – it disables some lint rules\n",[35,5725,5726],{"class":37,"line":45},[35,5727,5728],{"class":41},"# ruff: noqa: E402 E722 F601\n",[5019,5730,5732],{"id":5731},"background","Background",[12,5734,5735],{},"This tutorial demonstrates how to implement the Krylov Quantum Diagonalization Algorithm (KQD) within the context of Qiskit patterns. You will first learn about the theory behind the algorithm and then see a demonstration of its execution on a QPU.",[12,5737,5738,5739,5743,5744,5748,5749,5753,5754,5758,5759,5763],{},"Across disciplines, we're interested in learning ground state properties of quantum systems. Examples include understanding the fundamental nature of particles and forces, predicting and understanding the behavior of complex materials and understanding bio-chemical interactions and reactions. Because of the exponential growth of the Hilbert space and the correlation that arise in entangled systems, classical algorithm struggle to solve this problem for quantum systems of increasing size. At one end of the spectrum is the existing approach that takes advantage of the quantum hardware focus on variational quantum methods (for example, ",[19,5740,5742],{"href":5741},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Ftutorials\u002Fspin-chain-vqe","variational quantum eigensolver","). These techniques face challenges with current devices because of the high number of function calls required in the optimization process, which adds a large overhead in resources once advanced error mitigation techniques are introduced, thus limiting their efficacy to small systems. At the other end of the spectrum, there are fault-tolerant quantum methods with performance guarantees (for example, ",[19,5745,5747],{"href":5746},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fquant-ph\u002F0604193","quantum phase estimation","), which require deep circuits that can be executed only on a fault-tolerant device. For these reasons, we introduce here a quantum algorithm based on subspace methods (as described in this ",[19,5750,5752],{"href":5751},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2312.00178","review paper","), the Krylov quantum diagonalization (KQD) algorithm. This algorithm performs well at large scale ",[19,5755,5757],{"href":5756},"#references","[1]"," on existing quantum hardware, shares similar ",[19,5760,5762],{"href":5761},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2110.07492","performance guarantees"," as phase estimation, is compatible with advanced error mitigation techniques, and could provide results that are classically inaccessible.",[5019,5765,5767],{"id":5766},"requirements","Requirements",[12,5769,5770],{},"Before starting this tutorial, be sure you have the following installed:",[5772,5773,5774,5781],"ul",{},[5027,5775,5776,5777,5780],{},"Qiskit SDK v2.0 or later, with ",[19,5778,5691],{"href":5779},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fvisualization"," support",[5027,5782,5783,5784,5787],{},"Qiskit Runtime v0.22 or later ( ",[32,5785,5786],{},"pip install qiskit-ibm-runtime"," )",[5019,5789,5791],{"id":5790},"setup","Setup",[25,5793,5795],{"className":27,"code":5794,"language":29,"meta":30,"style":30},"import numpy as np\nimport scipy as sp\nimport matplotlib.pylab as plt\nfrom typing import Union, List\nimport itertools as it\nimport copy\nfrom sympy import Matrix\nimport warnings\n\nwarnings.filterwarnings(\"ignore\")\n\nfrom qiskit.quantum_info import SparsePauliOp, Pauli, StabilizerState\nfrom qiskit.circuit import Parameter, IfElseOp\nfrom qiskit import QuantumCircuit, QuantumRegister\nfrom qiskit.circuit.library import PauliEvolutionGate\nfrom qiskit.synthesis import LieTrotter\nfrom qiskit.transpiler import Target, CouplingMap\nfrom qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager\n\n\nfrom qiskit_ibm_runtime import (\n    QiskitRuntimeService,\n    EstimatorV2 as Estimator,\n)\n\n\ndef solve_regularized_gen_eig(\n    h: np.ndarray,\n    s: np.ndarray,\n    threshold: float,\n    k: int = 1,\n    return_dimn: bool = False,\n) -> Union[float, List[float]]:\n    \"\"\"\n    Method for solving the generalized eigenvalue problem with regularization\n\n    Args:\n        h (numpy.ndarray):\n            The effective representation of the matrix in the Krylov subspace\n        s (numpy.ndarray):\n            The matrix of overlaps between vectors of the Krylov subspace\n        threshold (float):\n            Cut-off value for the eigenvalue of s\n        k (int):\n            Number of eigenvalues to return\n        return_dimn (bool):\n            Whether to return the size of the regularized subspace\n\n    Returns:\n        lowest k-eigenvalue(s) that are the solution of the regularized generalized eigenvalue problem\n\n\n    \"\"\"\n    s_vals, s_vecs = sp.linalg.eigh(s)\n    s_vecs = s_vecs.T\n    good_vecs = np.array(\n        [vec for val, vec in zip(s_vals, s_vecs) if val > threshold]\n    )\n    h_reg = good_vecs.conj() @ h @ good_vecs.T\n    s_reg = good_vecs.conj() @ s @ good_vecs.T\n    if k == 1:\n        if return_dimn:\n            return sp.linalg.eigh(h_reg, s_reg)[0][0], len(good_vecs)\n        else:\n            return sp.linalg.eigh(h_reg, s_reg)[0][0]\n    else:\n        if return_dimn:\n            return sp.linalg.eigh(h_reg, s_reg)[0][:k], len(good_vecs)\n        else:\n            return sp.linalg.eigh(h_reg, s_reg)[0][:k]\n\n\ndef single_particle_gs(H_op, n_qubits):\n    \"\"\"\n    Find the ground state of the single particle(excitation) sector\n    \"\"\"\n    H_x = []\n    for p, coeff in H_op.to_list():\n        H_x.append(set([i for i, v in enumerate(Pauli(p).x) if v]))\n\n    H_z = []\n    for p, coeff in H_op.to_list():\n        H_z.append(set([i for i, v in enumerate(Pauli(p).z) if v]))\n\n    H_c = H_op.coeffs\n\n    print(\"n_sys_qubits\", n_qubits)\n\n    n_exc = 1\n    sub_dimn = int(sp.special.comb(n_qubits + 1, n_exc))\n    print(\"n_exc\", n_exc, \", subspace dimension\", sub_dimn)\n\n    few_particle_H = np.zeros((sub_dimn, sub_dimn), dtype=complex)\n\n    sparse_vecs = [\n        set(vec) for vec in it.combinations(range(n_qubits + 1), r=n_exc)\n    ]  # list all of the possible sets of n_exc indices of 1s in n_exc-particle states\n\n    m = 0\n    for i, i_set in enumerate(sparse_vecs):\n        for j, j_set in enumerate(sparse_vecs):\n            m += 1\n\n            if len(i_set.symmetric_difference(j_set)) \u003C= 2:\n                for p_x, p_z, coeff in zip(H_x, H_z, H_c):\n                    if i_set.symmetric_difference(j_set) == p_x:\n                        sgn = ((-1j) ** len(p_x.intersection(p_z))) * (\n                            (-1) ** len(i_set.intersection(p_z))\n                        )\n                    else:\n                        sgn = 0\n\n                    few_particle_H[i, j] += sgn * coeff\n\n    gs_en = min(np.linalg.eigvalsh(few_particle_H))\n    print(\"single particle ground state energy: \", gs_en)\n    return gs_en\n",[32,5796,5797,5807,5819,5830,5841,5853,5860,5872,5879,5883,5898,5902,5913,5925,5936,5948,5960,5972,5984,5988,5992,6004,6009,6019,6023,6027,6031,6040,6048,6055,6066,6082,6099,6114,6118,6123,6127,6132,6137,6142,6147,6152,6157,6162,6167,6172,6177,6182,6186,6191,6196,6200,6204,6208,6224,6234,6248,6276,6281,6307,6329,6341,6349,6374,6381,6399,6406,6412,6431,6437,6452,6456,6460,6480,6484,6489,6493,6502,6519,6556,6560,6569,6583,6615,6619,6629,6633,6646,6650,6659,6684,6702,6706,6729,6733,6742,6784,6792,6796,6806,6820,6833,6843,6847,6871,6886,6904,6940,6962,6967,6974,6982,6986,7001,7005,7023,7035],{"__ignoreMap":30},[35,5798,5799,5801,5803,5805],{"class":37,"line":38},[35,5800,110],{"class":109},[35,5802,162],{"class":113},[35,5804,165],{"class":109},[35,5806,168],{"class":113},[35,5808,5809,5811,5814,5816],{"class":37,"line":45},[35,5810,110],{"class":109},[35,5812,5813],{"class":113}," scipy ",[35,5815,165],{"class":109},[35,5817,5818],{"class":113}," sp\n",[35,5820,5821,5823,5826,5828],{"class":37,"line":51},[35,5822,110],{"class":109},[35,5824,5825],{"class":113}," matplotlib.pylab ",[35,5827,165],{"class":109},[35,5829,194],{"class":113},[35,5831,5832,5834,5836,5838],{"class":37,"line":57},[35,5833,128],{"class":109},[35,5835,144],{"class":113},[35,5837,110],{"class":109},[35,5839,5840],{"class":113}," Union, List\n",[35,5842,5843,5845,5848,5850],{"class":37,"line":63},[35,5844,110],{"class":109},[35,5846,5847],{"class":113}," itertools ",[35,5849,165],{"class":109},[35,5851,5852],{"class":113}," it\n",[35,5854,5855,5857],{"class":37,"line":69},[35,5856,110],{"class":109},[35,5858,5859],{"class":113}," copy\n",[35,5861,5862,5864,5867,5869],{"class":37,"line":75},[35,5863,128],{"class":109},[35,5865,5866],{"class":113}," sympy ",[35,5868,110],{"class":109},[35,5870,5871],{"class":113}," Matrix\n",[35,5873,5874,5876],{"class":37,"line":81},[35,5875,110],{"class":109},[35,5877,5878],{"class":113}," warnings\n",[35,5880,5881],{"class":37,"line":87},[35,5882,103],{"emptyLinePlaceholder":102},[35,5884,5885,5888,5891,5893,5896],{"class":37,"line":93},[35,5886,5887],{"class":113},"warnings.",[35,5889,5890],{"class":271},"filterwarnings",[35,5892,275],{"class":113},[35,5894,5895],{"class":234},"\"ignore\"",[35,5897,384],{"class":113},[35,5899,5900],{"class":37,"line":99},[35,5901,103],{"emptyLinePlaceholder":102},[35,5903,5904,5906,5908,5910],{"class":37,"line":106},[35,5905,128],{"class":109},[35,5907,5269],{"class":113},[35,5909,110],{"class":109},[35,5911,5912],{"class":113}," SparsePauliOp, Pauli, StabilizerState\n",[35,5914,5915,5917,5920,5922],{"class":37,"line":117},[35,5916,128],{"class":109},[35,5918,5919],{"class":113}," qiskit.circuit ",[35,5921,110],{"class":109},[35,5923,5924],{"class":113}," Parameter, IfElseOp\n",[35,5926,5927,5929,5931,5933],{"class":37,"line":125},[35,5928,128],{"class":109},[35,5930,5257],{"class":113},[35,5932,110],{"class":109},[35,5934,5935],{"class":113}," QuantumCircuit, QuantumRegister\n",[35,5937,5938,5940,5943,5945],{"class":37,"line":139},[35,5939,128],{"class":109},[35,5941,5942],{"class":113}," qiskit.circuit.library ",[35,5944,110],{"class":109},[35,5946,5947],{"class":113}," PauliEvolutionGate\n",[35,5949,5950,5952,5955,5957],{"class":37,"line":152},[35,5951,128],{"class":109},[35,5953,5954],{"class":113}," qiskit.synthesis ",[35,5956,110],{"class":109},[35,5958,5959],{"class":113}," LieTrotter\n",[35,5961,5962,5964,5967,5969],{"class":37,"line":157},[35,5963,128],{"class":109},[35,5965,5966],{"class":113}," qiskit.transpiler ",[35,5968,110],{"class":109},[35,5970,5971],{"class":113}," Target, CouplingMap\n",[35,5973,5974,5976,5979,5981],{"class":37,"line":171},[35,5975,128],{"class":109},[35,5977,5978],{"class":113}," qiskit.transpiler.preset_passmanagers ",[35,5980,110],{"class":109},[35,5982,5983],{"class":113}," generate_preset_pass_manager\n",[35,5985,5986],{"class":37,"line":184},[35,5987,103],{"emptyLinePlaceholder":102},[35,5989,5990],{"class":37,"line":197},[35,5991,103],{"emptyLinePlaceholder":102},[35,5993,5994,5996,5999,6001],{"class":37,"line":210},[35,5995,128],{"class":109},[35,5997,5998],{"class":113}," qiskit_ibm_runtime ",[35,6000,110],{"class":109},[35,6002,6003],{"class":113}," (\n",[35,6005,6006],{"class":37,"line":223},[35,6007,6008],{"class":113},"    QiskitRuntimeService,\n",[35,6010,6011,6014,6016],{"class":37,"line":228},[35,6012,6013],{"class":113},"    EstimatorV2 ",[35,6015,165],{"class":109},[35,6017,6018],{"class":113}," Estimator,\n",[35,6020,6021],{"class":37,"line":249},[35,6022,384],{"class":113},[35,6024,6025],{"class":37,"line":254},[35,6026,103],{"emptyLinePlaceholder":102},[35,6028,6029],{"class":37,"line":260},[35,6030,103],{"emptyLinePlaceholder":102},[35,6032,6033,6035,6038],{"class":37,"line":265},[35,6034,268],{"class":109},[35,6036,6037],{"class":271}," solve_regularized_gen_eig",[35,6039,903],{"class":113},[35,6041,6042,6045],{"class":37,"line":326},[35,6043,6044],{"class":278},"    h",[35,6046,6047],{"class":113},": np.ndarray,\n",[35,6049,6050,6053],{"class":37,"line":332},[35,6051,6052],{"class":278},"    s",[35,6054,6047],{"class":113},[35,6056,6057,6060,6062,6064],{"class":37,"line":338},[35,6058,6059],{"class":278},"    threshold",[35,6061,282],{"class":113},[35,6063,305],{"class":241},[35,6065,912],{"class":113},[35,6067,6068,6071,6073,6075,6077,6080],{"class":37,"line":344},[35,6069,6070],{"class":278},"    k",[35,6072,282],{"class":113},[35,6074,285],{"class":241},[35,6076,308],{"class":241},[35,6078,6079],{"class":245}," 1",[35,6081,912],{"class":113},[35,6083,6084,6087,6089,6092,6094,6097],{"class":37,"line":349},[35,6085,6086],{"class":278},"    return_dimn",[35,6088,282],{"class":113},[35,6090,6091],{"class":241},"bool",[35,6093,308],{"class":241},[35,6095,6096],{"class":245}," False",[35,6098,912],{"class":113},[35,6100,6101,6104,6106,6109,6111],{"class":37,"line":367},[35,6102,6103],{"class":113},") -> Union[",[35,6105,305],{"class":241},[35,6107,6108],{"class":113},", List[",[35,6110,305],{"class":241},[35,6112,6113],{"class":113},"]]:\n",[35,6115,6116],{"class":37,"line":387},[35,6117,329],{"class":234},[35,6119,6120],{"class":37,"line":404},[35,6121,6122],{"class":234},"    Method for solving the generalized eigenvalue problem with regularization\n",[35,6124,6125],{"class":37,"line":434},[35,6126,103],{"emptyLinePlaceholder":102},[35,6128,6129],{"class":37,"line":474},[35,6130,6131],{"class":234},"    Args:\n",[35,6133,6134],{"class":37,"line":546},[35,6135,6136],{"class":234},"        h (numpy.ndarray):\n",[35,6138,6139],{"class":37,"line":584},[35,6140,6141],{"class":234},"            The effective representation of the matrix in the Krylov subspace\n",[35,6143,6144],{"class":37,"line":589},[35,6145,6146],{"class":234},"        s (numpy.ndarray):\n",[35,6148,6149],{"class":37,"line":594},[35,6150,6151],{"class":234},"            The matrix of overlaps between vectors of the Krylov subspace\n",[35,6153,6154],{"class":37,"line":643},[35,6155,6156],{"class":234},"        threshold (float):\n",[35,6158,6159],{"class":37,"line":648},[35,6160,6161],{"class":234},"            Cut-off value for the eigenvalue of s\n",[35,6163,6164],{"class":37,"line":654},[35,6165,6166],{"class":234},"        k (int):\n",[35,6168,6169],{"class":37,"line":660},[35,6170,6171],{"class":234},"            Number of eigenvalues to return\n",[35,6173,6174],{"class":37,"line":665},[35,6175,6176],{"class":234},"        return_dimn (bool):\n",[35,6178,6179],{"class":37,"line":692},[35,6180,6181],{"class":234},"            Whether to return the size of the regularized subspace\n",[35,6183,6184],{"class":37,"line":714},[35,6185,103],{"emptyLinePlaceholder":102},[35,6187,6188],{"class":37,"line":729},[35,6189,6190],{"class":234},"    Returns:\n",[35,6192,6193],{"class":37,"line":737},[35,6194,6195],{"class":234},"        lowest k-eigenvalue(s) that are the solution of the regularized generalized eigenvalue problem\n",[35,6197,6198],{"class":37,"line":768},[35,6199,103],{"emptyLinePlaceholder":102},[35,6201,6202],{"class":37,"line":793},[35,6203,103],{"emptyLinePlaceholder":102},[35,6205,6206],{"class":37,"line":798},[35,6207,329],{"class":234},[35,6209,6210,6213,6215,6218,6221],{"class":37,"line":803},[35,6211,6212],{"class":113},"    s_vals, s_vecs ",[35,6214,242],{"class":241},[35,6216,6217],{"class":113}," sp.linalg.",[35,6219,6220],{"class":271},"eigh",[35,6222,6223],{"class":113},"(s)\n",[35,6225,6226,6229,6231],{"class":37,"line":809},[35,6227,6228],{"class":113},"    s_vecs ",[35,6230,242],{"class":241},[35,6232,6233],{"class":113}," s_vecs.T\n",[35,6235,6236,6239,6241,6243,6246],{"class":37,"line":814},[35,6237,6238],{"class":113},"    good_vecs ",[35,6240,242],{"class":241},[35,6242,2412],{"class":113},[35,6244,6245],{"class":271},"array",[35,6247,903],{"class":113},[35,6249,6250,6253,6255,6258,6260,6263,6266,6268,6271,6273],{"class":37,"line":833},[35,6251,6252],{"class":113},"        [vec ",[35,6254,1874],{"class":109},[35,6256,6257],{"class":113}," val, vec ",[35,6259,1663],{"class":109},[35,6261,6262],{"class":241}," zip",[35,6264,6265],{"class":113},"(s_vals, s_vecs) ",[35,6267,1492],{"class":109},[35,6269,6270],{"class":113}," val ",[35,6272,1497],{"class":241},[35,6274,6275],{"class":113}," threshold]\n",[35,6277,6278],{"class":37,"line":845},[35,6279,6280],{"class":113},"    )\n",[35,6282,6283,6286,6288,6291,6294,6296,6299,6302,6304],{"class":37,"line":854},[35,6284,6285],{"class":113},"    h_reg ",[35,6287,242],{"class":241},[35,6289,6290],{"class":113}," good_vecs.",[35,6292,6293],{"class":271},"conj",[35,6295,2519],{"class":113},[35,6297,6298],{"class":241},"@",[35,6300,6301],{"class":113}," h ",[35,6303,6298],{"class":241},[35,6305,6306],{"class":113}," good_vecs.T\n",[35,6308,6309,6312,6314,6316,6318,6320,6322,6325,6327],{"class":37,"line":865},[35,6310,6311],{"class":113},"    s_reg ",[35,6313,242],{"class":241},[35,6315,6290],{"class":113},[35,6317,6293],{"class":271},[35,6319,2519],{"class":113},[35,6321,6298],{"class":241},[35,6323,6324],{"class":113}," s ",[35,6326,6298],{"class":241},[35,6328,6306],{"class":113},[35,6330,6331,6333,6335,6337,6339],{"class":37,"line":873},[35,6332,352],{"class":109},[35,6334,395],{"class":113},[35,6336,358],{"class":241},[35,6338,6079],{"class":245},[35,6340,364],{"class":113},[35,6342,6343,6346],{"class":37,"line":878},[35,6344,6345],{"class":109},"        if",[35,6347,6348],{"class":113}," return_dimn:\n",[35,6350,6351,6354,6356,6358,6361,6363,6365,6367,6369,6371],{"class":37,"line":883},[35,6352,6353],{"class":109},"            return",[35,6355,6217],{"class":113},[35,6357,6220],{"class":271},[35,6359,6360],{"class":113},"(h_reg, s_reg)[",[35,6362,5312],{"class":245},[35,6364,1997],{"class":113},[35,6366,5312],{"class":245},[35,6368,3033],{"class":113},[35,6370,2420],{"class":241},[35,6372,6373],{"class":113},"(good_vecs)\n",[35,6375,6376,6379],{"class":37,"line":897},[35,6377,6378],{"class":109},"        else",[35,6380,364],{"class":113},[35,6382,6383,6385,6387,6389,6391,6393,6395,6397],{"class":37,"line":906},[35,6384,6353],{"class":109},[35,6386,6217],{"class":113},[35,6388,6220],{"class":271},[35,6390,6360],{"class":113},[35,6392,5312],{"class":245},[35,6394,1997],{"class":113},[35,6396,5312],{"class":245},[35,6398,1074],{"class":113},[35,6400,6401,6404],{"class":37,"line":915},[35,6402,6403],{"class":109},"    else",[35,6405,364],{"class":113},[35,6407,6408,6410],{"class":37,"line":923},[35,6409,6345],{"class":109},[35,6411,6348],{"class":113},[35,6413,6414,6416,6418,6420,6422,6424,6427,6429],{"class":37,"line":929},[35,6415,6353],{"class":109},[35,6417,6217],{"class":113},[35,6419,6220],{"class":271},[35,6421,6360],{"class":113},[35,6423,5312],{"class":245},[35,6425,6426],{"class":113},"][:k], ",[35,6428,2420],{"class":241},[35,6430,6373],{"class":113},[35,6432,6433,6435],{"class":37,"line":935},[35,6434,6378],{"class":109},[35,6436,364],{"class":113},[35,6438,6439,6441,6443,6445,6447,6449],{"class":37,"line":942},[35,6440,6353],{"class":109},[35,6442,6217],{"class":113},[35,6444,6220],{"class":271},[35,6446,6360],{"class":113},[35,6448,5312],{"class":245},[35,6450,6451],{"class":113},"][:k]\n",[35,6453,6454],{"class":37,"line":950},[35,6455,103],{"emptyLinePlaceholder":102},[35,6457,6458],{"class":37,"line":958},[35,6459,103],{"emptyLinePlaceholder":102},[35,6461,6462,6464,6467,6469,6472,6474,6477],{"class":37,"line":964},[35,6463,268],{"class":109},[35,6465,6466],{"class":271}," single_particle_gs",[35,6468,275],{"class":113},[35,6470,6471],{"class":278},"H_op",[35,6473,288],{"class":113},[35,6475,6476],{"class":278},"n_qubits",[35,6478,6479],{"class":113},"):\n",[35,6481,6482],{"class":37,"line":969},[35,6483,329],{"class":234},[35,6485,6486],{"class":37,"line":976},[35,6487,6488],{"class":234},"    Find the ground state of the single particle(excitation) sector\n",[35,6490,6491],{"class":37,"line":984},[35,6492,329],{"class":234},[35,6494,6495,6498,6500],{"class":37,"line":992},[35,6496,6497],{"class":113},"    H_x ",[35,6499,242],{"class":241},[35,6501,1850],{"class":113},[35,6503,6504,6506,6509,6511,6514,6517],{"class":37,"line":998},[35,6505,1657],{"class":109},[35,6507,6508],{"class":113}," p, coeff ",[35,6510,1663],{"class":109},[35,6512,6513],{"class":113}," H_op.",[35,6515,6516],{"class":271},"to_list",[35,6518,2148],{"class":113},[35,6520,6521,6524,6526,6528,6531,6534,6536,6539,6541,6543,6545,6548,6551,6553],{"class":37,"line":1003},[35,6522,6523],{"class":113},"        H_x.",[35,6525,1927],{"class":271},[35,6527,275],{"class":113},[35,6529,6530],{"class":241},"set",[35,6532,6533],{"class":113},"([i ",[35,6535,1874],{"class":109},[35,6537,6538],{"class":113}," i, v ",[35,6540,1663],{"class":109},[35,6542,2905],{"class":241},[35,6544,275],{"class":113},[35,6546,6547],{"class":271},"Pauli",[35,6549,6550],{"class":113},"(p).x) ",[35,6552,1492],{"class":109},[35,6554,6555],{"class":113}," v]))\n",[35,6557,6558],{"class":37,"line":1010},[35,6559,103],{"emptyLinePlaceholder":102},[35,6561,6562,6565,6567],{"class":37,"line":1018},[35,6563,6564],{"class":113},"    H_z ",[35,6566,242],{"class":241},[35,6568,1850],{"class":113},[35,6570,6571,6573,6575,6577,6579,6581],{"class":37,"line":1026},[35,6572,1657],{"class":109},[35,6574,6508],{"class":113},[35,6576,1663],{"class":109},[35,6578,6513],{"class":113},[35,6580,6516],{"class":271},[35,6582,2148],{"class":113},[35,6584,6585,6588,6590,6592,6594,6596,6598,6600,6602,6604,6606,6608,6611,6613],{"class":37,"line":1032},[35,6586,6587],{"class":113},"        H_z.",[35,6589,1927],{"class":271},[35,6591,275],{"class":113},[35,6593,6530],{"class":241},[35,6595,6533],{"class":113},[35,6597,1874],{"class":109},[35,6599,6538],{"class":113},[35,6601,1663],{"class":109},[35,6603,2905],{"class":241},[35,6605,275],{"class":113},[35,6607,6547],{"class":271},[35,6609,6610],{"class":113},"(p).z) ",[35,6612,1492],{"class":109},[35,6614,6555],{"class":113},[35,6616,6617],{"class":37,"line":1037},[35,6618,103],{"emptyLinePlaceholder":102},[35,6620,6621,6624,6626],{"class":37,"line":1044},[35,6622,6623],{"class":113},"    H_c ",[35,6625,242],{"class":241},[35,6627,6628],{"class":113}," H_op.coeffs\n",[35,6630,6631],{"class":37,"line":1052},[35,6632,103],{"emptyLinePlaceholder":102},[35,6634,6635,6638,6640,6643],{"class":37,"line":1060},[35,6636,6637],{"class":241},"    print",[35,6639,275],{"class":113},[35,6641,6642],{"class":234},"\"n_sys_qubits\"",[35,6644,6645],{"class":113},", n_qubits)\n",[35,6647,6648],{"class":37,"line":1066},[35,6649,103],{"emptyLinePlaceholder":102},[35,6651,6652,6655,6657],{"class":37,"line":1071},[35,6653,6654],{"class":113},"    n_exc ",[35,6656,242],{"class":241},[35,6658,734],{"class":245},[35,6660,6661,6664,6666,6668,6671,6674,6677,6679,6681],{"class":37,"line":1077},[35,6662,6663],{"class":113},"    sub_dimn ",[35,6665,242],{"class":241},[35,6667,1434],{"class":241},[35,6669,6670],{"class":113},"(sp.special.",[35,6672,6673],{"class":271},"comb",[35,6675,6676],{"class":113},"(n_qubits ",[35,6678,445],{"class":241},[35,6680,6079],{"class":245},[35,6682,6683],{"class":113},", n_exc))\n",[35,6685,6686,6688,6690,6693,6696,6699],{"class":37,"line":1082},[35,6687,6637],{"class":241},[35,6689,275],{"class":113},[35,6691,6692],{"class":234},"\"n_exc\"",[35,6694,6695],{"class":113},", n_exc, ",[35,6697,6698],{"class":234},"\", subspace dimension\"",[35,6700,6701],{"class":113},", sub_dimn)\n",[35,6703,6704],{"class":37,"line":1088},[35,6705,103],{"emptyLinePlaceholder":102},[35,6707,6708,6711,6713,6715,6718,6721,6724,6727],{"class":37,"line":1093},[35,6709,6710],{"class":113},"    few_particle_H ",[35,6712,242],{"class":241},[35,6714,2412],{"class":113},[35,6716,6717],{"class":271},"zeros",[35,6719,6720],{"class":113},"((sub_dimn, sub_dimn), ",[35,6722,6723],{"class":822},"dtype",[35,6725,6726],{"class":241},"=complex",[35,6728,384],{"class":113},[35,6730,6731],{"class":37,"line":1107},[35,6732,103],{"emptyLinePlaceholder":102},[35,6734,6735,6738,6740],{"class":37,"line":1120},[35,6736,6737],{"class":113},"    sparse_vecs ",[35,6739,242],{"class":241},[35,6741,894],{"class":113},[35,6743,6744,6747,6750,6752,6755,6757,6760,6763,6765,6768,6770,6772,6774,6776,6779,6781],{"class":37,"line":1133},[35,6745,6746],{"class":241},"        set",[35,6748,6749],{"class":113},"(vec) ",[35,6751,1874],{"class":109},[35,6753,6754],{"class":113}," vec ",[35,6756,1663],{"class":109},[35,6758,6759],{"class":113}," it.",[35,6761,6762],{"class":271},"combinations",[35,6764,275],{"class":113},[35,6766,6767],{"class":241},"range",[35,6769,6676],{"class":113},[35,6771,445],{"class":241},[35,6773,6079],{"class":245},[35,6775,685],{"class":113},[35,6777,6778],{"class":822},"r",[35,6780,242],{"class":241},[35,6782,6783],{"class":113},"n_exc)\n",[35,6785,6786,6789],{"class":37,"line":1161},[35,6787,6788],{"class":113},"    ]  ",[35,6790,6791],{"class":41},"# list all of the possible sets of n_exc indices of 1s in n_exc-particle states\n",[35,6793,6794],{"class":37,"line":1170},[35,6795,103],{"emptyLinePlaceholder":102},[35,6797,6798,6801,6803],{"class":37,"line":1183},[35,6799,6800],{"class":113},"    m ",[35,6802,242],{"class":241},[35,6804,6805],{"class":245}," 0\n",[35,6807,6808,6810,6813,6815,6817],{"class":37,"line":1194},[35,6809,1657],{"class":109},[35,6811,6812],{"class":113}," i, i_set ",[35,6814,1663],{"class":109},[35,6816,2905],{"class":241},[35,6818,6819],{"class":113},"(sparse_vecs):\n",[35,6821,6822,6824,6827,6829,6831],{"class":37,"line":1200},[35,6823,2087],{"class":109},[35,6825,6826],{"class":113}," j, j_set ",[35,6828,1663],{"class":109},[35,6830,2905],{"class":241},[35,6832,6819],{"class":113},[35,6834,6835,6838,6841],{"class":37,"line":1208},[35,6836,6837],{"class":113},"            m ",[35,6839,6840],{"class":241},"+=",[35,6842,734],{"class":245},[35,6844,6845],{"class":37,"line":1219},[35,6846,103],{"emptyLinePlaceholder":102},[35,6848,6849,6852,6855,6858,6861,6864,6867,6869],{"class":37,"line":1225},[35,6850,6851],{"class":109},"            if",[35,6853,6854],{"class":241}," len",[35,6856,6857],{"class":113},"(i_set.",[35,6859,6860],{"class":271},"symmetric_difference",[35,6862,6863],{"class":113},"(j_set)) ",[35,6865,6866],{"class":241},"\u003C=",[35,6868,424],{"class":245},[35,6870,364],{"class":113},[35,6872,6873,6876,6879,6881,6883],{"class":37,"line":1231},[35,6874,6875],{"class":109},"                for",[35,6877,6878],{"class":113}," p_x, p_z, coeff ",[35,6880,1663],{"class":109},[35,6882,6262],{"class":241},[35,6884,6885],{"class":113},"(H_x, H_z, H_c):\n",[35,6887,6888,6891,6894,6896,6899,6901],{"class":37,"line":1236},[35,6889,6890],{"class":109},"                    if",[35,6892,6893],{"class":113}," i_set.",[35,6895,6860],{"class":271},[35,6897,6898],{"class":113},"(j_set) ",[35,6900,358],{"class":241},[35,6902,6903],{"class":113}," p_x:\n",[35,6905,6906,6909,6911,6914,6916,6918,6921,6923,6925,6927,6930,6933,6936,6938],{"class":37,"line":1263},[35,6907,6908],{"class":113},"                        sgn ",[35,6910,242],{"class":241},[35,6912,6913],{"class":113}," ((",[35,6915,564],{"class":241},[35,6917,500],{"class":245},[35,6919,6920],{"class":109},"j",[35,6922,427],{"class":113},[35,6924,421],{"class":241},[35,6926,6854],{"class":241},[35,6928,6929],{"class":113},"(p_x.",[35,6931,6932],{"class":271},"intersection",[35,6934,6935],{"class":113},"(p_z))) ",[35,6937,484],{"class":241},[35,6939,6003],{"class":113},[35,6941,6942,6945,6947,6949,6951,6953,6955,6957,6959],{"class":37,"line":1268},[35,6943,6944],{"class":113},"                            (",[35,6946,564],{"class":241},[35,6948,500],{"class":245},[35,6950,427],{"class":113},[35,6952,421],{"class":241},[35,6954,6854],{"class":241},[35,6956,6857],{"class":113},[35,6958,6932],{"class":271},[35,6960,6961],{"class":113},"(p_z))\n",[35,6963,6964],{"class":37,"line":1273},[35,6965,6966],{"class":113},"                        )\n",[35,6968,6969,6972],{"class":37,"line":1279},[35,6970,6971],{"class":109},"                    else",[35,6973,364],{"class":113},[35,6975,6976,6978,6980],{"class":37,"line":1284},[35,6977,6908],{"class":113},[35,6979,242],{"class":241},[35,6981,6805],{"class":245},[35,6983,6984],{"class":37,"line":1317},[35,6985,103],{"emptyLinePlaceholder":102},[35,6987,6988,6991,6993,6996,6998],{"class":37,"line":1322},[35,6989,6990],{"class":113},"                    few_particle_H[i, j] ",[35,6992,6840],{"class":241},[35,6994,6995],{"class":113}," sgn ",[35,6997,484],{"class":241},[35,6999,7000],{"class":113}," coeff\n",[35,7002,7003],{"class":37,"line":1328},[35,7004,103],{"emptyLinePlaceholder":102},[35,7006,7007,7010,7012,7014,7017,7020],{"class":37,"line":1333},[35,7008,7009],{"class":113},"    gs_en ",[35,7011,242],{"class":241},[35,7013,673],{"class":241},[35,7015,7016],{"class":113},"(np.linalg.",[35,7018,7019],{"class":271},"eigvalsh",[35,7021,7022],{"class":113},"(few_particle_H))\n",[35,7024,7025,7027,7029,7032],{"class":37,"line":1368},[35,7026,6637],{"class":241},[35,7028,275],{"class":113},[35,7030,7031],{"class":234},"\"single particle ground state energy: \"",[35,7033,7034],{"class":113},", gs_en)\n",[35,7036,7037,7039],{"class":37,"line":1373},[35,7038,549],{"class":109},[35,7040,7041],{"class":113}," gs_en\n",[5019,7043,7045],{"id":7044},"step-1-map-classical-inputs-to-a-quantum-problem","Step 1: Map classical inputs to a quantum problem",[7047,7048,7050],"h3",{"id":7049},"the-krylov-space","The Krylov space",[12,7052,7053,7054,7147,7148,7177,7178,7207,7208,7267,7268,5122],{},"The Krylov space ",[35,7055,7058,7087],{"className":7056},[7057],"katex",[35,7059,7062],{"className":7060},[7061],"katex-mathml",[7063,7064,7066],"math",{"xmlns":7065},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[7067,7068,7069,7082],"semantics",{},[7070,7071,7072],"mrow",{},[7073,7074,7075,7080],"msup",{},[7076,7077,7079],"mi",{"mathvariant":7078},"script","K",[7076,7081,6778],{},[7083,7084,7086],"annotation",{"encoding":7085},"application\u002Fx-tex","\\mathcal{K}^r",[35,7088,7092],{"className":7089,"ariaHidden":7091},[7090],"katex-html","true",[35,7093,7096,7101],{"className":7094},[7095],"base",[35,7097],{"className":7098,"style":7100},[7099],"strut","height:0.6833em;",[35,7102,7105,7110],{"className":7103},[7104],"mord",[35,7106,7079],{"className":7107,"style":7109},[7104,7108],"mathcal","margin-right:0.0144em;",[35,7111,7114],{"className":7112},[7113],"msupsub",[35,7115,7118],{"className":7116},[7117],"vlist-t",[35,7119,7122],{"className":7120},[7121],"vlist-r",[35,7123,7127],{"className":7124,"style":7126},[7125],"vlist","height:0.6644em;",[35,7128,7130,7135],{"style":7129},"top:-3.063em;margin-right:0.05em;",[35,7131],{"className":7132,"style":7134},[7133],"pstrut","height:2.7em;",[35,7136,7142],{"className":7137},[7138,7139,7140,7141],"sizing","reset-size6","size3","mtight",[35,7143,6778],{"className":7144,"style":7146},[7104,7145,7141],"mathnormal","margin-right:0.0278em;"," of order ",[35,7149,7151,7164],{"className":7150},[7057],[35,7152,7154],{"className":7153},[7061],[7063,7155,7156],{"xmlns":7065},[7067,7157,7158,7162],{},[7070,7159,7160],{},[7076,7161,6778],{},[7083,7163,6778],{"encoding":7085},[35,7165,7167],{"className":7166,"ariaHidden":7091},[7090],[35,7168,7170,7174],{"className":7169},[7095],[35,7171],{"className":7172,"style":7173},[7099],"height:0.4306em;",[35,7175,6778],{"className":7176,"style":7146},[7104,7145]," is the space spanned by vectors obtained by multiplying higher powers of a matrix ",[35,7179,7181,7195],{"className":7180},[7057],[35,7182,7184],{"className":7183},[7061],[7063,7185,7186],{"xmlns":7065},[7067,7187,7188,7193],{},[7070,7189,7190],{},[7076,7191,7192],{},"A",[7083,7194,7192],{"encoding":7085},[35,7196,7198],{"className":7197,"ariaHidden":7091},[7090],[35,7199,7201,7204],{"className":7200},[7095],[35,7202],{"className":7203,"style":7100},[7099],[35,7205,7192],{"className":7206},[7104,7145],", up to ",[35,7209,7211,7232],{"className":7210},[7057],[35,7212,7214],{"className":7213},[7061],[7063,7215,7216],{"xmlns":7065},[7067,7217,7218,7229],{},[7070,7219,7220,7222,7226],{},[7076,7221,6778],{},[7223,7224,7225],"mo",{},"−",[7227,7228,500],"mn",{},[7083,7230,7231],{"encoding":7085},"r-1",[35,7233,7235,7257],{"className":7234,"ariaHidden":7091},[7090],[35,7236,7238,7242,7245,7250,7254],{"className":7237},[7095],[35,7239],{"className":7240,"style":7241},[7099],"height:0.6667em;vertical-align:-0.0833em;",[35,7243,6778],{"className":7244,"style":7146},[7104,7145],[35,7246],{"className":7247,"style":7249},[7248],"mspace","margin-right:0.2222em;",[35,7251,7225],{"className":7252},[7253],"mbin",[35,7255],{"className":7256,"style":7249},[7248],[35,7258,7260,7264],{"className":7259},[7095],[35,7261],{"className":7262,"style":7263},[7099],"height:0.6444em;",[35,7265,500],{"className":7266},[7104],", with a reference vector ",[35,7269,7271,7294],{"className":7270},[7057],[35,7272,7274],{"className":7273},[7061],[7063,7275,7276],{"xmlns":7065},[7067,7277,7278,7291],{},[7070,7279,7280,7284,7287],{},[7076,7281,7283],{"mathvariant":7282},"normal","∣",[7076,7285,7286],{},"v",[7223,7288,7290],{"stretchy":7289},"false","⟩",[7083,7292,7293],{"encoding":7085},"\\vert v \\rangle",[35,7295,7297],{"className":7296,"ariaHidden":7091},[7090],[35,7298,7300,7304,7307,7311],{"className":7299},[7095],[35,7301],{"className":7302,"style":7303},[7099],"height:1em;vertical-align:-0.25em;",[35,7305,7283],{"className":7306},[7104],[35,7308,7286],{"className":7309,"style":7310},[7104,7145],"margin-right:0.0359em;",[35,7312,7290],{"className":7313},[7314],"mclose",[35,7316,7319],{"className":7317},[7318],"katex-display",[35,7320,7322,7407],{"className":7321},[7057],[35,7323,7325],{"className":7324},[7061],[7063,7326,7328],{"xmlns":7065,"display":7327},"block",[7067,7329,7330,7404],{},[7070,7331,7332,7338,7340],{},[7073,7333,7334,7336],{},[7076,7335,7079],{"mathvariant":7078},[7076,7337,6778],{},[7223,7339,242],{},[7070,7341,7342,7344,7346,7348,7350,7352,7354,7356,7358,7360,7362,7368,7370,7372,7374,7376,7378,7380,7382,7384,7396,7398,7400,7402],{},[7223,7343,1681],{"fence":7091},[7076,7345,7283],{"mathvariant":7282},[7076,7347,7286],{},[7223,7349,7290],{"stretchy":7289},[7223,7351,2877],{"separator":7091},[7076,7353,7192],{},[7076,7355,7283],{"mathvariant":7282},[7076,7357,7286],{},[7223,7359,7290],{"stretchy":7289},[7223,7361,2877],{"separator":7091},[7073,7363,7364,7366],{},[7076,7365,7192],{},[7227,7367,460],{},[7076,7369,7283],{"mathvariant":7282},[7076,7371,7286],{},[7223,7373,7290],{"stretchy":7289},[7223,7375,2877],{"separator":7091},[7076,7377,5122],{"mathvariant":7282},[7076,7379,5122],{"mathvariant":7282},[7076,7381,5122],{"mathvariant":7282},[7223,7383,2877],{"separator":7091},[7073,7385,7386,7388],{},[7076,7387,7192],{},[7070,7389,7390,7392,7394],{},[7076,7391,6778],{},[7223,7393,7225],{},[7227,7395,500],{},[7076,7397,7283],{"mathvariant":7282},[7076,7399,7286],{},[7223,7401,7290],{"stretchy":7289},[7223,7403,1697],{"fence":7091},[7083,7405,7406],{"encoding":7085},"\\mathcal{K}^r = \\left\\{ \\vert v \\rangle, A \\vert v \\rangle, A^2 \\vert v \\rangle, ..., A^{r-1} \\vert v \\rangle \\right\\}",[35,7408,7410,7458],{"className":7409,"ariaHidden":7091},[7090],[35,7411,7413,7417,7447,7451,7455],{"className":7412},[7095],[35,7414],{"className":7415,"style":7416},[7099],"height:0.7144em;",[35,7418,7420,7423],{"className":7419},[7104],[35,7421,7079],{"className":7422,"style":7109},[7104,7108],[35,7424,7426],{"className":7425},[7113],[35,7427,7429],{"className":7428},[7117],[35,7430,7432],{"className":7431},[7121],[35,7433,7435],{"className":7434,"style":7416},[7125],[35,7436,7438,7441],{"style":7437},"top:-3.113em;margin-right:0.05em;",[35,7439],{"className":7440,"style":7134},[7133],[35,7442,7444],{"className":7443},[7138,7139,7140,7141],[35,7445,6778],{"className":7446,"style":7146},[7104,7145,7141],[35,7448],{"className":7449,"style":7450},[7248],"margin-right:0.2778em;",[35,7452,242],{"className":7453},[7454],"mrel",[35,7456],{"className":7457,"style":7450},[7248],[35,7459,7461,7465],{"className":7460},[7095],[35,7462],{"className":7463,"style":7464},[7099],"height:1.2141em;vertical-align:-0.35em;",[35,7466,7469,7480,7483,7486,7489,7493,7497,7500,7503,7506,7509,7512,7515,7545,7548,7551,7554,7557,7560,7564,7567,7570,7608,7611,7614,7617],{"className":7467},[7468],"minner",[35,7470,7475],{"className":7471,"style":7474},[7472,7473],"mopen","delimcenter","top:0em;",[35,7476,1681],{"className":7477},[7478,7479],"delimsizing","size1",[35,7481,7283],{"className":7482},[7104],[35,7484,7286],{"className":7485,"style":7310},[7104,7145],[35,7487,7290],{"className":7488},[7314],[35,7490,2877],{"className":7491},[7492],"mpunct",[35,7494],{"className":7495,"style":7496},[7248],"margin-right:0.1667em;",[35,7498,7192],{"className":7499},[7104,7145],[35,7501,7283],{"className":7502},[7104],[35,7504,7286],{"className":7505,"style":7310},[7104,7145],[35,7507,7290],{"className":7508},[7314],[35,7510,2877],{"className":7511},[7492],[35,7513],{"className":7514,"style":7496},[7248],[35,7516,7518,7521],{"className":7517},[7104],[35,7519,7192],{"className":7520},[7104,7145],[35,7522,7524],{"className":7523},[7113],[35,7525,7527],{"className":7526},[7117],[35,7528,7530],{"className":7529},[7121],[35,7531,7534],{"className":7532,"style":7533},[7125],"height:0.8641em;",[35,7535,7536,7539],{"style":7437},[35,7537],{"className":7538,"style":7134},[7133],[35,7540,7542],{"className":7541},[7138,7139,7140,7141],[35,7543,460],{"className":7544},[7104,7141],[35,7546,7283],{"className":7547},[7104],[35,7549,7286],{"className":7550,"style":7310},[7104,7145],[35,7552,7290],{"className":7553},[7314],[35,7555,2877],{"className":7556},[7492],[35,7558],{"className":7559,"style":7496},[7248],[35,7561,7563],{"className":7562},[7104],"...",[35,7565,2877],{"className":7566},[7492],[35,7568],{"className":7569,"style":7496},[7248],[35,7571,7573,7576],{"className":7572},[7104],[35,7574,7192],{"className":7575},[7104,7145],[35,7577,7579],{"className":7578},[7113],[35,7580,7582],{"className":7581},[7117],[35,7583,7585],{"className":7584},[7121],[35,7586,7588],{"className":7587,"style":7533},[7125],[35,7589,7590,7593],{"style":7437},[35,7591],{"className":7592,"style":7134},[7133],[35,7594,7596],{"className":7595},[7138,7139,7140,7141],[35,7597,7599,7602,7605],{"className":7598},[7104,7141],[35,7600,6778],{"className":7601,"style":7146},[7104,7145,7141],[35,7603,7225],{"className":7604},[7253,7141],[35,7606,500],{"className":7607},[7104,7141],[35,7609,7283],{"className":7610},[7104],[35,7612,7286],{"className":7613,"style":7310},[7104,7145],[35,7615,7290],{"className":7616},[7314],[35,7618,7620],{"className":7619,"style":7474},[7314,7473],[35,7621,1697],{"className":7622},[7478,7479],[12,7624,7625,7626,7654,7655,7685,7686,7766,7767,7795,7796,7903,7904,7974,7975,8003,8004,8105,8106,8109,8110,8138,8139,5122],{},"If the matrix ",[35,7627,7629,7642],{"className":7628},[7057],[35,7630,7632],{"className":7631},[7061],[7063,7633,7634],{"xmlns":7065},[7067,7635,7636,7640],{},[7070,7637,7638],{},[7076,7639,7192],{},[7083,7641,7192],{"encoding":7085},[35,7643,7645],{"className":7644,"ariaHidden":7091},[7090],[35,7646,7648,7651],{"className":7647},[7095],[35,7649],{"className":7650,"style":7100},[7099],[35,7652,7192],{"className":7653},[7104,7145]," is the Hamiltonian ",[35,7656,7658,7672],{"className":7657},[7057],[35,7659,7661],{"className":7660},[7061],[7063,7662,7663],{"xmlns":7065},[7067,7664,7665,7670],{},[7070,7666,7667],{},[7076,7668,7669],{},"H",[7083,7671,7669],{"encoding":7085},[35,7673,7675],{"className":7674,"ariaHidden":7091},[7090],[35,7676,7678,7681],{"className":7677},[7095],[35,7679],{"className":7680,"style":7100},[7099],[35,7682,7669],{"className":7683,"style":7684},[7104,7145],"margin-right:0.0813em;",", we'll refer to the corresponding space as the power Krylov space ",[35,7687,7689,7709],{"className":7688},[7057],[35,7690,7692],{"className":7691},[7061],[7063,7693,7694],{"xmlns":7065},[7067,7695,7696,7706],{},[7070,7697,7698],{},[7699,7700,7701,7703],"msub",{},[7076,7702,7079],{"mathvariant":7078},[7076,7704,7705],{},"P",[7083,7707,7708],{"encoding":7085},"\\mathcal{K}_P",[35,7710,7712],{"className":7711,"ariaHidden":7091},[7090],[35,7713,7715,7719],{"className":7714},[7095],[35,7716],{"className":7717,"style":7718},[7099],"height:0.8333em;vertical-align:-0.15em;",[35,7720,7722,7725],{"className":7721},[7104],[35,7723,7079],{"className":7724,"style":7109},[7104,7108],[35,7726,7728],{"className":7727},[7113],[35,7729,7732,7757],{"className":7730},[7117,7731],"vlist-t2",[35,7733,7735,7752],{"className":7734},[7121],[35,7736,7739],{"className":7737,"style":7738},[7125],"height:0.3283em;",[35,7740,7742,7745],{"style":7741},"top:-2.55em;margin-left:-0.0144em;margin-right:0.05em;",[35,7743],{"className":7744,"style":7134},[7133],[35,7746,7748],{"className":7747},[7138,7139,7140,7141],[35,7749,7705],{"className":7750,"style":7751},[7104,7145,7141],"margin-right:0.1389em;",[35,7753,7756],{"className":7754},[7755],"vlist-s","​",[35,7758,7760],{"className":7759},[7121],[35,7761,7764],{"className":7762,"style":7763},[7125],"height:0.15em;",[35,7765],{},". In the case where ",[35,7768,7770,7783],{"className":7769},[7057],[35,7771,7773],{"className":7772},[7061],[7063,7774,7775],{"xmlns":7065},[7067,7776,7777,7781],{},[7070,7778,7779],{},[7076,7780,7192],{},[7083,7782,7192],{"encoding":7085},[35,7784,7786],{"className":7785,"ariaHidden":7091},[7090],[35,7787,7789,7792],{"className":7788},[7095],[35,7790],{"className":7791,"style":7100},[7099],[35,7793,7192],{"className":7794},[7104,7145]," is the time-evolution operator generated by the Hamiltonian ",[35,7797,7799,7833],{"className":7798},[7057],[35,7800,7802],{"className":7801},[7061],[7063,7803,7804],{"xmlns":7065},[7067,7805,7806,7830],{},[7070,7807,7808,7811,7813],{},[7076,7809,7810],{},"U",[7223,7812,242],{},[7073,7814,7815,7818],{},[7076,7816,7817],{},"e",[7070,7819,7820,7822,7825,7827],{},[7223,7821,7225],{},[7076,7823,7824],{},"i",[7076,7826,7669],{},[7076,7828,7829],{},"t",[7083,7831,7832],{"encoding":7085},"U=e^{-iHt}",[35,7834,7836,7855],{"className":7835,"ariaHidden":7091},[7090],[35,7837,7839,7842,7846,7849,7852],{"className":7838},[7095],[35,7840],{"className":7841,"style":7100},[7099],[35,7843,7810],{"className":7844,"style":7845},[7104,7145],"margin-right:0.109em;",[35,7847],{"className":7848,"style":7450},[7248],[35,7850,242],{"className":7851},[7454],[35,7853],{"className":7854,"style":7450},[7248],[35,7856,7858,7862],{"className":7857},[7095],[35,7859],{"className":7860,"style":7861},[7099],"height:0.8413em;",[35,7863,7865,7868],{"className":7864},[7104],[35,7866,7817],{"className":7867},[7104,7145],[35,7869,7871],{"className":7870},[7113],[35,7872,7874],{"className":7873},[7117],[35,7875,7877],{"className":7876},[7121],[35,7878,7880],{"className":7879,"style":7861},[7125],[35,7881,7882,7885],{"style":7129},[35,7883],{"className":7884,"style":7134},[7133],[35,7886,7888],{"className":7887},[7138,7139,7140,7141],[35,7889,7891,7894,7897,7900],{"className":7890},[7104,7141],[35,7892,7225],{"className":7893},[7104,7141],[35,7895,7824],{"className":7896},[7104,7145,7141],[35,7898,7669],{"className":7899,"style":7684},[7104,7145,7141],[35,7901,7829],{"className":7902},[7104,7145,7141],", we'll refer to the space as the unitary Krylov space ",[35,7905,7907,7925],{"className":7906},[7057],[35,7908,7910],{"className":7909},[7061],[7063,7911,7912],{"xmlns":7065},[7067,7913,7914,7922],{},[7070,7915,7916],{},[7699,7917,7918,7920],{},[7076,7919,7079],{"mathvariant":7078},[7076,7921,7810],{},[7083,7923,7924],{"encoding":7085},"\\mathcal{K}_U",[35,7926,7928],{"className":7927,"ariaHidden":7091},[7090],[35,7929,7931,7934],{"className":7930},[7095],[35,7932],{"className":7933,"style":7718},[7099],[35,7935,7937,7940],{"className":7936},[7104],[35,7938,7079],{"className":7939,"style":7109},[7104,7108],[35,7941,7943],{"className":7942},[7113],[35,7944,7946,7966],{"className":7945},[7117,7731],[35,7947,7949,7963],{"className":7948},[7121],[35,7950,7952],{"className":7951,"style":7738},[7125],[35,7953,7954,7957],{"style":7741},[35,7955],{"className":7956,"style":7134},[7133],[35,7958,7960],{"className":7959},[7138,7139,7140,7141],[35,7961,7810],{"className":7962,"style":7845},[7104,7145,7141],[35,7964,7756],{"className":7965},[7755],[35,7967,7969],{"className":7968},[7121],[35,7970,7972],{"className":7971,"style":7763},[7125],[35,7973],{},". The power Krylov subspace that we use classically cannot be generated directly on a quantum computer as ",[35,7976,7978,7991],{"className":7977},[7057],[35,7979,7981],{"className":7980},[7061],[7063,7982,7983],{"xmlns":7065},[7067,7984,7985,7989],{},[7070,7986,7987],{},[7076,7988,7669],{},[7083,7990,7669],{"encoding":7085},[35,7992,7994],{"className":7993,"ariaHidden":7091},[7090],[35,7995,7997,8000],{"className":7996},[7095],[35,7998],{"className":7999,"style":7100},[7099],[35,8001,7669],{"className":8002,"style":7684},[7104,7145]," is not a unitary operator. Instead, we can use the time-evolution operator ",[35,8005,8007,8037],{"className":8006},[7057],[35,8008,8010],{"className":8009},[7061],[7063,8011,8012],{"xmlns":7065},[7067,8013,8014,8034],{},[7070,8015,8016,8018,8020],{},[7076,8017,7810],{},[7223,8019,242],{},[7073,8021,8022,8024],{},[7076,8023,7817],{},[7070,8025,8026,8028,8030,8032],{},[7223,8027,7225],{},[7076,8029,7824],{},[7076,8031,7669],{},[7076,8033,7829],{},[7083,8035,8036],{"encoding":7085},"U = e^{-iHt}",[35,8038,8040,8058],{"className":8039,"ariaHidden":7091},[7090],[35,8041,8043,8046,8049,8052,8055],{"className":8042},[7095],[35,8044],{"className":8045,"style":7100},[7099],[35,8047,7810],{"className":8048,"style":7845},[7104,7145],[35,8050],{"className":8051,"style":7450},[7248],[35,8053,242],{"className":8054},[7454],[35,8056],{"className":8057,"style":7450},[7248],[35,8059,8061,8064],{"className":8060},[7095],[35,8062],{"className":8063,"style":7861},[7099],[35,8065,8067,8070],{"className":8066},[7104],[35,8068,7817],{"className":8069},[7104,7145],[35,8071,8073],{"className":8072},[7113],[35,8074,8076],{"className":8075},[7117],[35,8077,8079],{"className":8078},[7121],[35,8080,8082],{"className":8081,"style":7861},[7125],[35,8083,8084,8087],{"style":7129},[35,8085],{"className":8086,"style":7134},[7133],[35,8088,8090],{"className":8089},[7138,7139,7140,7141],[35,8091,8093,8096,8099,8102],{"className":8092},[7104,7141],[35,8094,7225],{"className":8095},[7104,7141],[35,8097,7824],{"className":8098},[7104,7145,7141],[35,8100,7669],{"className":8101,"style":7684},[7104,7145,7141],[35,8103,7829],{"className":8104},[7104,7145,7141]," which can be shown to give similar ",[19,8107,8108],{"href":5761},"convergence guarantees"," as the power method. Powers of ",[35,8111,8113,8126],{"className":8112},[7057],[35,8114,8116],{"className":8115},[7061],[7063,8117,8118],{"xmlns":7065},[7067,8119,8120,8124],{},[7070,8121,8122],{},[7076,8123,7810],{},[7083,8125,7810],{"encoding":7085},[35,8127,8129],{"className":8128,"ariaHidden":7091},[7090],[35,8130,8132,8135],{"className":8131},[7095],[35,8133],{"className":8134,"style":7100},[7099],[35,8136,7810],{"className":8137,"style":7845},[7104,7145]," then become different time steps ",[35,8140,8142,8182],{"className":8141},[7057],[35,8143,8145],{"className":8144},[7061],[7063,8146,8147],{"xmlns":7065},[7067,8148,8149,8179],{},[7070,8150,8151,8157,8159],{},[7073,8152,8153,8155],{},[7076,8154,7810],{},[7076,8156,279],{},[7223,8158,242],{},[7073,8160,8161,8163],{},[7076,8162,7817],{},[7070,8164,8165,8167,8169,8171,8173,8175,8177],{},[7223,8166,7225],{},[7076,8168,7824],{},[7076,8170,7669],{},[7223,8172,275],{"stretchy":7289},[7076,8174,279],{},[7076,8176,7829],{},[7223,8178,1694],{"stretchy":7289},[7083,8180,8181],{"encoding":7085},"U^k = e^{-iH(kt)}",[35,8183,8185,8231],{"className":8184,"ariaHidden":7091},[7090],[35,8186,8188,8192,8222,8225,8228],{"className":8187},[7095],[35,8189],{"className":8190,"style":8191},[7099],"height:0.8491em;",[35,8193,8195,8198],{"className":8194},[7104],[35,8196,7810],{"className":8197,"style":7845},[7104,7145],[35,8199,8201],{"className":8200},[7113],[35,8202,8204],{"className":8203},[7117],[35,8205,8207],{"className":8206},[7121],[35,8208,8210],{"className":8209,"style":8191},[7125],[35,8211,8212,8215],{"style":7129},[35,8213],{"className":8214,"style":7134},[7133],[35,8216,8218],{"className":8217},[7138,7139,7140,7141],[35,8219,279],{"className":8220,"style":8221},[7104,7145,7141],"margin-right:0.0315em;",[35,8223],{"className":8224,"style":7450},[7248],[35,8226,242],{"className":8227},[7454],[35,8229],{"className":8230,"style":7450},[7248],[35,8232,8234,8238],{"className":8233},[7095],[35,8235],{"className":8236,"style":8237},[7099],"height:0.888em;",[35,8239,8241,8244],{"className":8240},[7104],[35,8242,7817],{"className":8243},[7104,7145],[35,8245,8247],{"className":8246},[7113],[35,8248,8250],{"className":8249},[7117],[35,8251,8253],{"className":8252},[7121],[35,8254,8256],{"className":8255,"style":8237},[7125],[35,8257,8258,8261],{"style":7129},[35,8259],{"className":8260,"style":7134},[7133],[35,8262,8264],{"className":8263},[7138,7139,7140,7141],[35,8265,8267,8270,8273,8276,8279,8282,8285],{"className":8266},[7104,7141],[35,8268,7225],{"className":8269},[7104,7141],[35,8271,7824],{"className":8272},[7104,7145,7141],[35,8274,7669],{"className":8275,"style":7684},[7104,7145,7141],[35,8277,275],{"className":8278},[7472,7141],[35,8280,279],{"className":8281,"style":8221},[7104,7145,7141],[35,8283,7829],{"className":8284},[7104,7145,7141],[35,8286,1694],{"className":8287},[7314,7141],[35,8289,8291],{"className":8290},[7318],[35,8292,8294,8382],{"className":8293},[7057],[35,8295,8297],{"className":8296},[7061],[7063,8298,8299],{"xmlns":7065,"display":7327},[7067,8300,8301,8379],{},[7070,8302,8303,8312,8314],{},[8304,8305,8306,8308,8310],"msubsup",{},[7076,8307,7079],{"mathvariant":7078},[7076,8309,7810],{},[7076,8311,6778],{},[7223,8313,242],{},[7070,8315,8316,8318,8320,8323,8325,8327,8329,8331,8333,8335,8337,8343,8345,8347,8349,8351,8353,8355,8357,8359,8371,8373,8375,8377],{},[7223,8317,1681],{"fence":7091},[7076,8319,7283],{"mathvariant":7282},[7076,8321,8322],{},"ψ",[7223,8324,7290],{"stretchy":7289},[7223,8326,2877],{"separator":7091},[7076,8328,7810],{},[7076,8330,7283],{"mathvariant":7282},[7076,8332,8322],{},[7223,8334,7290],{"stretchy":7289},[7223,8336,2877],{"separator":7091},[7073,8338,8339,8341],{},[7076,8340,7810],{},[7227,8342,460],{},[7076,8344,7283],{"mathvariant":7282},[7076,8346,8322],{},[7223,8348,7290],{"stretchy":7289},[7223,8350,2877],{"separator":7091},[7076,8352,5122],{"mathvariant":7282},[7076,8354,5122],{"mathvariant":7282},[7076,8356,5122],{"mathvariant":7282},[7223,8358,2877],{"separator":7091},[7073,8360,8361,8363],{},[7076,8362,7810],{},[7070,8364,8365,8367,8369],{},[7076,8366,6778],{},[7223,8368,7225],{},[7227,8370,500],{},[7076,8372,7283],{"mathvariant":7282},[7076,8374,8322],{},[7223,8376,7290],{"stretchy":7289},[7223,8378,1697],{"fence":7091},[7083,8380,8381],{"encoding":7085},"\\mathcal{K}_U^r = \\left\\{ \\vert \\psi \\rangle, U \\vert \\psi \\rangle, U^2 \\vert \\psi \\rangle, ..., U^{r-1} \\vert \\psi \\rangle \\right\\}",[35,8383,8385,8454],{"className":8384,"ariaHidden":7091},[7090],[35,8386,8388,8392,8445,8448,8451],{"className":8387},[7095],[35,8389],{"className":8390,"style":8391},[7099],"height:0.9614em;vertical-align:-0.247em;",[35,8393,8395,8398],{"className":8394},[7104],[35,8396,7079],{"className":8397,"style":7109},[7104,7108],[35,8399,8401],{"className":8400},[7113],[35,8402,8404,8436],{"className":8403},[7117,7731],[35,8405,8407,8433],{"className":8406},[7121],[35,8408,8410,8422],{"className":8409,"style":7416},[7125],[35,8411,8413,8416],{"style":8412},"top:-2.453em;margin-left:-0.0144em;margin-right:0.05em;",[35,8414],{"className":8415,"style":7134},[7133],[35,8417,8419],{"className":8418},[7138,7139,7140,7141],[35,8420,7810],{"className":8421,"style":7845},[7104,7145,7141],[35,8423,8424,8427],{"style":7437},[35,8425],{"className":8426,"style":7134},[7133],[35,8428,8430],{"className":8429},[7138,7139,7140,7141],[35,8431,6778],{"className":8432,"style":7146},[7104,7145,7141],[35,8434,7756],{"className":8435},[7755],[35,8437,8439],{"className":8438},[7121],[35,8440,8443],{"className":8441,"style":8442},[7125],"height:0.247em;",[35,8444],{},[35,8446],{"className":8447,"style":7450},[7248],[35,8449,242],{"className":8450},[7454],[35,8452],{"className":8453,"style":7450},[7248],[35,8455,8457,8460],{"className":8456},[7095],[35,8458],{"className":8459,"style":7464},[7099],[35,8461,8463,8469,8472,8475,8478,8481,8484,8487,8490,8493,8496,8499,8502,8531,8534,8537,8540,8543,8546,8549,8552,8555,8593,8596,8599,8602],{"className":8462},[7468],[35,8464,8466],{"className":8465,"style":7474},[7472,7473],[35,8467,1681],{"className":8468},[7478,7479],[35,8470,7283],{"className":8471},[7104],[35,8473,8322],{"className":8474,"style":7310},[7104,7145],[35,8476,7290],{"className":8477},[7314],[35,8479,2877],{"className":8480},[7492],[35,8482],{"className":8483,"style":7496},[7248],[35,8485,7810],{"className":8486,"style":7845},[7104,7145],[35,8488,7283],{"className":8489},[7104],[35,8491,8322],{"className":8492,"style":7310},[7104,7145],[35,8494,7290],{"className":8495},[7314],[35,8497,2877],{"className":8498},[7492],[35,8500],{"className":8501,"style":7496},[7248],[35,8503,8505,8508],{"className":8504},[7104],[35,8506,7810],{"className":8507,"style":7845},[7104,7145],[35,8509,8511],{"className":8510},[7113],[35,8512,8514],{"className":8513},[7117],[35,8515,8517],{"className":8516},[7121],[35,8518,8520],{"className":8519,"style":7533},[7125],[35,8521,8522,8525],{"style":7437},[35,8523],{"className":8524,"style":7134},[7133],[35,8526,8528],{"className":8527},[7138,7139,7140,7141],[35,8529,460],{"className":8530},[7104,7141],[35,8532,7283],{"className":8533},[7104],[35,8535,8322],{"className":8536,"style":7310},[7104,7145],[35,8538,7290],{"className":8539},[7314],[35,8541,2877],{"className":8542},[7492],[35,8544],{"className":8545,"style":7496},[7248],[35,8547,7563],{"className":8548},[7104],[35,8550,2877],{"className":8551},[7492],[35,8553],{"className":8554,"style":7496},[7248],[35,8556,8558,8561],{"className":8557},[7104],[35,8559,7810],{"className":8560,"style":7845},[7104,7145],[35,8562,8564],{"className":8563},[7113],[35,8565,8567],{"className":8566},[7117],[35,8568,8570],{"className":8569},[7121],[35,8571,8573],{"className":8572,"style":7533},[7125],[35,8574,8575,8578],{"style":7437},[35,8576],{"className":8577,"style":7134},[7133],[35,8579,8581],{"className":8580},[7138,7139,7140,7141],[35,8582,8584,8587,8590],{"className":8583},[7104,7141],[35,8585,6778],{"className":8586,"style":7146},[7104,7145,7141],[35,8588,7225],{"className":8589},[7253,7141],[35,8591,500],{"className":8592},[7104,7141],[35,8594,7283],{"className":8595},[7104],[35,8597,8322],{"className":8598,"style":7310},[7104,7145],[35,8600,7290],{"className":8601},[7314],[35,8603,8605],{"className":8604,"style":7474},[7314,7473],[35,8606,1697],{"className":8607},[7478,7479],[12,8609,8610],{},"See the Appendix for a detailed derivation of how the unitary Krylov space allows to represents low-energy eigenstates accurately.",[7047,8612,8614],{"id":8613},"krylov-quantum-diagonalization-algorithm","Krylov quantum diagonalization algorithm",[12,8616,8617,8618,8646,8647,8716,8717,8786,8787,8857,8858,8918],{},"Given an Hamiltonian ",[35,8619,8621,8634],{"className":8620},[7057],[35,8622,8624],{"className":8623},[7061],[7063,8625,8626],{"xmlns":7065},[7067,8627,8628,8632],{},[7070,8629,8630],{},[7076,8631,7669],{},[7083,8633,7669],{"encoding":7085},[35,8635,8637],{"className":8636,"ariaHidden":7091},[7090],[35,8638,8640,8643],{"className":8639},[7095],[35,8641],{"className":8642,"style":7100},[7099],[35,8644,7669],{"className":8645,"style":7684},[7104,7145]," that we wish to diagonalize, first we consider the corresponding unitary Krylov space ",[35,8648,8650,8667],{"className":8649},[7057],[35,8651,8653],{"className":8652},[7061],[7063,8654,8655],{"xmlns":7065},[7067,8656,8657,8665],{},[7070,8658,8659],{},[7699,8660,8661,8663],{},[7076,8662,7079],{"mathvariant":7078},[7076,8664,7810],{},[7083,8666,7924],{"encoding":7085},[35,8668,8670],{"className":8669,"ariaHidden":7091},[7090],[35,8671,8673,8676],{"className":8672},[7095],[35,8674],{"className":8675,"style":7718},[7099],[35,8677,8679,8682],{"className":8678},[7104],[35,8680,7079],{"className":8681,"style":7109},[7104,7108],[35,8683,8685],{"className":8684},[7113],[35,8686,8688,8708],{"className":8687},[7117,7731],[35,8689,8691,8705],{"className":8690},[7121],[35,8692,8694],{"className":8693,"style":7738},[7125],[35,8695,8696,8699],{"style":7741},[35,8697],{"className":8698,"style":7134},[7133],[35,8700,8702],{"className":8701},[7138,7139,7140,7141],[35,8703,7810],{"className":8704,"style":7845},[7104,7145,7141],[35,8706,7756],{"className":8707},[7755],[35,8709,8711],{"className":8710},[7121],[35,8712,8714],{"className":8713,"style":7763},[7125],[35,8715],{},". The goal is to find a compact representation of the Hamiltonian in ",[35,8718,8720,8737],{"className":8719},[7057],[35,8721,8723],{"className":8722},[7061],[7063,8724,8725],{"xmlns":7065},[7067,8726,8727,8735],{},[7070,8728,8729],{},[7699,8730,8731,8733],{},[7076,8732,7079],{"mathvariant":7078},[7076,8734,7810],{},[7083,8736,7924],{"encoding":7085},[35,8738,8740],{"className":8739,"ariaHidden":7091},[7090],[35,8741,8743,8746],{"className":8742},[7095],[35,8744],{"className":8745,"style":7718},[7099],[35,8747,8749,8752],{"className":8748},[7104],[35,8750,7079],{"className":8751,"style":7109},[7104,7108],[35,8753,8755],{"className":8754},[7113],[35,8756,8758,8778],{"className":8757},[7117,7731],[35,8759,8761,8775],{"className":8760},[7121],[35,8762,8764],{"className":8763,"style":7738},[7125],[35,8765,8766,8769],{"style":7741},[35,8767],{"className":8768,"style":7134},[7133],[35,8770,8772],{"className":8771},[7138,7139,7140,7141],[35,8773,7810],{"className":8774,"style":7845},[7104,7145,7141],[35,8776,7756],{"className":8777},[7755],[35,8779,8781],{"className":8780},[7121],[35,8782,8784],{"className":8783,"style":7763},[7125],[35,8785],{},", which we'll refer to as ",[35,8788,8790,8810],{"className":8789},[7057],[35,8791,8793],{"className":8792},[7061],[7063,8794,8795],{"xmlns":7065},[7067,8796,8797,8807],{},[7070,8798,8799],{},[8800,8801,8802,8804],"mover",{"accent":7091},[7076,8803,7669],{},[7223,8805,8806],{},"~",[7083,8808,8809],{"encoding":7085},"\\tilde{H}",[35,8811,8813],{"className":8812,"ariaHidden":7091},[7090],[35,8814,8816,8820],{"className":8815},[7095],[35,8817],{"className":8818,"style":8819},[7099],"height:0.9202em;",[35,8821,8824],{"className":8822},[7104,8823],"accent",[35,8825,8827],{"className":8826},[7117],[35,8828,8830],{"className":8829},[7121],[35,8831,8833,8843],{"className":8832,"style":8819},[7125],[35,8834,8836,8840],{"style":8835},"top:-3em;",[35,8837],{"className":8838,"style":8839},[7133],"height:3em;",[35,8841,7669],{"className":8842,"style":7684},[7104,7145],[35,8844,8846,8849],{"style":8845},"top:-3.6023em;",[35,8847],{"className":8848,"style":8839},[7133],[35,8850,8854],{"className":8851,"style":8853},[8852],"accent-body","left:-0.1944em;",[35,8855,8806],{"className":8856},[7104],". The matrix elements of ",[35,8859,8861,8878],{"className":8860},[7057],[35,8862,8864],{"className":8863},[7061],[7063,8865,8866],{"xmlns":7065},[7067,8867,8868,8876],{},[7070,8869,8870],{},[8800,8871,8872,8874],{"accent":7091},[7076,8873,7669],{},[7223,8875,8806],{},[7083,8877,8809],{"encoding":7085},[35,8879,8881],{"className":8880,"ariaHidden":7091},[7090],[35,8882,8884,8887],{"className":8883},[7095],[35,8885],{"className":8886,"style":8819},[7099],[35,8888,8890],{"className":8889},[7104,8823],[35,8891,8893],{"className":8892},[7117],[35,8894,8896],{"className":8895},[7121],[35,8897,8899,8907],{"className":8898,"style":8819},[7125],[35,8900,8901,8904],{"style":8835},[35,8902],{"className":8903,"style":8839},[7133],[35,8905,7669],{"className":8906,"style":7684},[7104,7145],[35,8908,8909,8912],{"style":8845},[35,8910],{"className":8911,"style":8839},[7133],[35,8913,8915],{"className":8914,"style":8853},[8852],[35,8916,8806],{"className":8917},[7104],", the projection of the Hamiltonian in the Krylov space, can be calculated by calculating the following expectation values",[35,8920,8922],{"className":8921},[7318],[35,8923,8925,8979],{"className":8924},[7057],[35,8926,8928],{"className":8927},[7061],[7063,8929,8930],{"xmlns":7065,"display":7327},[7067,8931,8932,8976],{},[7070,8933,8934,8949,8951,8954,8960,8962,8964,8966,8972,8974],{},[7699,8935,8936,8942],{},[8800,8937,8938,8940],{"accent":7091},[7076,8939,7669],{},[7223,8941,8806],{},[7070,8943,8944,8947],{},[7076,8945,8946],{},"m",[7076,8948,291],{},[7223,8950,242],{},[7223,8952,8953],{"stretchy":7289},"⟨",[7699,8955,8956,8958],{},[7076,8957,8322],{},[7076,8959,8946],{},[7076,8961,7283],{"mathvariant":7282},[7076,8963,7669],{},[7076,8965,7283],{"mathvariant":7282},[7699,8967,8968,8970],{},[7076,8969,8322],{},[7076,8971,291],{},[7223,8973,7290],{"stretchy":7289},[7223,8975,242],{},[7083,8977,8978],{"encoding":7085},"\\tilde{H}_{mn} = \\langle \\psi_m \\vert H \\vert \\psi_n \\rangle =",[35,8980,8982,9071],{"className":8981,"ariaHidden":7091},[7090],[35,8983,8985,8989,9062,9065,9068],{"className":8984},[7095],[35,8986],{"className":8987,"style":8988},[7099],"height:1.0702em;vertical-align:-0.15em;",[35,8990,8992,9023],{"className":8991},[7104],[35,8993,8995],{"className":8994},[7104,8823],[35,8996,8998],{"className":8997},[7117],[35,8999,9001],{"className":9000},[7121],[35,9002,9004,9012],{"className":9003,"style":8819},[7125],[35,9005,9006,9009],{"style":8835},[35,9007],{"className":9008,"style":8839},[7133],[35,9010,7669],{"className":9011,"style":7684},[7104,7145],[35,9013,9014,9017],{"style":8845},[35,9015],{"className":9016,"style":8839},[7133],[35,9018,9020],{"className":9019,"style":8853},[8852],[35,9021,8806],{"className":9022},[7104],[35,9024,9026],{"className":9025},[7113],[35,9027,9029,9054],{"className":9028},[7117,7731],[35,9030,9032,9051],{"className":9031},[7121],[35,9033,9036],{"className":9034,"style":9035},[7125],"height:0.1514em;",[35,9037,9039,9042],{"style":9038},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[35,9040],{"className":9041,"style":7134},[7133],[35,9043,9045],{"className":9044},[7138,7139,7140,7141],[35,9046,9048],{"className":9047},[7104,7141],[35,9049,7227],{"className":9050},[7104,7145,7141],[35,9052,7756],{"className":9053},[7755],[35,9055,9057],{"className":9056},[7121],[35,9058,9060],{"className":9059,"style":7763},[7125],[35,9061],{},[35,9063],{"className":9064,"style":7450},[7248],[35,9066,242],{"className":9067},[7454],[35,9069],{"className":9070,"style":7450},[7248],[35,9072,9074,9077,9080,9121,9124,9127,9130,9170,9173,9176],{"className":9073},[7095],[35,9075],{"className":9076,"style":7303},[7099],[35,9078,8953],{"className":9079},[7472],[35,9081,9083,9086],{"className":9082},[7104],[35,9084,8322],{"className":9085,"style":7310},[7104,7145],[35,9087,9089],{"className":9088},[7113],[35,9090,9092,9113],{"className":9091},[7117,7731],[35,9093,9095,9110],{"className":9094},[7121],[35,9096,9098],{"className":9097,"style":9035},[7125],[35,9099,9101,9104],{"style":9100},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[35,9102],{"className":9103,"style":7134},[7133],[35,9105,9107],{"className":9106},[7138,7139,7140,7141],[35,9108,8946],{"className":9109},[7104,7145,7141],[35,9111,7756],{"className":9112},[7755],[35,9114,9116],{"className":9115},[7121],[35,9117,9119],{"className":9118,"style":7763},[7125],[35,9120],{},[35,9122,7283],{"className":9123},[7104],[35,9125,7669],{"className":9126,"style":7684},[7104,7145],[35,9128,7283],{"className":9129},[7104],[35,9131,9133,9136],{"className":9132},[7104],[35,9134,8322],{"className":9135,"style":7310},[7104,7145],[35,9137,9139],{"className":9138},[7113],[35,9140,9142,9162],{"className":9141},[7117,7731],[35,9143,9145,9159],{"className":9144},[7121],[35,9146,9148],{"className":9147,"style":9035},[7125],[35,9149,9150,9153],{"style":9100},[35,9151],{"className":9152,"style":7134},[7133],[35,9154,9156],{"className":9155},[7138,7139,7140,7141],[35,9157,291],{"className":9158},[7104,7145,7141],[35,9160,7756],{"className":9161},[7755],[35,9163,9165],{"className":9164},[7121],[35,9166,9168],{"className":9167,"style":7763},[7125],[35,9169],{},[35,9171,7290],{"className":9172},[7314],[35,9174],{"className":9175,"style":7450},[7248],[35,9177,242],{"className":9178},[7454],[35,9180,9182],{"className":9181},[7318],[35,9183,9185,9247],{"className":9184},[7057],[35,9186,9188],{"className":9187},[7061],[7063,9189,9190],{"xmlns":7065,"display":7327},[7067,9191,9192,9244],{},[7070,9193,9194,9196,9198,9200,9202,9218,9220,9238,9240,9242],{},[7223,9195,242],{},[7223,9197,8953],{"stretchy":7289},[7076,9199,8322],{},[7076,9201,7283],{"mathvariant":7282},[7073,9203,9204,9206],{},[7076,9205,7817],{},[7070,9207,9208,9210,9212],{},[7076,9209,7824],{},[7076,9211,7669],{},[7699,9213,9214,9216],{},[7076,9215,7829],{},[7076,9217,8946],{},[7076,9219,7669],{},[7073,9221,9222,9224],{},[7076,9223,7817],{},[7070,9225,9226,9228,9230,9232],{},[7223,9227,7225],{},[7076,9229,7824],{},[7076,9231,7669],{},[7699,9233,9234,9236],{},[7076,9235,7829],{},[7076,9237,291],{},[7076,9239,7283],{"mathvariant":7282},[7076,9241,8322],{},[7223,9243,7290],{"stretchy":7289},[7083,9245,9246],{"encoding":7085},"= \\langle \\psi \\vert  e^{i H t_m}   H e^{-i H t_n} \\vert \\psi \\rangle",[35,9248,9250,9263],{"className":9249,"ariaHidden":7091},[7090],[35,9251,9253,9257,9260],{"className":9252},[7095],[35,9254],{"className":9255,"style":9256},[7099],"height:0.3669em;",[35,9258,242],{"className":9259},[7454],[35,9261],{"className":9262,"style":7450},[7248],[35,9264,9266,9270,9273,9276,9279,9360,9363,9441,9444,9447],{"className":9265},[7095],[35,9267],{"className":9268,"style":9269},[7099],"height:1.1413em;vertical-align:-0.25em;",[35,9271,8953],{"className":9272},[7472],[35,9274,8322],{"className":9275,"style":7310},[7104,7145],[35,9277,7283],{"className":9278},[7104],[35,9280,9282,9285],{"className":9281},[7104],[35,9283,7817],{"className":9284},[7104,7145],[35,9286,9288],{"className":9287},[7113],[35,9289,9291],{"className":9290},[7117],[35,9292,9294],{"className":9293},[7121],[35,9295,9298],{"className":9296,"style":9297},[7125],"height:0.8913em;",[35,9299,9300,9303],{"style":7437},[35,9301],{"className":9302,"style":7134},[7133],[35,9304,9306],{"className":9305},[7138,7139,7140,7141],[35,9307,9309,9312,9315],{"className":9308},[7104,7141],[35,9310,7824],{"className":9311},[7104,7145,7141],[35,9313,7669],{"className":9314,"style":7684},[7104,7145,7141],[35,9316,9318,9321],{"className":9317},[7104,7141],[35,9319,7829],{"className":9320},[7104,7145,7141],[35,9322,9324],{"className":9323},[7113],[35,9325,9327,9351],{"className":9326},[7117,7731],[35,9328,9330,9348],{"className":9329},[7121],[35,9331,9334],{"className":9332,"style":9333},[7125],"height:0.1645em;",[35,9335,9337,9341],{"style":9336},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[35,9338],{"className":9339,"style":9340},[7133],"height:2.5em;",[35,9342,9345],{"className":9343},[7138,9344,7479,7141],"reset-size3",[35,9346,8946],{"className":9347},[7104,7145,7141],[35,9349,7756],{"className":9350},[7755],[35,9352,9354],{"className":9353},[7121],[35,9355,9358],{"className":9356,"style":9357},[7125],"height:0.143em;",[35,9359],{},[35,9361,7669],{"className":9362,"style":7684},[7104,7145],[35,9364,9366,9369],{"className":9365},[7104],[35,9367,7817],{"className":9368},[7104,7145],[35,9370,9372],{"className":9371},[7113],[35,9373,9375],{"className":9374},[7117],[35,9376,9378],{"className":9377},[7121],[35,9379,9381],{"className":9380,"style":9297},[7125],[35,9382,9383,9386],{"style":7437},[35,9384],{"className":9385,"style":7134},[7133],[35,9387,9389],{"className":9388},[7138,7139,7140,7141],[35,9390,9392,9395,9398,9401],{"className":9391},[7104,7141],[35,9393,7225],{"className":9394},[7104,7141],[35,9396,7824],{"className":9397},[7104,7145,7141],[35,9399,7669],{"className":9400,"style":7684},[7104,7145,7141],[35,9402,9404,9407],{"className":9403},[7104,7141],[35,9405,7829],{"className":9406},[7104,7145,7141],[35,9408,9410],{"className":9409},[7113],[35,9411,9413,9433],{"className":9412},[7117,7731],[35,9414,9416,9430],{"className":9415},[7121],[35,9417,9419],{"className":9418,"style":9333},[7125],[35,9420,9421,9424],{"style":9336},[35,9422],{"className":9423,"style":9340},[7133],[35,9425,9427],{"className":9426},[7138,9344,7479,7141],[35,9428,291],{"className":9429},[7104,7145,7141],[35,9431,7756],{"className":9432},[7755],[35,9434,9436],{"className":9435},[7121],[35,9437,9439],{"className":9438,"style":9357},[7125],[35,9440],{},[35,9442,7283],{"className":9443},[7104],[35,9445,8322],{"className":9446,"style":7310},[7104,7145],[35,9448,7290],{"className":9449},[7314],[35,9451,9453],{"className":9452},[7318],[35,9454,9456,9519],{"className":9455},[7057],[35,9457,9459],{"className":9458},[7061],[7063,9460,9461],{"xmlns":7065,"display":7327},[7067,9462,9463,9516],{},[7070,9464,9465,9467,9469,9471,9473,9490,9492,9510,9512,9514],{},[7223,9466,242],{},[7223,9468,8953],{"stretchy":7289},[7076,9470,8322],{},[7076,9472,7283],{"mathvariant":7282},[7073,9474,9475,9477],{},[7076,9476,7817],{},[7070,9478,9479,9481,9483,9485,9488],{},[7076,9480,7824],{},[7076,9482,7669],{},[7076,9484,8946],{},[7076,9486,9487],{},"d",[7076,9489,7829],{},[7076,9491,7669],{},[7073,9493,9494,9496],{},[7076,9495,7817],{},[7070,9497,9498,9500,9502,9504,9506,9508],{},[7223,9499,7225],{},[7076,9501,7824],{},[7076,9503,7669],{},[7076,9505,291],{},[7076,9507,9487],{},[7076,9509,7829],{},[7076,9511,7283],{"mathvariant":7282},[7076,9513,8322],{},[7223,9515,7290],{"stretchy":7289},[7083,9517,9518],{"encoding":7085},"= \\langle \\psi \\vert  e^{i H m dt}   H e^{-i H n dt} \\vert \\psi \\rangle",[35,9520,9522,9534],{"className":9521,"ariaHidden":7091},[7090],[35,9523,9525,9528,9531],{"className":9524},[7095],[35,9526],{"className":9527,"style":9256},[7099],[35,9529,242],{"className":9530},[7454],[35,9532],{"className":9533,"style":7450},[7248],[35,9535,9537,9541,9544,9547,9550,9595,9598,9645,9648,9651],{"className":9536},[7095],[35,9538],{"className":9539,"style":9540},[7099],"height:1.1491em;vertical-align:-0.25em;",[35,9542,8953],{"className":9543},[7472],[35,9545,8322],{"className":9546,"style":7310},[7104,7145],[35,9548,7283],{"className":9549},[7104],[35,9551,9553,9556],{"className":9552},[7104],[35,9554,7817],{"className":9555},[7104,7145],[35,9557,9559],{"className":9558},[7113],[35,9560,9562],{"className":9561},[7117],[35,9563,9565],{"className":9564},[7121],[35,9566,9569],{"className":9567,"style":9568},[7125],"height:0.8991em;",[35,9570,9571,9574],{"style":7437},[35,9572],{"className":9573,"style":7134},[7133],[35,9575,9577],{"className":9576},[7138,7139,7140,7141],[35,9578,9580,9583,9586,9589,9592],{"className":9579},[7104,7141],[35,9581,7824],{"className":9582},[7104,7145,7141],[35,9584,7669],{"className":9585,"style":7684},[7104,7145,7141],[35,9587,8946],{"className":9588},[7104,7145,7141],[35,9590,9487],{"className":9591},[7104,7145,7141],[35,9593,7829],{"className":9594},[7104,7145,7141],[35,9596,7669],{"className":9597,"style":7684},[7104,7145],[35,9599,9601,9604],{"className":9600},[7104],[35,9602,7817],{"className":9603},[7104,7145],[35,9605,9607],{"className":9606},[7113],[35,9608,9610],{"className":9609},[7117],[35,9611,9613],{"className":9612},[7121],[35,9614,9616],{"className":9615,"style":9568},[7125],[35,9617,9618,9621],{"style":7437},[35,9619],{"className":9620,"style":7134},[7133],[35,9622,9624],{"className":9623},[7138,7139,7140,7141],[35,9625,9627,9630,9633,9636,9639,9642],{"className":9626},[7104,7141],[35,9628,7225],{"className":9629},[7104,7141],[35,9631,7824],{"className":9632},[7104,7145,7141],[35,9634,7669],{"className":9635,"style":7684},[7104,7145,7141],[35,9637,291],{"className":9638},[7104,7145,7141],[35,9640,9487],{"className":9641},[7104,7145,7141],[35,9643,7829],{"className":9644},[7104,7145,7141],[35,9646,7283],{"className":9647},[7104],[35,9649,8322],{"className":9650,"style":7310},[7104,7145],[35,9652,7290],{"className":9653},[7314],[12,9655,9656,9657,9866,9867,9972,9973,10007,10008,10077,10078,10106,10107,10159,10160,10188,10189,10245,10246,10306,10307,10367,10368],{},"Where ",[35,9658,9660,9708],{"className":9659},[7057],[35,9661,9663],{"className":9662},[7061],[7063,9664,9665],{"xmlns":7065},[7067,9666,9667,9705],{},[7070,9668,9669,9671,9677,9679,9681,9699,9701,9703],{},[7076,9670,7283],{"mathvariant":7282},[7699,9672,9673,9675],{},[7076,9674,8322],{},[7076,9676,291],{},[7223,9678,7290],{"stretchy":7289},[7223,9680,242],{},[7073,9682,9683,9685],{},[7076,9684,7817],{},[7070,9686,9687,9689,9691,9693],{},[7223,9688,7225],{},[7076,9690,7824],{},[7076,9692,7669],{},[7699,9694,9695,9697],{},[7076,9696,7829],{},[7076,9698,291],{},[7076,9700,7283],{"mathvariant":7282},[7076,9702,8322],{},[7223,9704,7290],{"stretchy":7289},[7083,9706,9707],{"encoding":7085},"\\vert \\psi_n \\rangle = e^{-i H t_n} \\vert \\psi \\rangle",[35,9709,9711,9772],{"className":9710,"ariaHidden":7091},[7090],[35,9712,9714,9717,9720,9760,9763,9766,9769],{"className":9713},[7095],[35,9715],{"className":9716,"style":7303},[7099],[35,9718,7283],{"className":9719},[7104],[35,9721,9723,9726],{"className":9722},[7104],[35,9724,8322],{"className":9725,"style":7310},[7104,7145],[35,9727,9729],{"className":9728},[7113],[35,9730,9732,9752],{"className":9731},[7117,7731],[35,9733,9735,9749],{"className":9734},[7121],[35,9736,9738],{"className":9737,"style":9035},[7125],[35,9739,9740,9743],{"style":9100},[35,9741],{"className":9742,"style":7134},[7133],[35,9744,9746],{"className":9745},[7138,7139,7140,7141],[35,9747,291],{"className":9748},[7104,7145,7141],[35,9750,7756],{"className":9751},[7755],[35,9753,9755],{"className":9754},[7121],[35,9756,9758],{"className":9757,"style":7763},[7125],[35,9759],{},[35,9761,7290],{"className":9762},[7314],[35,9764],{"className":9765,"style":7450},[7248],[35,9767,242],{"className":9768},[7454],[35,9770],{"className":9771,"style":7450},[7248],[35,9773,9775,9779,9857,9860,9863],{"className":9774},[7095],[35,9776],{"className":9777,"style":9778},[7099],"height:1.0913em;vertical-align:-0.25em;",[35,9780,9782,9785],{"className":9781},[7104],[35,9783,7817],{"className":9784},[7104,7145],[35,9786,9788],{"className":9787},[7113],[35,9789,9791],{"className":9790},[7117],[35,9792,9794],{"className":9793},[7121],[35,9795,9797],{"className":9796,"style":7861},[7125],[35,9798,9799,9802],{"style":7129},[35,9800],{"className":9801,"style":7134},[7133],[35,9803,9805],{"className":9804},[7138,7139,7140,7141],[35,9806,9808,9811,9814,9817],{"className":9807},[7104,7141],[35,9809,7225],{"className":9810},[7104,7141],[35,9812,7824],{"className":9813},[7104,7145,7141],[35,9815,7669],{"className":9816,"style":7684},[7104,7145,7141],[35,9818,9820,9823],{"className":9819},[7104,7141],[35,9821,7829],{"className":9822},[7104,7145,7141],[35,9824,9826],{"className":9825},[7113],[35,9827,9829,9849],{"className":9828},[7117,7731],[35,9830,9832,9846],{"className":9831},[7121],[35,9833,9835],{"className":9834,"style":9333},[7125],[35,9836,9837,9840],{"style":9336},[35,9838],{"className":9839,"style":9340},[7133],[35,9841,9843],{"className":9842},[7138,9344,7479,7141],[35,9844,291],{"className":9845},[7104,7145,7141],[35,9847,7756],{"className":9848},[7755],[35,9850,9852],{"className":9851},[7121],[35,9853,9855],{"className":9854,"style":9357},[7125],[35,9856],{},[35,9858,7283],{"className":9859},[7104],[35,9861,8322],{"className":9862,"style":7310},[7104,7145],[35,9864,7290],{"className":9865},[7314]," are the vectors of the unitary Krylov space and ",[35,9868,9870,9896],{"className":9869},[7057],[35,9871,9873],{"className":9872},[7061],[7063,9874,9875],{"xmlns":7065},[7067,9876,9877,9893],{},[7070,9878,9879,9885,9887,9889,9891],{},[7699,9880,9881,9883],{},[7076,9882,7829],{},[7076,9884,291],{},[7223,9886,242],{},[7076,9888,291],{},[7076,9890,9487],{},[7076,9892,7829],{},[7083,9894,9895],{"encoding":7085},"t_n = n dt",[35,9897,9899,9956],{"className":9898,"ariaHidden":7091},[7090],[35,9900,9902,9906,9947,9950,9953],{"className":9901},[7095],[35,9903],{"className":9904,"style":9905},[7099],"height:0.7651em;vertical-align:-0.15em;",[35,9907,9909,9912],{"className":9908},[7104],[35,9910,7829],{"className":9911},[7104,7145],[35,9913,9915],{"className":9914},[7113],[35,9916,9918,9939],{"className":9917},[7117,7731],[35,9919,9921,9936],{"className":9920},[7121],[35,9922,9924],{"className":9923,"style":9035},[7125],[35,9925,9927,9930],{"style":9926},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[35,9928],{"className":9929,"style":7134},[7133],[35,9931,9933],{"className":9932},[7138,7139,7140,7141],[35,9934,291],{"className":9935},[7104,7145,7141],[35,9937,7756],{"className":9938},[7755],[35,9940,9942],{"className":9941},[7121],[35,9943,9945],{"className":9944,"style":7763},[7125],[35,9946],{},[35,9948],{"className":9949,"style":7450},[7248],[35,9951,242],{"className":9952},[7454],[35,9954],{"className":9955,"style":7450},[7248],[35,9957,9959,9963,9966,9969],{"className":9958},[7095],[35,9960],{"className":9961,"style":9962},[7099],"height:0.6944em;",[35,9964,291],{"className":9965},[7104,7145],[35,9967,9487],{"className":9968},[7104,7145],[35,9970,7829],{"className":9971},[7104,7145]," are the multiples of the time step ",[35,9974,9976,9992],{"className":9975},[7057],[35,9977,9979],{"className":9978},[7061],[7063,9980,9981],{"xmlns":7065},[7067,9982,9983,9989],{},[7070,9984,9985,9987],{},[7076,9986,9487],{},[7076,9988,7829],{},[7083,9990,9991],{"encoding":7085},"dt",[35,9993,9995],{"className":9994,"ariaHidden":7091},[7090],[35,9996,9998,10001,10004],{"className":9997},[7095],[35,9999],{"className":10000,"style":9962},[7099],[35,10002,9487],{"className":10003},[7104,7145],[35,10005,7829],{"className":10006},[7104,7145]," chosen. On a quantum computer, the calculation of each matrix elements can be done with any algorithm which allows to obtain overlap between quantum states. This tutorial focuses on the Hadamard test. Given that the ",[35,10009,10011,10028],{"className":10010},[7057],[35,10012,10014],{"className":10013},[7061],[7063,10015,10016],{"xmlns":7065},[7067,10017,10018,10026],{},[7070,10019,10020],{},[7699,10021,10022,10024],{},[7076,10023,7079],{"mathvariant":7078},[7076,10025,7810],{},[7083,10027,7924],{"encoding":7085},[35,10029,10031],{"className":10030,"ariaHidden":7091},[7090],[35,10032,10034,10037],{"className":10033},[7095],[35,10035],{"className":10036,"style":7718},[7099],[35,10038,10040,10043],{"className":10039},[7104],[35,10041,7079],{"className":10042,"style":7109},[7104,7108],[35,10044,10046],{"className":10045},[7113],[35,10047,10049,10069],{"className":10048},[7117,7731],[35,10050,10052,10066],{"className":10051},[7121],[35,10053,10055],{"className":10054,"style":7738},[7125],[35,10056,10057,10060],{"style":7741},[35,10058],{"className":10059,"style":7134},[7133],[35,10061,10063],{"className":10062},[7138,7139,7140,7141],[35,10064,7810],{"className":10065,"style":7845},[7104,7145,7141],[35,10067,7756],{"className":10068},[7755],[35,10070,10072],{"className":10071},[7121],[35,10073,10075],{"className":10074,"style":7763},[7125],[35,10076],{}," has dimension ",[35,10079,10081,10094],{"className":10080},[7057],[35,10082,10084],{"className":10083},[7061],[7063,10085,10086],{"xmlns":7065},[7067,10087,10088,10092],{},[7070,10089,10090],{},[7076,10091,6778],{},[7083,10093,6778],{"encoding":7085},[35,10095,10097],{"className":10096,"ariaHidden":7091},[7090],[35,10098,10100,10103],{"className":10099},[7095],[35,10101],{"className":10102,"style":7173},[7099],[35,10104,6778],{"className":10105,"style":7146},[7104,7145],", the Hamiltonian projected into the subspace will have dimensions ",[35,10108,10110,10129],{"className":10109},[7057],[35,10111,10113],{"className":10112},[7061],[7063,10114,10115],{"xmlns":7065},[7067,10116,10117,10126],{},[7070,10118,10119,10121,10124],{},[7076,10120,6778],{},[7223,10122,10123],{},"×",[7076,10125,6778],{},[7083,10127,10128],{"encoding":7085},"r \\times r",[35,10130,10132,10150],{"className":10131,"ariaHidden":7091},[7090],[35,10133,10135,10138,10141,10144,10147],{"className":10134},[7095],[35,10136],{"className":10137,"style":7241},[7099],[35,10139,6778],{"className":10140,"style":7146},[7104,7145],[35,10142],{"className":10143,"style":7249},[7248],[35,10145,10123],{"className":10146},[7253],[35,10148],{"className":10149,"style":7249},[7248],[35,10151,10153,10156],{"className":10152},[7095],[35,10154],{"className":10155,"style":7173},[7099],[35,10157,6778],{"className":10158,"style":7146},[7104,7145],". With ",[35,10161,10163,10176],{"className":10162},[7057],[35,10164,10166],{"className":10165},[7061],[7063,10167,10168],{"xmlns":7065},[7067,10169,10170,10174],{},[7070,10171,10172],{},[7076,10173,6778],{},[7083,10175,6778],{"encoding":7085},[35,10177,10179],{"className":10178,"ariaHidden":7091},[7090],[35,10180,10182,10185],{"className":10181},[7095],[35,10183],{"className":10184,"style":7173},[7099],[35,10186,6778],{"className":10187,"style":7146},[7104,7145]," small enough (generally ",[35,10190,10192,10213],{"className":10191},[7057],[35,10193,10195],{"className":10194},[7061],[7063,10196,10197],{"xmlns":7065},[7067,10198,10199,10210],{},[7070,10200,10201,10203,10206,10208],{},[7076,10202,6778],{},[7223,10204,10205],{},"\u003C",[7223,10207,10205],{},[7227,10209,1691],{},[7083,10211,10212],{"encoding":7085},"r\u003C\u003C100",[35,10214,10216,10236],{"className":10215,"ariaHidden":7091},[7090],[35,10217,10219,10223,10226,10229,10233],{"className":10218},[7095],[35,10220],{"className":10221,"style":10222},[7099],"height:0.5782em;vertical-align:-0.0391em;",[35,10224,6778],{"className":10225,"style":7146},[7104,7145],[35,10227],{"className":10228,"style":7450},[7248],[35,10230,10232],{"className":10231},[7454],"\u003C\u003C",[35,10234],{"className":10235,"style":7450},[7248],[35,10237,10239,10242],{"className":10238},[7095],[35,10240],{"className":10241,"style":7263},[7099],[35,10243,1691],{"className":10244},[7104]," is sufficient to obtain convergence of estimates of eigenenergies) we can then easily diagonalize the projected Hamiltonian ",[35,10247,10249,10266],{"className":10248},[7057],[35,10250,10252],{"className":10251},[7061],[7063,10253,10254],{"xmlns":7065},[7067,10255,10256,10264],{},[7070,10257,10258],{},[8800,10259,10260,10262],{"accent":7091},[7076,10261,7669],{},[7223,10263,8806],{},[7083,10265,8809],{"encoding":7085},[35,10267,10269],{"className":10268,"ariaHidden":7091},[7090],[35,10270,10272,10275],{"className":10271},[7095],[35,10273],{"className":10274,"style":8819},[7099],[35,10276,10278],{"className":10277},[7104,8823],[35,10279,10281],{"className":10280},[7117],[35,10282,10284],{"className":10283},[7121],[35,10285,10287,10295],{"className":10286,"style":8819},[7125],[35,10288,10289,10292],{"style":8835},[35,10290],{"className":10291,"style":8839},[7133],[35,10293,7669],{"className":10294,"style":7684},[7104,7145],[35,10296,10297,10300],{"style":8845},[35,10298],{"className":10299,"style":8839},[7133],[35,10301,10303],{"className":10302,"style":8853},[8852],[35,10304,8806],{"className":10305},[7104],". However, we cannot directly diagonalize ",[35,10308,10310,10327],{"className":10309},[7057],[35,10311,10313],{"className":10312},[7061],[7063,10314,10315],{"xmlns":7065},[7067,10316,10317,10325],{},[7070,10318,10319],{},[8800,10320,10321,10323],{"accent":7091},[7076,10322,7669],{},[7223,10324,8806],{},[7083,10326,8809],{"encoding":7085},[35,10328,10330],{"className":10329,"ariaHidden":7091},[7090],[35,10331,10333,10336],{"className":10332},[7095],[35,10334],{"className":10335,"style":8819},[7099],[35,10337,10339],{"className":10338},[7104,8823],[35,10340,10342],{"className":10341},[7117],[35,10343,10345],{"className":10344},[7121],[35,10346,10348,10356],{"className":10347,"style":8819},[7125],[35,10349,10350,10353],{"style":8835},[35,10351],{"className":10352,"style":8839},[7133],[35,10354,7669],{"className":10355,"style":7684},[7104,7145],[35,10357,10358,10361],{"style":8845},[35,10359],{"className":10360,"style":8839},[7133],[35,10362,10364],{"className":10363,"style":8853},[8852],[35,10365,8806],{"className":10366},[7104]," because of the non-orthogonality of the Krylov space vectors. We'll have to measure their overlaps and construct a matrix ",[35,10369,10371,10390],{"className":10370},[7057],[35,10372,10374],{"className":10373},[7061],[7063,10375,10376],{"xmlns":7065},[7067,10377,10378,10387],{},[7070,10379,10380],{},[8800,10381,10382,10385],{"accent":7091},[7076,10383,10384],{},"S",[7223,10386,8806],{},[7083,10388,10389],{"encoding":7085},"\\tilde{S}",[35,10391,10393],{"className":10392,"ariaHidden":7091},[7090],[35,10394,10396,10399],{"className":10395},[7095],[35,10397],{"className":10398,"style":8819},[7099],[35,10400,10402],{"className":10401},[7104,8823],[35,10403,10405],{"className":10404},[7117],[35,10406,10408],{"className":10407},[7121],[35,10409,10411,10420],{"className":10410,"style":8819},[7125],[35,10412,10413,10416],{"style":8835},[35,10414],{"className":10415,"style":8839},[7133],[35,10417,10384],{"className":10418,"style":10419},[7104,7145],"margin-right:0.0576em;",[35,10421,10422,10425],{"style":8845},[35,10423],{"className":10424,"style":8839},[7133],[35,10426,10429],{"className":10427,"style":10428},[8852],"left:-0.1667em;",[35,10430,8806],{"className":10431},[7104],[35,10433,10435],{"className":10434},[7318],[35,10436,10438,10484],{"className":10437},[7057],[35,10439,10441],{"className":10440},[7061],[7063,10442,10443],{"xmlns":7065,"display":7327},[7067,10444,10445,10481],{},[7070,10446,10447,10461,10463,10465,10471,10473,10479],{},[7699,10448,10449,10455],{},[8800,10450,10451,10453],{"accent":7091},[7076,10452,10384],{},[7223,10454,8806],{},[7070,10456,10457,10459],{},[7076,10458,8946],{},[7076,10460,291],{},[7223,10462,242],{},[7223,10464,8953],{"stretchy":7289},[7699,10466,10467,10469],{},[7076,10468,8322],{},[7076,10470,8946],{},[7076,10472,7283],{"mathvariant":7282},[7699,10474,10475,10477],{},[7076,10476,8322],{},[7076,10478,291],{},[7223,10480,7290],{"stretchy":7289},[7083,10482,10483],{"encoding":7085},"\\tilde{S}_{mn} = \\langle \\psi_m \\vert \\psi_n \\rangle",[35,10485,10487,10574],{"className":10486,"ariaHidden":7091},[7090],[35,10488,10490,10493,10565,10568,10571],{"className":10489},[7095],[35,10491],{"className":10492,"style":8988},[7099],[35,10494,10496,10527],{"className":10495},[7104],[35,10497,10499],{"className":10498},[7104,8823],[35,10500,10502],{"className":10501},[7117],[35,10503,10505],{"className":10504},[7121],[35,10506,10508,10516],{"className":10507,"style":8819},[7125],[35,10509,10510,10513],{"style":8835},[35,10511],{"className":10512,"style":8839},[7133],[35,10514,10384],{"className":10515,"style":10419},[7104,7145],[35,10517,10518,10521],{"style":8845},[35,10519],{"className":10520,"style":8839},[7133],[35,10522,10524],{"className":10523,"style":10428},[8852],[35,10525,8806],{"className":10526},[7104],[35,10528,10530],{"className":10529},[7113],[35,10531,10533,10557],{"className":10532},[7117,7731],[35,10534,10536,10554],{"className":10535},[7121],[35,10537,10539],{"className":10538,"style":9035},[7125],[35,10540,10542,10545],{"style":10541},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[35,10543],{"className":10544,"style":7134},[7133],[35,10546,10548],{"className":10547},[7138,7139,7140,7141],[35,10549,10551],{"className":10550},[7104,7141],[35,10552,7227],{"className":10553},[7104,7145,7141],[35,10555,7756],{"className":10556},[7755],[35,10558,10560],{"className":10559},[7121],[35,10561,10563],{"className":10562,"style":7763},[7125],[35,10564],{},[35,10566],{"className":10567,"style":7450},[7248],[35,10569,242],{"className":10570},[7454],[35,10572],{"className":10573,"style":7450},[7248],[35,10575,10577,10580,10583,10623,10626,10666],{"className":10576},[7095],[35,10578],{"className":10579,"style":7303},[7099],[35,10581,8953],{"className":10582},[7472],[35,10584,10586,10589],{"className":10585},[7104],[35,10587,8322],{"className":10588,"style":7310},[7104,7145],[35,10590,10592],{"className":10591},[7113],[35,10593,10595,10615],{"className":10594},[7117,7731],[35,10596,10598,10612],{"className":10597},[7121],[35,10599,10601],{"className":10600,"style":9035},[7125],[35,10602,10603,10606],{"style":9100},[35,10604],{"className":10605,"style":7134},[7133],[35,10607,10609],{"className":10608},[7138,7139,7140,7141],[35,10610,8946],{"className":10611},[7104,7145,7141],[35,10613,7756],{"className":10614},[7755],[35,10616,10618],{"className":10617},[7121],[35,10619,10621],{"className":10620,"style":7763},[7125],[35,10622],{},[35,10624,7283],{"className":10625},[7104],[35,10627,10629,10632],{"className":10628},[7104],[35,10630,8322],{"className":10631,"style":7310},[7104,7145],[35,10633,10635],{"className":10634},[7113],[35,10636,10638,10658],{"className":10637},[7117,7731],[35,10639,10641,10655],{"className":10640},[7121],[35,10642,10644],{"className":10643,"style":9035},[7125],[35,10645,10646,10649],{"style":9100},[35,10647],{"className":10648,"style":7134},[7133],[35,10650,10652],{"className":10651},[7138,7139,7140,7141],[35,10653,291],{"className":10654},[7104,7145,7141],[35,10656,7756],{"className":10657},[7755],[35,10659,10661],{"className":10660},[7121],[35,10662,10664],{"className":10663,"style":7763},[7125],[35,10665],{},[35,10667,7290],{"className":10668},[7314],[12,10670,10671],{},"This allows us to solve the eigenvalue problem in a non-orthogonal space (also called generalized eigenvalue problem)",[35,10673,10675],{"className":10674},[7318],[35,10676,10678,10729],{"className":10677},[7057],[35,10679,10681],{"className":10680},[7061],[7063,10682,10683],{"xmlns":7065,"display":7327},[7067,10684,10685,10726],{},[7070,10686,10687,10693,10697,10705,10707,10710,10712,10718,10720],{},[8800,10688,10689,10691],{"accent":7091},[7076,10690,7669],{},[7223,10692,8806],{},[10694,10695,10696],"mtext",{}," ",[8800,10698,10699,10702],{"accent":7091},[7076,10700,10701],{},"c",[7223,10703,10704],{},"⃗",[7223,10706,242],{},[7076,10708,10709],{},"E",[10694,10711,10696],{},[8800,10713,10714,10716],{"accent":7091},[7076,10715,10384],{},[7223,10717,8806],{},[10694,10719,10696],{},[8800,10721,10722,10724],{"accent":7091},[7076,10723,10701],{},[7223,10725,10704],{},[7083,10727,10728],{"encoding":7085},"\\tilde{H} \\ \\vec{c} = E \\ \\tilde{S} \\ 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can then obtain estimates of the eigenvalues and eigenstates of ",[35,10914,10916,10929],{"className":10915},[7057],[35,10917,10919],{"className":10918},[7061],[7063,10920,10921],{"xmlns":7065},[7067,10922,10923,10927],{},[7070,10924,10925],{},[7076,10926,7669],{},[7083,10928,7669],{"encoding":7085},[35,10930,10932],{"className":10931,"ariaHidden":7091},[7090],[35,10933,10935,10938],{"className":10934},[7095],[35,10936],{"className":10937,"style":7100},[7099],[35,10939,7669],{"className":10940,"style":7684},[7104,7145]," by looking at the ones of ",[35,10943,10945,10962],{"className":10944},[7057],[35,10946,10948],{"className":10947},[7061],[7063,10949,10950],{"xmlns":7065},[7067,10951,10952,10960],{},[7070,10953,10954],{},[8800,10955,10956,10958],{"accent":7091},[7076,10957,7669],{},[7223,10959,8806],{},[7083,10961,8809],{"encoding":7085},[35,10963,10965],{"className":10964,"ariaHidden":7091},[7090],[35,10966,10968,10971],{"className":10967},[7095],[35,10969],{"className":10970,"style":8819},[7099],[35,10972,10974],{"className":10973},[7104,8823],[35,10975,10977],{"className":10976},[7117],[35,10978,10980],{"className":10979},[7121],[35,10981,10983,10991],{"className":10982,"style":8819},[7125],[35,10984,10985,10988],{"style":8835},[35,10986],{"className":10987,"style":8839},[7133],[35,10989,7669],{"className":10990,"style":7684},[7104,7145],[35,10992,10993,10996],{"style":8845},[35,10994],{"className":10995,"style":8839},[7133],[35,10997,10999],{"className":10998,"style":8853},[8852],[35,11000,8806],{"className":11001},[7104],". For example, the estimate of the ground state energy is obtained by taking the smallest eigenvalue ",[35,11004,11006,11019],{"className":11005},[7057],[35,11007,11009],{"className":11008},[7061],[7063,11010,11011],{"xmlns":7065},[7067,11012,11013,11017],{},[7070,11014,11015],{},[7076,11016,10701],{},[7083,11018,10701],{"encoding":7085},[35,11020,11022],{"className":11021,"ariaHidden":7091},[7090],[35,11023,11025,11028],{"className":11024},[7095],[35,11026],{"className":11027,"style":7173},[7099],[35,11029,10701],{"className":11030},[7104,7145]," and the ground state from the corresponding eigenvector ",[35,11033,11035,11053],{"className":11034},[7057],[35,11036,11038],{"className":11037},[7061],[7063,11039,11040],{"xmlns":7065},[7067,11041,11042,11050],{},[7070,11043,11044],{},[8800,11045,11046,11048],{"accent":7091},[7076,11047,10701],{},[7223,11049,10704],{},[7083,11051,11052],{"encoding":7085},"\\vec{c}",[35,11054,11056],{"className":11055,"ariaHidden":7091},[7090],[35,11057,11059,11062],{"className":11058},[7095],[35,11060],{"className":11061,"style":10784},[7099],[35,11063,11065],{"className":11064},[7104,8823],[35,11066,11068],{"className":11067},[7117],[35,11069,11071],{"className":11070},[7121],[35,11072,11074,11082],{"className":11073,"style":10784},[7125],[35,11075,11076,11079],{"style":8835},[35,11077],{"className":11078,"style":8839},[7133],[35,11080,10701],{"className":11081},[7104,7145],[35,11083,11084,11087],{"style":8835},[35,11085],{"className":11086,"style":8839},[7133],[35,11088,11090],{"className":11089,"style":10801},[8852],[35,11091,11093],{"className":11092,"style":10806},[10805],[10808,11094,11095],{"xmlns":10810,"width":10811,"height":10812,"style":10813,"viewBox":10814,"preserveAspectRatio":10815},[10817,11096],{"d":10819},". The coefficients in ",[35,11099,11101,11118],{"className":11100},[7057],[35,11102,11104],{"className":11103},[7061],[7063,11105,11106],{"xmlns":7065},[7067,11107,11108,11116],{},[7070,11109,11110],{},[8800,11111,11112,11114],{"accent":7091},[7076,11113,10701],{},[7223,11115,10704],{},[7083,11117,11052],{"encoding":7085},[35,11119,11121],{"className":11120,"ariaHidden":7091},[7090],[35,11122,11124,11127],{"className":11123},[7095],[35,11125],{"className":11126,"style":10784},[7099],[35,11128,11130],{"className":11129},[7104,8823],[35,11131,11133],{"className":11132},[7117],[35,11134,11136],{"className":11135},[7121],[35,11137,11139,11147],{"className":11138,"style":10784},[7125],[35,11140,11141,11144],{"style":8835},[35,11142],{"className":11143,"style":8839},[7133],[35,11145,10701],{"className":11146},[7104,7145],[35,11148,11149,11152],{"style":8835},[35,11150],{"className":11151,"style":8839},[7133],[35,11153,11155],{"className":11154,"style":10801},[8852],[35,11156,11158],{"className":11157,"style":10806},[10805],[10808,11159,11160],{"xmlns":10810,"width":10811,"height":10812,"style":10813,"viewBox":10814,"preserveAspectRatio":10815},[10817,11161],{"d":10819}," determines the contribution of the different vectors that span ",[35,11164,11166,11183],{"className":11165},[7057],[35,11167,11169],{"className":11168},[7061],[7063,11170,11171],{"xmlns":7065},[7067,11172,11173,11181],{},[7070,11174,11175],{},[7699,11176,11177,11179],{},[7076,11178,7079],{"mathvariant":7078},[7076,11180,7810],{},[7083,11182,7924],{"encoding":7085},[35,11184,11186],{"className":11185,"ariaHidden":7091},[7090],[35,11187,11189,11192],{"className":11188},[7095],[35,11190],{"className":11191,"style":7718},[7099],[35,11193,11195,11198],{"className":11194},[7104],[35,11196,7079],{"className":11197,"style":7109},[7104,7108],[35,11199,11201],{"className":11200},[7113],[35,11202,11204,11224],{"className":11203},[7117,7731],[35,11205,11207,11221],{"className":11206},[7121],[35,11208,11210],{"className":11209,"style":7738},[7125],[35,11211,11212,11215],{"style":7741},[35,11213],{"className":11214,"style":7134},[7133],[35,11216,11218],{"className":11217},[7138,7139,7140,7141],[35,11219,7810],{"className":11220,"style":7845},[7104,7145,7141],[35,11222,7756],{"className":11223},[7755],[35,11225,11227],{"className":11226},[7121],[35,11228,11230],{"className":11229,"style":7763},[7125],[35,11231],{},[12,11233,11234],{},[11235,11236],"img",{"alt":11237,"src":11238},"fig1.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-06.avif",[12,11240,11241,11242,11363,11364,288,11444,11525,11526,288,11612,11696,11697,11757,11758,11840,11841,11920,11921,12003,12004,12032,12033,12184],{},"The Figure shows a circuit representation of the modified Hadamard test, a method that is used to compute the overlap between different quantum states. For each matrix element ",[35,11243,11245,11273],{"className":11244},[7057],[35,11246,11248],{"className":11247},[7061],[7063,11249,11250],{"xmlns":7065},[7067,11251,11252,11270],{},[7070,11253,11254],{},[7699,11255,11256,11262],{},[8800,11257,11258,11260],{"accent":7091},[7076,11259,7669],{},[7223,11261,8806],{},[7070,11263,11264,11266,11268],{},[7076,11265,7824],{},[7223,11267,2877],{"separator":7091},[7076,11269,6920],{},[7083,11271,11272],{"encoding":7085},"\\tilde{H}_{i,j}",[35,11274,11276],{"className":11275,"ariaHidden":7091},[7090],[35,11277,11279,11283],{"className":11278},[7095],[35,11280],{"className":11281,"style":11282},[7099],"height:1.2063em;vertical-align:-0.2861em;",[35,11284,11286,11317],{"className":11285},[7104],[35,11287,11289],{"className":11288},[7104,8823],[35,11290,11292],{"className":11291},[7117],[35,11293,11295],{"className":11294},[7121],[35,11296,11298,11306],{"className":11297,"style":8819},[7125],[35,11299,11300,11303],{"style":8835},[35,11301],{"className":11302,"style":8839},[7133],[35,11304,7669],{"className":11305,"style":7684},[7104,7145],[35,11307,11308,11311],{"style":8845},[35,11309],{"className":11310,"style":8839},[7133],[35,11312,11314],{"className":11313,"style":8853},[8852],[35,11315,8806],{"className":11316},[7104],[35,11318,11320],{"className":11319},[7113],[35,11321,11323,11354],{"className":11322},[7117,7731],[35,11324,11326,11351],{"className":11325},[7121],[35,11327,11330],{"className":11328,"style":11329},[7125],"height:0.3117em;",[35,11331,11332,11335],{"style":9038},[35,11333],{"className":11334,"style":7134},[7133],[35,11336,11338],{"className":11337},[7138,7139,7140,7141],[35,11339,11341,11344,11347],{"className":11340},[7104,7141],[35,11342,7824],{"className":11343},[7104,7145,7141],[35,11345,2877],{"className":11346},[7492,7141],[35,11348,6920],{"className":11349,"style":11350},[7104,7145,7141],"margin-right:0.0572em;",[35,11352,7756],{"className":11353},[7755],[35,11355,11357],{"className":11356},[7121],[35,11358,11361],{"className":11359,"style":11360},[7125],"height:0.2861em;",[35,11362],{},", a Hadamard test between the state ",[35,11365,11367,11389],{"className":11366},[7057],[35,11368,11370],{"className":11369},[7061],[7063,11371,11372],{"xmlns":7065},[7067,11373,11374,11386],{},[7070,11375,11376,11378,11384],{},[7076,11377,7283],{"mathvariant":7282},[7699,11379,11380,11382],{},[7076,11381,8322],{},[7076,11383,7824],{},[7223,11385,7290],{"stretchy":7289},[7083,11387,11388],{"encoding":7085},"\\vert \\psi_i \\rangle",[35,11390,11392],{"className":11391,"ariaHidden":7091},[7090],[35,11393,11395,11398,11401,11441],{"className":11394},[7095],[35,11396],{"className":11397,"style":7303},[7099],[35,11399,7283],{"className":11400},[7104],[35,11402,11404,11407],{"className":11403},[7104],[35,11405,8322],{"className":11406,"style":7310},[7104,7145],[35,11408,11410],{"className":11409},[7113],[35,11411,11413,11433],{"className":11412},[7117,7731],[35,11414,11416,11430],{"className":11415},[7121],[35,11417,11419],{"className":11418,"style":11329},[7125],[35,11420,11421,11424],{"style":9100},[35,11422],{"className":11423,"style":7134},[7133],[35,11425,11427],{"className":11426},[7138,7139,7140,7141],[35,11428,7824],{"className":11429},[7104,7145,7141],[35,11431,7756],{"className":11432},[7755],[35,11434,11436],{"className":11435},[7121],[35,11437,11439],{"className":11438,"style":7763},[7125],[35,11440],{},[35,11442,7290],{"className":11443},[7314],[35,11445,11447,11469],{"className":11446},[7057],[35,11448,11450],{"className":11449},[7061],[7063,11451,11452],{"xmlns":7065},[7067,11453,11454,11466],{},[7070,11455,11456,11458,11464],{},[7076,11457,7283],{"mathvariant":7282},[7699,11459,11460,11462],{},[7076,11461,8322],{},[7076,11463,6920],{},[7223,11465,7290],{"stretchy":7289},[7083,11467,11468],{"encoding":7085},"\\vert \\psi_j \\rangle",[35,11470,11472],{"className":11471,"ariaHidden":7091},[7090],[35,11473,11475,11479,11482,11522],{"className":11474},[7095],[35,11476],{"className":11477,"style":11478},[7099],"height:1.0361em;vertical-align:-0.2861em;",[35,11480,7283],{"className":11481},[7104],[35,11483,11485,11488],{"className":11484},[7104],[35,11486,8322],{"className":11487,"style":7310},[7104,7145],[35,11489,11491],{"className":11490},[7113],[35,11492,11494,11514],{"className":11493},[7117,7731],[35,11495,11497,11511],{"className":11496},[7121],[35,11498,11500],{"className":11499,"style":11329},[7125],[35,11501,11502,11505],{"style":9100},[35,11503],{"className":11504,"style":7134},[7133],[35,11506,11508],{"className":11507},[7138,7139,7140,7141],[35,11509,6920],{"className":11510,"style":11350},[7104,7145,7141],[35,11512,7756],{"className":11513},[7755],[35,11515,11517],{"className":11516},[7121],[35,11518,11520],{"className":11519,"style":11360},[7125],[35,11521],{},[35,11523,7290],{"className":11524},[7314]," is carried out. This is highlighted in the figure by the color scheme for the matrix elements and the corresponding ",[35,11527,11529,11553],{"className":11528},[7057],[35,11530,11532],{"className":11531},[7061],[7063,11533,11534],{"xmlns":7065},[7067,11535,11536,11550],{},[7070,11537,11538,11541,11544],{},[10694,11539,11540],{},"Prep",[10694,11542,11543],{},"  ",[7699,11545,11546,11548],{},[7076,11547,8322],{},[7076,11549,7824],{},[7083,11551,11552],{"encoding":7085},"\\text{Prep} \\; \\psi_i",[35,11554,11556],{"className":11555,"ariaHidden":7091},[7090],[35,11557,11559,11563,11569,11572],{"className":11558},[7095],[35,11560],{"className":11561,"style":11562},[7099],"height:0.8889em;vertical-align:-0.1944em;",[35,11564,11566],{"className":11565},[7104,4991],[35,11567,11540],{"className":11568},[7104],[35,11570],{"className":11571,"style":7450},[7248],[35,11573,11575,11578],{"className":11574},[7104],[35,11576,8322],{"className":11577,"style":7310},[7104,7145],[35,11579,11581],{"className":11580},[7113],[35,11582,11584,11604],{"className":11583},[7117,7731],[35,11585,11587,11601],{"className":11586},[7121],[35,11588,11590],{"className":11589,"style":11329},[7125],[35,11591,11592,11595],{"style":9100},[35,11593],{"className":11594,"style":7134},[7133],[35,11596,11598],{"className":11597},[7138,7139,7140,7141],[35,11599,7824],{"className":11600},[7104,7145,7141],[35,11602,7756],{"className":11603},[7755],[35,11605,11607],{"className":11606},[7121],[35,11608,11610],{"className":11609,"style":7763},[7125],[35,11611],{},[35,11613,11615,11637],{"className":11614},[7057],[35,11616,11618],{"className":11617},[7061],[7063,11619,11620],{"xmlns":7065},[7067,11621,11622,11634],{},[7070,11623,11624,11626,11628],{},[10694,11625,11540],{},[10694,11627,11543],{},[7699,11629,11630,11632],{},[7076,11631,8322],{},[7076,11633,6920],{},[7083,11635,11636],{"encoding":7085},"\\text{Prep} \\; \\psi_j",[35,11638,11640],{"className":11639,"ariaHidden":7091},[7090],[35,11641,11643,11647,11653,11656],{"className":11642},[7095],[35,11644],{"className":11645,"style":11646},[7099],"height:0.9805em;vertical-align:-0.2861em;",[35,11648,11650],{"className":11649},[7104,4991],[35,11651,11540],{"className":11652},[7104],[35,11654],{"className":11655,"style":7450},[7248],[35,11657,11659,11662],{"className":11658},[7104],[35,11660,8322],{"className":11661,"style":7310},[7104,7145],[35,11663,11665],{"className":11664},[7113],[35,11666,11668,11688],{"className":11667},[7117,7731],[35,11669,11671,11685],{"className":11670},[7121],[35,11672,11674],{"className":11673,"style":11329},[7125],[35,11675,11676,11679],{"style":9100},[35,11677],{"className":11678,"style":7134},[7133],[35,11680,11682],{"className":11681},[7138,7139,7140,7141],[35,11683,6920],{"className":11684,"style":11350},[7104,7145,7141],[35,11686,7756],{"className":11687},[7755],[35,11689,11691],{"className":11690},[7121],[35,11692,11694],{"className":11693,"style":11360},[7125],[35,11695],{}," operations. Thus, a set of Hadamard tests for all the possible combinations of Krylov space vectors is required to compute all the matrix elements of the projected Hamiltonian ",[35,11698,11700,11717],{"className":11699},[7057],[35,11701,11703],{"className":11702},[7061],[7063,11704,11705],{"xmlns":7065},[7067,11706,11707,11715],{},[7070,11708,11709],{},[8800,11710,11711,11713],{"accent":7091},[7076,11712,7669],{},[7223,11714,8806],{},[7083,11716,8809],{"encoding":7085},[35,11718,11720],{"className":11719,"ariaHidden":7091},[7090],[35,11721,11723,11726],{"className":11722},[7095],[35,11724],{"className":11725,"style":8819},[7099],[35,11727,11729],{"className":11728},[7104,8823],[35,11730,11732],{"className":11731},[7117],[35,11733,11735],{"className":11734},[7121],[35,11736,11738,11746],{"className":11737,"style":8819},[7125],[35,11739,11740,11743],{"style":8835},[35,11741],{"className":11742,"style":8839},[7133],[35,11744,7669],{"className":11745,"style":7684},[7104,7145],[35,11747,11748,11751],{"style":8845},[35,11749],{"className":11750,"style":8839},[7133],[35,11752,11754],{"className":11753,"style":8853},[8852],[35,11755,8806],{"className":11756},[7104],". The top wire in the Hadamard test circuit is an ancilla qubit which is measured either in the X or Y basis, its expectation value determines the value of the overlap between the states. The bottom wire represents all the qubits of the system Hamiltonian. The ",[35,11759,11761,11782],{"className":11760},[7057],[35,11762,11764],{"className":11763},[7061],[7063,11765,11766],{"xmlns":7065},[7067,11767,11768,11780],{},[7070,11769,11770,11772,11774],{},[10694,11771,11540],{},[10694,11773,11543],{},[7699,11775,11776,11778],{},[7076,11777,8322],{},[7076,11779,7824],{},[7083,11781,11552],{"encoding":7085},[35,11783,11785],{"className":11784,"ariaHidden":7091},[7090],[35,11786,11788,11791,11797,11800],{"className":11787},[7095],[35,11789],{"className":11790,"style":11562},[7099],[35,11792,11794],{"className":11793},[7104,4991],[35,11795,11540],{"className":11796},[7104],[35,11798],{"className":11799,"style":7450},[7248],[35,11801,11803,11806],{"className":11802},[7104],[35,11804,8322],{"className":11805,"style":7310},[7104,7145],[35,11807,11809],{"className":11808},[7113],[35,11810,11812,11832],{"className":11811},[7117,7731],[35,11813,11815,11829],{"className":11814},[7121],[35,11816,11818],{"className":11817,"style":11329},[7125],[35,11819,11820,11823],{"style":9100},[35,11821],{"className":11822,"style":7134},[7133],[35,11824,11826],{"className":11825},[7138,7139,7140,7141],[35,11827,7824],{"className":11828},[7104,7145,7141],[35,11830,7756],{"className":11831},[7755],[35,11833,11835],{"className":11834},[7121],[35,11836,11838],{"className":11837,"style":7763},[7125],[35,11839],{}," operation prepares the system qubit in the state ",[35,11842,11844,11865],{"className":11843},[7057],[35,11845,11847],{"className":11846},[7061],[7063,11848,11849],{"xmlns":7065},[7067,11850,11851,11863],{},[7070,11852,11853,11855,11861],{},[7076,11854,7283],{"mathvariant":7282},[7699,11856,11857,11859],{},[7076,11858,8322],{},[7076,11860,7824],{},[7223,11862,7290],{"stretchy":7289},[7083,11864,11388],{"encoding":7085},[35,11866,11868],{"className":11867,"ariaHidden":7091},[7090],[35,11869,11871,11874,11877,11917],{"className":11870},[7095],[35,11872],{"className":11873,"style":7303},[7099],[35,11875,7283],{"className":11876},[7104],[35,11878,11880,11883],{"className":11879},[7104],[35,11881,8322],{"className":11882,"style":7310},[7104,7145],[35,11884,11886],{"className":11885},[7113],[35,11887,11889,11909],{"className":11888},[7117,7731],[35,11890,11892,11906],{"className":11891},[7121],[35,11893,11895],{"className":11894,"style":11329},[7125],[35,11896,11897,11900],{"style":9100},[35,11898],{"className":11899,"style":7134},[7133],[35,11901,11903],{"className":11902},[7138,7139,7140,7141],[35,11904,7824],{"className":11905},[7104,7145,7141],[35,11907,7756],{"className":11908},[7755],[35,11910,11912],{"className":11911},[7121],[35,11913,11915],{"className":11914,"style":7763},[7125],[35,11916],{},[35,11918,7290],{"className":11919},[7314]," controlled by the state of the ancilla qubit (similarly for ",[35,11922,11924,11945],{"className":11923},[7057],[35,11925,11927],{"className":11926},[7061],[7063,11928,11929],{"xmlns":7065},[7067,11930,11931,11943],{},[7070,11932,11933,11935,11937],{},[10694,11934,11540],{},[10694,11936,11543],{},[7699,11938,11939,11941],{},[7076,11940,8322],{},[7076,11942,6920],{},[7083,11944,11636],{"encoding":7085},[35,11946,11948],{"className":11947,"ariaHidden":7091},[7090],[35,11949,11951,11954,11960,11963],{"className":11950},[7095],[35,11952],{"className":11953,"style":11646},[7099],[35,11955,11957],{"className":11956},[7104,4991],[35,11958,11540],{"className":11959},[7104],[35,11961],{"className":11962,"style":7450},[7248],[35,11964,11966,11969],{"className":11965},[7104],[35,11967,8322],{"className":11968,"style":7310},[7104,7145],[35,11970,11972],{"className":11971},[7113],[35,11973,11975,11995],{"className":11974},[7117,7731],[35,11976,11978,11992],{"className":11977},[7121],[35,11979,11981],{"className":11980,"style":11329},[7125],[35,11982,11983,11986],{"style":9100},[35,11984],{"className":11985,"style":7134},[7133],[35,11987,11989],{"className":11988},[7138,7139,7140,7141],[35,11990,6920],{"className":11991,"style":11350},[7104,7145,7141],[35,11993,7756],{"className":11994},[7755],[35,11996,11998],{"className":11997},[7121],[35,11999,12001],{"className":12000,"style":11360},[7125],[35,12002],{},") and the operation ",[35,12005,12007,12020],{"className":12006},[7057],[35,12008,12010],{"className":12009},[7061],[7063,12011,12012],{"xmlns":7065},[7067,12013,12014,12018],{},[7070,12015,12016],{},[7076,12017,7705],{},[7083,12019,7705],{"encoding":7085},[35,12021,12023],{"className":12022,"ariaHidden":7091},[7090],[35,12024,12026,12029],{"className":12025},[7095],[35,12027],{"className":12028,"style":7100},[7099],[35,12030,7705],{"className":12031,"style":7751},[7104,7145]," represents Pauli decomposition of the system Hamiltonian ",[35,12034,12036,12065],{"className":12035},[7057],[35,12037,12039],{"className":12038},[7061],[7063,12040,12041],{"xmlns":7065},[7067,12042,12043,12062],{},[7070,12044,12045,12047,12049,12056],{},[7076,12046,7669],{},[7223,12048,242],{},[7699,12050,12051,12054],{},[7223,12052,12053],{},"∑",[7076,12055,7824],{},[7699,12057,12058,12060],{},[7076,12059,7705],{},[7076,12061,7824],{},[7083,12063,12064],{"encoding":7085},"H = \\sum_i P_i",[35,12066,12068,12086],{"className":12067,"ariaHidden":7091},[7090],[35,12069,12071,12074,12077,12080,12083],{"className":12070},[7095],[35,12072],{"className":12073,"style":7100},[7099],[35,12075,7669],{"className":12076,"style":7684},[7104,7145],[35,12078],{"className":12079,"style":7450},[7248],[35,12081,242],{"className":12082},[7454],[35,12084],{"className":12085,"style":7450},[7248],[35,12087,12089,12093,12140,12143],{"className":12088},[7095],[35,12090],{"className":12091,"style":12092},[7099],"height:1.0497em;vertical-align:-0.2997em;",[35,12094,12097,12103],{"className":12095},[12096],"mop",[35,12098,12053],{"className":12099,"style":12102},[12096,12100,12101],"op-symbol","small-op","position:relative;top:0em;",[35,12104,12106],{"className":12105},[7113],[35,12107,12109,12131],{"className":12108},[7117,7731],[35,12110,12112,12128],{"className":12111},[7121],[35,12113,12116],{"className":12114,"style":12115},[7125],"height:0.162em;",[35,12117,12119,12122],{"style":12118},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[35,12120],{"className":12121,"style":7134},[7133],[35,12123,12125],{"className":12124},[7138,7139,7140,7141],[35,12126,7824],{"className":12127},[7104,7145,7141],[35,12129,7756],{"className":12130},[7755],[35,12132,12134],{"className":12133},[7121],[35,12135,12138],{"className":12136,"style":12137},[7125],"height:0.2997em;",[35,12139],{},[35,12141],{"className":12142,"style":7496},[7248],[35,12144,12146,12149],{"className":12145},[7104],[35,12147,7705],{"className":12148,"style":7751},[7104,7145],[35,12150,12152],{"className":12151},[7113],[35,12153,12155,12176],{"className":12154},[7117,7731],[35,12156,12158,12173],{"className":12157},[7121],[35,12159,12161],{"className":12160,"style":11329},[7125],[35,12162,12164,12167],{"style":12163},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[35,12165],{"className":12166,"style":7134},[7133],[35,12168,12170],{"className":12169},[7138,7139,7140,7141],[35,12171,7824],{"className":12172},[7104,7145,7141],[35,12174,7756],{"className":12175},[7755],[35,12177,12179],{"className":12178},[7121],[35,12180,12182],{"className":12181,"style":7763},[7125],[35,12183],{},". A more detailed derivation of the operations calculated by the Hadamard test is given below.",[12186,12187,12189],"h4",{"id":12188},"define-hamiltonian","Define Hamiltonian",[12,12191,12192,12193,12222,12223],{},"Let's consider the Heisenberg Hamiltonian for ",[35,12194,12196,12210],{"className":12195},[7057],[35,12197,12199],{"className":12198},[7061],[7063,12200,12201],{"xmlns":7065},[7067,12202,12203,12208],{},[7070,12204,12205],{},[7076,12206,12207],{},"N",[7083,12209,12207],{"encoding":7085},[35,12211,12213],{"className":12212,"ariaHidden":7091},[7090],[35,12214,12216,12219],{"className":12215},[7095],[35,12217],{"className":12218,"style":7100},[7099],[35,12220,12207],{"className":12221,"style":7845},[7104,7145]," qubits on a linear chain: ",[35,12224,12226,12302],{"className":12225},[7057],[35,12227,12229],{"className":12228},[7061],[7063,12230,12231],{"xmlns":7065},[7067,12232,12233,12299],{},[7070,12234,12235,12237,12239,12253,12260,12266,12268,12275,12281,12283,12286,12293],{},[7076,12236,7669],{},[7223,12238,242],{},[8304,12240,12241,12243,12251],{},[7223,12242,12053],{},[7070,12244,12245,12247,12249],{},[7076,12246,7824],{},[7223,12248,2877],{"separator":7091},[7076,12250,6920],{},[7076,12252,12207],{},[7699,12254,12255,12258],{},[7076,12256,12257],{},"X",[7076,12259,7824],{},[7699,12261,12262,12264],{},[7076,12263,12257],{},[7076,12265,6920],{},[7223,12267,445],{},[7699,12269,12270,12273],{},[7076,12271,12272],{},"Y",[7076,12274,7824],{},[7699,12276,12277,12279],{},[7076,12278,12272],{},[7076,12280,6920],{},[7223,12282,7225],{},[7076,12284,12285],{},"J",[7699,12287,12288,12291],{},[7076,12289,12290],{},"Z",[7076,12292,7824],{},[7699,12294,12295,12297],{},[7076,12296,12290],{},[7076,12298,6920],{},[7083,12300,12301],{"encoding":7085},"H= \\sum_{i,j}^N X_i X_j + Y_i Y_j - J Z_i Z_j",[35,12303,12305,12323,12487,12584],{"className":12304,"ariaHidden":7091},[7090],[35,12306,12308,12311,12314,12317,12320],{"className":12307},[7095],[35,12309],{"className":12310,"style":7100},[7099],[35,12312,7669],{"className":12313,"style":7684},[7104,7145],[35,12315],{"className":12316,"style":7450},[7248],[35,12318,242],{"className":12319},[7454],[35,12321],{"className":12322,"style":7450},[7248],[35,12324,12326,12330,12393,12396,12438,12478,12481,12484],{"className":12325},[7095],[35,12327],{"className":12328,"style":12329},[7099],"height:1.417em;vertical-align:-0.4358em;",[35,12331,12333,12336],{"className":12332},[12096],[35,12334,12053],{"className":12335,"style":12102},[12096,12100,12101],[35,12337,12339],{"className":12338},[7113],[35,12340,12342,12384],{"className":12341},[7117,7731],[35,12343,12345,12381],{"className":12344},[7121],[35,12346,12349,12369],{"className":12347,"style":12348},[7125],"height:0.9812em;",[35,12350,12351,12354],{"style":12118},[35,12352],{"className":12353,"style":7134},[7133],[35,12355,12357],{"className":12356},[7138,7139,7140,7141],[35,12358,12360,12363,12366],{"className":12359},[7104,7141],[35,12361,7824],{"className":12362},[7104,7145,7141],[35,12364,2877],{"className":12365},[7492,7141],[35,12367,6920],{"className":12368,"style":11350},[7104,7145,7141],[35,12370,12372,12375],{"style":12371},"top:-3.2029em;margin-right:0.05em;",[35,12373],{"className":12374,"style":7134},[7133],[35,12376,12378],{"className":12377},[7138,7139,7140,7141],[35,12379,12207],{"className":12380,"style":7845},[7104,7145,7141],[35,12382,7756],{"className":12383},[7755],[35,12385,12387],{"className":12386},[7121],[35,12388,12391],{"className":12389,"style":12390},[7125],"height:0.4358em;",[35,12392],{},[35,12394],{"className":12395,"style":7496},[7248],[35,12397,12399,12403],{"className":12398},[7104],[35,12400,12257],{"className":12401,"style":12402},[7104,7145],"margin-right:0.0785em;",[35,12404,12406],{"className":12405},[7113],[35,12407,12409,12430],{"className":12408},[7117,7731],[35,12410,12412,12427],{"className":12411},[7121],[35,12413,12415],{"className":12414,"style":11329},[7125],[35,12416,12418,12421],{"style":12417},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[35,12419],{"className":12420,"style":7134},[7133],[35,12422,12424],{"className":12423},[7138,7139,7140,7141],[35,12425,7824],{"className":12426},[7104,7145,7141],[35,12428,7756],{"className":12429},[7755],[35,12431,12433],{"className":12432},[7121],[35,12434,12436],{"className":12435,"style":7763},[7125],[35,12437],{},[35,12439,12441,12444],{"className":12440},[7104],[35,12442,12257],{"className":12443,"style":12402},[7104,7145],[35,12445,12447],{"className":12446},[7113],[35,12448,12450,12470],{"className":12449},[7117,7731],[35,12451,12453,12467],{"className":12452},[7121],[35,12454,12456],{"className":12455,"style":11329},[7125],[35,12457,12458,12461],{"style":12417},[35,12459],{"className":12460,"style":7134},[7133],[35,12462,12464],{"className":12463},[7138,7139,7140,7141],[35,12465,6920],{"className":12466,"style":11350},[7104,7145,7141],[35,12468,7756],{"className":12469},[7755],[35,12471,12473],{"className":12472},[7121],[35,12474,12476],{"className":12475,"style":11360},[7125],[35,12477],{},[35,12479],{"className":12480,"style":7249},[7248],[35,12482,445],{"className":12483},[7253],[35,12485],{"className":12486,"style":7249},[7248],[35,12488,12490,12494,12535,12575,12578,12581],{"className":12489},[7095],[35,12491],{"className":12492,"style":12493},[7099],"height:0.9694em;vertical-align:-0.2861em;",[35,12495,12497,12500],{"className":12496},[7104],[35,12498,12272],{"className":12499,"style":7249},[7104,7145],[35,12501,12503],{"className":12502},[7113],[35,12504,12506,12527],{"className":12505},[7117,7731],[35,12507,12509,12524],{"className":12508},[7121],[35,12510,12512],{"className":12511,"style":11329},[7125],[35,12513,12515,12518],{"style":12514},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[35,12516],{"className":12517,"style":7134},[7133],[35,12519,12521],{"className":12520},[7138,7139,7140,7141],[35,12522,7824],{"className":12523},[7104,7145,7141],[35,12525,7756],{"className":12526},[7755],[35,12528,12530],{"className":12529},[7121],[35,12531,12533],{"className":12532,"style":7763},[7125],[35,12534],{},[35,12536,12538,12541],{"className":12537},[7104],[35,12539,12272],{"className":12540,"style":7249},[7104,7145],[35,12542,12544],{"className":12543},[7113],[35,12545,12547,12567],{"className":12546},[7117,7731],[35,12548,12550,12564],{"className":12549},[7121],[35,12551,12553],{"className":12552,"style":11329},[7125],[35,12554,12555,12558],{"style":12514},[35,12556],{"className":12557,"style":7134},[7133],[35,12559,12561],{"className":12560},[7138,7139,7140,7141],[35,12562,6920],{"className":12563,"style":11350},[7104,7145,7141],[35,12565,7756],{"className":12566},[7755],[35,12568,12570],{"className":12569},[7121],[35,12571,12573],{"className":12572,"style":11360},[7125],[35,12574],{},[35,12576],{"className":12577,"style":7249},[7248],[35,12579,7225],{"className":12580},[7253],[35,12582],{"className":12583,"style":7249},[7248],[35,12585,12587,12590,12594,12636],{"className":12586},[7095],[35,12588],{"className":12589,"style":12493},[7099],[35,12591,12285],{"className":12592,"style":12593},[7104,7145],"margin-right:0.0962em;",[35,12595,12597,12601],{"className":12596},[7104],[35,12598,12290],{"className":12599,"style":12600},[7104,7145],"margin-right:0.0715em;",[35,12602,12604],{"className":12603},[7113],[35,12605,12607,12628],{"className":12606},[7117,7731],[35,12608,12610,12625],{"className":12609},[7121],[35,12611,12613],{"className":12612,"style":11329},[7125],[35,12614,12616,12619],{"style":12615},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[35,12617],{"className":12618,"style":7134},[7133],[35,12620,12622],{"className":12621},[7138,7139,7140,7141],[35,12623,7824],{"className":12624},[7104,7145,7141],[35,12626,7756],{"className":12627},[7755],[35,12629,12631],{"className":12630},[7121],[35,12632,12634],{"className":12633,"style":7763},[7125],[35,12635],{},[35,12637,12639,12642],{"className":12638},[7104],[35,12640,12290],{"className":12641,"style":12600},[7104,7145],[35,12643,12645],{"className":12644},[7113],[35,12646,12648,12668],{"className":12647},[7117,7731],[35,12649,12651,12665],{"className":12650},[7121],[35,12652,12654],{"className":12653,"style":11329},[7125],[35,12655,12656,12659],{"style":12615},[35,12657],{"className":12658,"style":7134},[7133],[35,12660,12662],{"className":12661},[7138,7139,7140,7141],[35,12663,6920],{"className":12664,"style":11350},[7104,7145,7141],[35,12666,7756],{"className":12667},[7755],[35,12669,12671],{"className":12670},[7121],[35,12672,12674],{"className":12673,"style":11360},[7125],[35,12675],{},[25,12677,12679],{"className":27,"code":12678,"language":29,"meta":30,"style":30},"# Define problem Hamiltonian.\nn_qubits = 30\nJ = 1  # coupling strength for ZZ interaction\n\n# Define the Hamiltonian:\nH_int = [[\"I\"] * n_qubits for _ in range(3 * (n_qubits - 1))]\nfor i in range(n_qubits - 1):\n    H_int[i][i] = \"Z\"\n    H_int[i][i + 1] = \"Z\"\nfor i in range(n_qubits - 1):\n    H_int[n_qubits - 1 + i][i] = \"X\"\n    H_int[n_qubits - 1 + i][i + 1] = \"X\"\nfor i in range(n_qubits - 1):\n    H_int[2 * (n_qubits - 1) + i][i] = \"Y\"\n    H_int[2 * (n_qubits - 1) + i][i + 1] = \"Y\"\nH_int = [\"\".join(term) for term in H_int]\nH_tot = [(term, J) if term.count(\"Z\") == 2 else (term, 1) for term in H_int]\n\n# Get operator\nH_op = SparsePauliOp.from_list(H_tot)\nprint(H_tot)\n",[32,12680,12681,12686,12696,12708,12712,12717,12764,12782,12792,12807,12825,12844,12867,12885,12911,12941,12970,13016,13020,13025,13041],{"__ignoreMap":30},[35,12682,12683],{"class":37,"line":38},[35,12684,12685],{"class":41},"# Define problem Hamiltonian.\n",[35,12687,12688,12691,12693],{"class":37,"line":45},[35,12689,12690],{"class":113},"n_qubits ",[35,12692,242],{"class":241},[35,12694,12695],{"class":245}," 30\n",[35,12697,12698,12701,12703,12705],{"class":37,"line":51},[35,12699,12700],{"class":113},"J ",[35,12702,242],{"class":241},[35,12704,6079],{"class":245},[35,12706,12707],{"class":41},"  # coupling strength for ZZ interaction\n",[35,12709,12710],{"class":37,"line":57},[35,12711,103],{"emptyLinePlaceholder":102},[35,12713,12714],{"class":37,"line":63},[35,12715,12716],{"class":41},"# Define the Hamiltonian:\n",[35,12718,12719,12722,12724,12727,12730,12732,12734,12737,12739,12742,12744,12747,12749,12752,12754,12757,12759,12761],{"class":37,"line":69},[35,12720,12721],{"class":113},"H_int ",[35,12723,242],{"class":241},[35,12725,12726],{"class":113}," [[",[35,12728,12729],{"class":234},"\"I\"",[35,12731,238],{"class":113},[35,12733,484],{"class":241},[35,12735,12736],{"class":113}," n_qubits ",[35,12738,1874],{"class":109},[35,12740,12741],{"class":113}," _ ",[35,12743,1663],{"class":109},[35,12745,12746],{"class":241}," range",[35,12748,275],{"class":113},[35,12750,12751],{"class":245},"3",[35,12753,463],{"class":241},[35,12755,12756],{"class":113}," (n_qubits ",[35,12758,564],{"class":241},[35,12760,6079],{"class":245},[35,12762,12763],{"class":113},"))]\n",[35,12765,12766,12768,12770,12772,12774,12776,12778,12780],{"class":37,"line":75},[35,12767,1874],{"class":109},[35,12769,2958],{"class":113},[35,12771,1663],{"class":109},[35,12773,12746],{"class":241},[35,12775,6676],{"class":113},[35,12777,564],{"class":241},[35,12779,6079],{"class":245},[35,12781,6479],{"class":113},[35,12783,12784,12787,12789],{"class":37,"line":81},[35,12785,12786],{"class":113},"    H_int[i][i] ",[35,12788,242],{"class":241},[35,12790,12791],{"class":234}," \"Z\"\n",[35,12793,12794,12797,12799,12801,12803,12805],{"class":37,"line":87},[35,12795,12796],{"class":113},"    H_int[i][i ",[35,12798,445],{"class":241},[35,12800,6079],{"class":245},[35,12802,238],{"class":113},[35,12804,242],{"class":241},[35,12806,12791],{"class":234},[35,12808,12809,12811,12813,12815,12817,12819,12821,12823],{"class":37,"line":93},[35,12810,1874],{"class":109},[35,12812,2958],{"class":113},[35,12814,1663],{"class":109},[35,12816,12746],{"class":241},[35,12818,6676],{"class":113},[35,12820,564],{"class":241},[35,12822,6079],{"class":245},[35,12824,6479],{"class":113},[35,12826,12827,12830,12832,12834,12836,12839,12841],{"class":37,"line":99},[35,12828,12829],{"class":113},"    H_int[n_qubits ",[35,12831,564],{"class":241},[35,12833,6079],{"class":245},[35,12835,415],{"class":241},[35,12837,12838],{"class":113}," i][i] ",[35,12840,242],{"class":241},[35,12842,12843],{"class":234}," \"X\"\n",[35,12845,12846,12848,12850,12852,12854,12857,12859,12861,12863,12865],{"class":37,"line":106},[35,12847,12829],{"class":113},[35,12849,564],{"class":241},[35,12851,6079],{"class":245},[35,12853,415],{"class":241},[35,12855,12856],{"class":113}," i][i ",[35,12858,445],{"class":241},[35,12860,6079],{"class":245},[35,12862,238],{"class":113},[35,12864,242],{"class":241},[35,12866,12843],{"class":234},[35,12868,12869,12871,12873,12875,12877,12879,12881,12883],{"class":37,"line":117},[35,12870,1874],{"class":109},[35,12872,2958],{"class":113},[35,12874,1663],{"class":109},[35,12876,12746],{"class":241},[35,12878,6676],{"class":113},[35,12880,564],{"class":241},[35,12882,6079],{"class":245},[35,12884,6479],{"class":113},[35,12886,12887,12890,12892,12894,12896,12898,12900,12902,12904,12906,12908],{"class":37,"line":125},[35,12888,12889],{"class":113},"    H_int[",[35,12891,460],{"class":245},[35,12893,463],{"class":241},[35,12895,12756],{"class":113},[35,12897,564],{"class":241},[35,12899,6079],{"class":245},[35,12901,427],{"class":113},[35,12903,445],{"class":241},[35,12905,12838],{"class":113},[35,12907,242],{"class":241},[35,12909,12910],{"class":234}," \"Y\"\n",[35,12912,12913,12915,12917,12919,12921,12923,12925,12927,12929,12931,12933,12935,12937,12939],{"class":37,"line":139},[35,12914,12889],{"class":113},[35,12916,460],{"class":245},[35,12918,463],{"class":241},[35,12920,12756],{"class":113},[35,12922,564],{"class":241},[35,12924,6079],{"class":245},[35,12926,427],{"class":113},[35,12928,445],{"class":241},[35,12930,12856],{"class":113},[35,12932,445],{"class":241},[35,12934,6079],{"class":245},[35,12936,238],{"class":113},[35,12938,242],{"class":241},[35,12940,12910],{"class":234},[35,12942,12943,12945,12947,12949,12952,12954,12957,12960,12962,12965,12967],{"class":37,"line":152},[35,12944,12721],{"class":113},[35,12946,242],{"class":241},[35,12948,2871],{"class":113},[35,12950,12951],{"class":234},"\"\"",[35,12953,5122],{"class":113},[35,12955,12956],{"class":271},"join",[35,12958,12959],{"class":113},"(term) ",[35,12961,1874],{"class":109},[35,12963,12964],{"class":113}," term ",[35,12966,1663],{"class":109},[35,12968,12969],{"class":113}," H_int]\n",[35,12971,12972,12975,12977,12980,12982,12985,12988,12990,12993,12995,12997,12999,13001,13004,13006,13008,13010,13012,13014],{"class":37,"line":157},[35,12973,12974],{"class":113},"H_tot ",[35,12976,242],{"class":241},[35,12978,12979],{"class":113}," [(term, J) ",[35,12981,1492],{"class":109},[35,12983,12984],{"class":113}," term.",[35,12986,12987],{"class":271},"count",[35,12989,275],{"class":113},[35,12991,12992],{"class":234},"\"Z\"",[35,12994,427],{"class":113},[35,12996,358],{"class":241},[35,12998,424],{"class":245},[35,13000,1502],{"class":109},[35,13002,13003],{"class":113}," (term, ",[35,13005,500],{"class":245},[35,13007,427],{"class":113},[35,13009,1874],{"class":109},[35,13011,12964],{"class":113},[35,13013,1663],{"class":109},[35,13015,12969],{"class":113},[35,13017,13018],{"class":37,"line":171},[35,13019,103],{"emptyLinePlaceholder":102},[35,13021,13022],{"class":37,"line":184},[35,13023,13024],{"class":41},"# Get operator\n",[35,13026,13027,13030,13032,13035,13038],{"class":37,"line":197},[35,13028,13029],{"class":113},"H_op ",[35,13031,242],{"class":241},[35,13033,13034],{"class":113}," SparsePauliOp.",[35,13036,13037],{"class":271},"from_list",[35,13039,13040],{"class":113},"(H_tot)\n",[35,13042,13043,13045],{"class":37,"line":210},[35,13044,1861],{"class":241},[35,13046,13040],{"class":113},[25,13048,13051],{"className":13049,"code":13050,"language":4991,"meta":30},[4989],"[('ZZIIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IZZIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIZZIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIZZIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIZZIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIZZIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIZZIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIZZIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIZZIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIZZIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIZZIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIZZIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIZZIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIZZIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIZZIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIZZIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIZZIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIZZIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIZZIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIZZIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIZZIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIZZIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIZZIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIZZIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIZZIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIZZIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIZZII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIZZI', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIIZZ', 1), ('XXIIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IXXIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIXXIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIXXIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIXXIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIXXIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIXXIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIXXIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIXXIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIXXIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIXXIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIXXIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIXXIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIXXIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIXXIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIXXIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIXXIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIXXIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIXXIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIXXIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIXXIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIXXIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIXXIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIXXIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIXXIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIXXIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIXXII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIXXI', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIIXX', 1), ('YYIIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IYYIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIYYIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIYYIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIYYIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIYYIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIYYIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIYYIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIYYIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIYYIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIYYIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIYYIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIYYIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIYYIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIYYIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIYYIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIYYIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIYYIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIYYIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIYYIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIYYIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIYYIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIYYIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIYYIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIYYIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIYYIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIYYII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIYYI', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIIYY', 1)]\n",[32,13052,13050],{"__ignoreMap":30},[12186,13054,13056],{"id":13055},"set-parameters-for-the-algorithm","Set parameters for the algorithm",[12,13058,13059,13060,13062,13063,13066,13067,13120,13121,13154],{},"We heuristically choose a value for the time-step ",[32,13061,9991],{}," (based on upper bounds on the Hamiltonian norm). Ref ",[19,13064,13065],{"href":5756},"[2]","  showed that a sufficiently small timestep is ",[35,13068,13070,13097],{"className":13069},[7057],[35,13071,13073],{"className":13072},[7061],[7063,13074,13075],{"xmlns":7065},[7067,13076,13077,13094],{},[7070,13078,13079,13082,13084,13086,13088,13090,13092],{},[7076,13080,13081],{},"π",[7076,13083,398],{"mathvariant":7282},[7076,13085,7283],{"mathvariant":7282},[7076,13087,7283],{"mathvariant":7282},[7076,13089,7669],{},[7076,13091,7283],{"mathvariant":7282},[7076,13093,7283],{"mathvariant":7282},[7083,13095,13096],{"encoding":7085},"\\pi\u002F\\vert \\vert H \\vert \\vert",[35,13098,13100],{"className":13099,"ariaHidden":7091},[7090],[35,13101,13103,13106,13109,13113,13116],{"className":13102},[7095],[35,13104],{"className":13105,"style":7303},[7099],[35,13107,13081],{"className":13108,"style":7310},[7104,7145],[35,13110,13112],{"className":13111},[7104],"\u002F∣∣",[35,13114,7669],{"className":13115,"style":7684},[7104,7145],[35,13117,13119],{"className":13118},[7104],"∣∣",", and that it is preferable up to a point to underestimate this value rather than overestimate, since overestimating can allow contributions from high-energy states to corrupt even the optimal state in the Krylov space. On the other hand, choosing ",[35,13122,13124,13139],{"className":13123},[7057],[35,13125,13127],{"className":13126},[7061],[7063,13128,13129],{"xmlns":7065},[7067,13130,13131,13137],{},[7070,13132,13133,13135],{},[7076,13134,9487],{},[7076,13136,7829],{},[7083,13138,9991],{"encoding":7085},[35,13140,13142],{"className":13141,"ariaHidden":7091},[7090],[35,13143,13145,13148,13151],{"className":13144},[7095],[35,13146],{"className":13147,"style":9962},[7099],[35,13149,9487],{"className":13150},[7104,7145],[35,13152,7829],{"className":13153},[7104,7145]," to be too small leads to worse conditioning of the Krylov subspace, since the Krylov basis vectors differ less from timestep to timestep.",[25,13156,13158],{"className":27,"code":13157,"language":29,"meta":30,"style":30},"# Get Hamiltonian restricted to single-particle states\nsingle_particle_H = np.zeros((n_qubits, n_qubits))\nfor i in range(n_qubits):\n    for j in range(i + 1):\n        for p, coeff in H_op.to_list():\n            p_x = Pauli(p).x\n            p_z = Pauli(p).z\n            if all(\n                p_x[k] == ((i == k) + (j == k)) % 2 for k in range(n_qubits)\n            ):\n                sgn = (\n                    (-1j) ** sum(p_z[k] and p_x[k] for k in range(n_qubits))\n                ) * ((-1) ** p_z[i])\n            else:\n                sgn = 0\n            single_particle_H[i, j] += sgn * coeff\nfor i in range(n_qubits):\n    for j in range(i + 1, n_qubits):\n        single_particle_H[i, j] = np.conj(single_particle_H[j, i])\n\n# Set dt according to spectral norm\ndt = np.pi \u002F np.linalg.norm(single_particle_H, ord=2)\ndt\n",[32,13159,13160,13165,13179,13192,13212,13226,13239,13251,13260,13302,13307,13316,13354,13374,13381,13389,13402,13414,13433,13447,13451,13456,13486],{"__ignoreMap":30},[35,13161,13162],{"class":37,"line":38},[35,13163,13164],{"class":41},"# Get Hamiltonian restricted to single-particle states\n",[35,13166,13167,13170,13172,13174,13176],{"class":37,"line":45},[35,13168,13169],{"class":113},"single_particle_H ",[35,13171,242],{"class":241},[35,13173,2412],{"class":113},[35,13175,6717],{"class":271},[35,13177,13178],{"class":113},"((n_qubits, n_qubits))\n",[35,13180,13181,13183,13185,13187,13189],{"class":37,"line":51},[35,13182,1874],{"class":109},[35,13184,2958],{"class":113},[35,13186,1663],{"class":109},[35,13188,12746],{"class":241},[35,13190,13191],{"class":113},"(n_qubits):\n",[35,13193,13194,13196,13199,13201,13203,13206,13208,13210],{"class":37,"line":57},[35,13195,1657],{"class":109},[35,13197,13198],{"class":113}," j ",[35,13200,1663],{"class":109},[35,13202,12746],{"class":241},[35,13204,13205],{"class":113},"(i ",[35,13207,445],{"class":241},[35,13209,6079],{"class":245},[35,13211,6479],{"class":113},[35,13213,13214,13216,13218,13220,13222,13224],{"class":37,"line":63},[35,13215,2087],{"class":109},[35,13217,6508],{"class":113},[35,13219,1663],{"class":109},[35,13221,6513],{"class":113},[35,13223,6516],{"class":271},[35,13225,2148],{"class":113},[35,13227,13228,13231,13233,13236],{"class":37,"line":69},[35,13229,13230],{"class":113},"            p_x ",[35,13232,242],{"class":241},[35,13234,13235],{"class":271}," Pauli",[35,13237,13238],{"class":113},"(p).x\n",[35,13240,13241,13244,13246,13248],{"class":37,"line":75},[35,13242,13243],{"class":113},"            p_z ",[35,13245,242],{"class":241},[35,13247,13235],{"class":271},[35,13249,13250],{"class":113},"(p).z\n",[35,13252,13253,13255,13258],{"class":37,"line":81},[35,13254,6851],{"class":109},[35,13256,13257],{"class":241}," all",[35,13259,903],{"class":113},[35,13261,13262,13265,13267,13270,13272,13275,13277,13280,13282,13285,13288,13290,13293,13295,13297,13299],{"class":37,"line":87},[35,13263,13264],{"class":113},"                p_x[k] ",[35,13266,358],{"class":241},[35,13268,13269],{"class":113}," ((i ",[35,13271,358],{"class":241},[35,13273,13274],{"class":113}," k) ",[35,13276,445],{"class":241},[35,13278,13279],{"class":113}," (j ",[35,13281,358],{"class":241},[35,13283,13284],{"class":113}," k)) ",[35,13286,13287],{"class":241},"%",[35,13289,424],{"class":245},[35,13291,13292],{"class":109}," for",[35,13294,395],{"class":113},[35,13296,1663],{"class":109},[35,13298,12746],{"class":241},[35,13300,13301],{"class":113},"(n_qubits)\n",[35,13303,13304],{"class":37,"line":93},[35,13305,13306],{"class":113},"            ):\n",[35,13308,13309,13312,13314],{"class":37,"line":99},[35,13310,13311],{"class":113},"                sgn ",[35,13313,242],{"class":241},[35,13315,6003],{"class":113},[35,13317,13318,13321,13323,13325,13327,13329,13331,13334,13337,13340,13343,13345,13347,13349,13351],{"class":37,"line":106},[35,13319,13320],{"class":113},"                    (",[35,13322,564],{"class":241},[35,13324,500],{"class":245},[35,13326,6920],{"class":109},[35,13328,427],{"class":113},[35,13330,421],{"class":241},[35,13332,13333],{"class":241}," sum",[35,13335,13336],{"class":113},"(p_z[k] ",[35,13338,13339],{"class":109},"and",[35,13341,13342],{"class":113}," p_x[k] ",[35,13344,1874],{"class":109},[35,13346,395],{"class":113},[35,13348,1663],{"class":109},[35,13350,12746],{"class":241},[35,13352,13353],{"class":113},"(n_qubits))\n",[35,13355,13356,13359,13361,13363,13365,13367,13369,13371],{"class":37,"line":117},[35,13357,13358],{"class":113},"                ) ",[35,13360,484],{"class":241},[35,13362,6913],{"class":113},[35,13364,564],{"class":241},[35,13366,500],{"class":245},[35,13368,427],{"class":113},[35,13370,421],{"class":241},[35,13372,13373],{"class":113}," p_z[i])\n",[35,13375,13376,13379],{"class":37,"line":125},[35,13377,13378],{"class":109},"            else",[35,13380,364],{"class":113},[35,13382,13383,13385,13387],{"class":37,"line":139},[35,13384,13311],{"class":113},[35,13386,242],{"class":241},[35,13388,6805],{"class":245},[35,13390,13391,13394,13396,13398,13400],{"class":37,"line":152},[35,13392,13393],{"class":113},"            single_particle_H[i, j] ",[35,13395,6840],{"class":241},[35,13397,6995],{"class":113},[35,13399,484],{"class":241},[35,13401,7000],{"class":113},[35,13403,13404,13406,13408,13410,13412],{"class":37,"line":157},[35,13405,1874],{"class":109},[35,13407,2958],{"class":113},[35,13409,1663],{"class":109},[35,13411,12746],{"class":241},[35,13413,13191],{"class":113},[35,13415,13416,13418,13420,13422,13424,13426,13428,13430],{"class":37,"line":171},[35,13417,1657],{"class":109},[35,13419,13198],{"class":113},[35,13421,1663],{"class":109},[35,13423,12746],{"class":241},[35,13425,13205],{"class":113},[35,13427,445],{"class":241},[35,13429,6079],{"class":245},[35,13431,13432],{"class":113},", n_qubits):\n",[35,13434,13435,13438,13440,13442,13444],{"class":37,"line":184},[35,13436,13437],{"class":113},"        single_particle_H[i, j] ",[35,13439,242],{"class":241},[35,13441,2412],{"class":113},[35,13443,6293],{"class":271},[35,13445,13446],{"class":113},"(single_particle_H[j, i])\n",[35,13448,13449],{"class":37,"line":197},[35,13450,103],{"emptyLinePlaceholder":102},[35,13452,13453],{"class":37,"line":210},[35,13454,13455],{"class":41},"# Set dt according to spectral norm\n",[35,13457,13458,13461,13463,13466,13468,13471,13474,13477,13480,13482,13484],{"class":37,"line":223},[35,13459,13460],{"class":113},"dt ",[35,13462,242],{"class":241},[35,13464,13465],{"class":113}," np.pi ",[35,13467,398],{"class":241},[35,13469,13470],{"class":113}," np.linalg.",[35,13472,13473],{"class":271},"norm",[35,13475,13476],{"class":113},"(single_particle_H, ",[35,13478,13479],{"class":822},"ord",[35,13481,242],{"class":241},[35,13483,460],{"class":245},[35,13485,384],{"class":113},[35,13487,13488],{"class":37,"line":228},[35,13489,13490],{"class":113},"dt\n",[25,13492,13495],{"className":13493,"code":13494,"language":4991,"meta":30},[4989],"np.float64(0.10833078115826875)\n",[32,13496,13494],{"__ignoreMap":30},[12,13498,13499],{},"And set other parameters of the algorithm. For the sake of this tutorial, we'll limit ourselves to using a Krylov space with only five dimensions, which is quite limiting.",[25,13501,13503],{"className":27,"code":13502,"language":29,"meta":30,"style":30},"# Set parameters for quantum Krylov algorithm\nkrylov_dim = 5  # size of Krylov subspace\nnum_trotter_steps = 6\ndt_circ = dt \u002F num_trotter_steps\n",[32,13504,13505,13510,13523,13533],{"__ignoreMap":30},[35,13506,13507],{"class":37,"line":38},[35,13508,13509],{"class":41},"# Set parameters for quantum Krylov algorithm\n",[35,13511,13512,13515,13517,13520],{"class":37,"line":45},[35,13513,13514],{"class":113},"krylov_dim ",[35,13516,242],{"class":241},[35,13518,13519],{"class":245}," 5",[35,13521,13522],{"class":41},"  # size of Krylov subspace\n",[35,13524,13525,13528,13530],{"class":37,"line":51},[35,13526,13527],{"class":113},"num_trotter_steps ",[35,13529,242],{"class":241},[35,13531,13532],{"class":245}," 6\n",[35,13534,13535,13538,13540,13543,13545],{"class":37,"line":57},[35,13536,13537],{"class":113},"dt_circ ",[35,13539,242],{"class":241},[35,13541,13542],{"class":113}," dt ",[35,13544,398],{"class":241},[35,13546,13547],{"class":113}," num_trotter_steps\n",[12186,13549,13551],{"id":13550},"state-preparation","State preparation",[12,13553,13554,13555,13594,13595,13633],{},"Pick a reference state ",[35,13556,13558,13576],{"className":13557},[7057],[35,13559,13561],{"className":13560},[7061],[7063,13562,13563],{"xmlns":7065},[7067,13564,13565,13573],{},[7070,13566,13567,13569,13571],{},[7076,13568,7283],{"mathvariant":7282},[7076,13570,8322],{},[7223,13572,7290],{"stretchy":7289},[7083,13574,13575],{"encoding":7085},"\\vert \\psi \\rangle",[35,13577,13579],{"className":13578,"ariaHidden":7091},[7090],[35,13580,13582,13585,13588,13591],{"className":13581},[7095],[35,13583],{"className":13584,"style":7303},[7099],[35,13586,7283],{"className":13587},[7104],[35,13589,8322],{"className":13590,"style":7310},[7104,7145],[35,13592,7290],{"className":13593},[7314]," that has some overlap with the ground state. For this Hamiltonian, We use the a state with an excitation in the middle qubit ",[35,13596,13598,13617],{"className":13597},[7057],[35,13599,13601],{"className":13600},[7061],[7063,13602,13603],{"xmlns":7065},[7067,13604,13605,13614],{},[7070,13606,13607,13609,13612],{},[7076,13608,7283],{"mathvariant":7282},[7227,13610,13611],{},"00..010...00",[7223,13613,7290],{"stretchy":7289},[7083,13615,13616],{"encoding":7085},"\\vert 00..010...00 \\rangle",[35,13618,13620],{"className":13619,"ariaHidden":7091},[7090],[35,13621,13623,13626,13630],{"className":13622},[7095],[35,13624],{"className":13625,"style":7303},[7099],[35,13627,13629],{"className":13628},[7104],"∣00..010...00",[35,13631,7290],{"className":13632},[7314]," as our reference state.",[25,13635,13637],{"className":27,"code":13636,"language":29,"meta":30,"style":30},"qc_state_prep = QuantumCircuit(n_qubits)\nqc_state_prep.x(int(n_qubits \u002F 2) + 1)\nqc_state_prep.draw(\"mpl\", scale=0.5)\n",[32,13638,13639,13650,13676],{"__ignoreMap":30},[35,13640,13641,13644,13646,13648],{"class":37,"line":38},[35,13642,13643],{"class":113},"qc_state_prep ",[35,13645,242],{"class":241},[35,13647,5293],{"class":271},[35,13649,13301],{"class":113},[35,13651,13652,13655,13658,13660,13662,13664,13666,13668,13670,13672,13674],{"class":37,"line":45},[35,13653,13654],{"class":113},"qc_state_prep.",[35,13656,13657],{"class":271},"x",[35,13659,275],{"class":113},[35,13661,285],{"class":241},[35,13663,6676],{"class":113},[35,13665,398],{"class":241},[35,13667,424],{"class":245},[35,13669,427],{"class":113},[35,13671,445],{"class":241},[35,13673,6079],{"class":245},[35,13675,384],{"class":113},[35,13677,13678,13680,13683,13685,13688,13690,13693,13695,13698],{"class":37,"line":51},[35,13679,13654],{"class":113},[35,13681,13682],{"class":271},"draw",[35,13684,275],{"class":113},[35,13686,13687],{"class":234},"\"mpl\"",[35,13689,288],{"class":113},[35,13691,13692],{"class":822},"scale",[35,13694,242],{"class":241},[35,13696,13697],{"class":245},"0.5",[35,13699,384],{"class":113},[4995,13701],{"alt":13702,"src":13703},"Output of the previous code cell","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-01.avif",[12186,13705,13707],{"id":13706},"time-evolution","Time evolution",[12,13709,13710,13711,13811,13812,5122],{},"We can realize the time-evolution operator generated by a given Hamiltonian: ",[35,13712,13714,13743],{"className":13713},[7057],[35,13715,13717],{"className":13716},[7061],[7063,13718,13719],{"xmlns":7065},[7067,13720,13721,13741],{},[7070,13722,13723,13725,13727],{},[7076,13724,7810],{},[7223,13726,242],{},[7073,13728,13729,13731],{},[7076,13730,7817],{},[7070,13732,13733,13735,13737,13739],{},[7223,13734,7225],{},[7076,13736,7824],{},[7076,13738,7669],{},[7076,13740,7829],{},[7083,13742,7832],{"encoding":7085},[35,13744,13746,13764],{"className":13745,"ariaHidden":7091},[7090],[35,13747,13749,13752,13755,13758,13761],{"className":13748},[7095],[35,13750],{"className":13751,"style":7100},[7099],[35,13753,7810],{"className":13754,"style":7845},[7104,7145],[35,13756],{"className":13757,"style":7450},[7248],[35,13759,242],{"className":13760},[7454],[35,13762],{"className":13763,"style":7450},[7248],[35,13765,13767,13770],{"className":13766},[7095],[35,13768],{"className":13769,"style":7861},[7099],[35,13771,13773,13776],{"className":13772},[7104],[35,13774,7817],{"className":13775},[7104,7145],[35,13777,13779],{"className":13778},[7113],[35,13780,13782],{"className":13781},[7117],[35,13783,13785],{"className":13784},[7121],[35,13786,13788],{"className":13787,"style":7861},[7125],[35,13789,13790,13793],{"style":7129},[35,13791],{"className":13792,"style":7134},[7133],[35,13794,13796],{"className":13795},[7138,7139,7140,7141],[35,13797,13799,13802,13805,13808],{"className":13798},[7104,7141],[35,13800,7225],{"className":13801},[7104,7141],[35,13803,7824],{"className":13804},[7104,7145,7141],[35,13806,7669],{"className":13807,"style":7684},[7104,7145,7141],[35,13809,7829],{"className":13810},[7104,7145,7141]," via the ",[19,13813,13815],{"href":13814},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fqiskit.synthesis.LieTrotter","Lie-Trotter approximation",[25,13817,13819],{"className":27,"code":13818,"language":29,"meta":30,"style":30},"t = Parameter(\"t\")\n\n## Create the time-evo op circuit\nevol_gate = PauliEvolutionGate(\n    H_op, time=t, synthesis=LieTrotter(reps=num_trotter_steps)\n)\n\nqr = QuantumRegister(n_qubits)\nqc_evol = QuantumCircuit(qr)\nqc_evol.append(evol_gate, qargs=qr)\n",[32,13820,13821,13838,13842,13847,13859,13890,13894,13898,13910,13922],{"__ignoreMap":30},[35,13822,13823,13826,13828,13831,13833,13836],{"class":37,"line":38},[35,13824,13825],{"class":113},"t ",[35,13827,242],{"class":241},[35,13829,13830],{"class":271}," Parameter",[35,13832,275],{"class":113},[35,13834,13835],{"class":234},"\"t\"",[35,13837,384],{"class":113},[35,13839,13840],{"class":37,"line":45},[35,13841,103],{"emptyLinePlaceholder":102},[35,13843,13844],{"class":37,"line":51},[35,13845,13846],{"class":41},"## Create the time-evo op circuit\n",[35,13848,13849,13852,13854,13857],{"class":37,"line":57},[35,13850,13851],{"class":113},"evol_gate ",[35,13853,242],{"class":241},[35,13855,13856],{"class":271}," PauliEvolutionGate",[35,13858,903],{"class":113},[35,13860,13861,13864,13867,13869,13872,13875,13877,13880,13882,13885,13887],{"class":37,"line":63},[35,13862,13863],{"class":113},"    H_op, ",[35,13865,13866],{"class":822},"time",[35,13868,242],{"class":241},[35,13870,13871],{"class":113},"t, ",[35,13873,13874],{"class":822},"synthesis",[35,13876,242],{"class":241},[35,13878,13879],{"class":271},"LieTrotter",[35,13881,275],{"class":113},[35,13883,13884],{"class":822},"reps",[35,13886,242],{"class":241},[35,13888,13889],{"class":113},"num_trotter_steps)\n",[35,13891,13892],{"class":37,"line":69},[35,13893,384],{"class":113},[35,13895,13896],{"class":37,"line":75},[35,13897,103],{"emptyLinePlaceholder":102},[35,13899,13900,13903,13905,13908],{"class":37,"line":81},[35,13901,13902],{"class":113},"qr ",[35,13904,242],{"class":241},[35,13906,13907],{"class":271}," QuantumRegister",[35,13909,13301],{"class":113},[35,13911,13912,13915,13917,13919],{"class":37,"line":87},[35,13913,13914],{"class":113},"qc_evol ",[35,13916,242],{"class":241},[35,13918,5293],{"class":271},[35,13920,13921],{"class":113},"(qr)\n",[35,13923,13924,13927,13929,13932,13935,13937],{"class":37,"line":93},[35,13925,13926],{"class":113},"qc_evol.",[35,13928,1927],{"class":271},[35,13930,13931],{"class":113},"(evol_gate, ",[35,13933,13934],{"class":822},"qargs",[35,13936,242],{"class":241},[35,13938,13939],{"class":113},"qr)\n",[25,13941,13944],{"className":13942,"code":13943,"language":4991,"meta":30},[4989],"\u003Cqiskit.circuit.instructionset.InstructionSet at 0x11eef9be0>\n",[32,13945,13943],{"__ignoreMap":30},[12186,13947,13949],{"id":13948},"hadamard-test","Hadamard 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   |0\\rangle|0\\rangle^N \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle + |1\\rangle \\Big)|0\\rangle^N \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle|0\\rangle^N+|1\\rangle |\\psi_i\\rangle\\Big) \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle |0\\rangle^N+|1\\rangle P |\\psi_i\\rangle\\Big) \\quad\\longrightarrow\\quad\\frac{1}{\\sqrt{2}}\\Big(|0\\rangle |\\psi_j\\rangle+|1\\rangle 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are controlled operations that prepare 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vectors of the unitary Krylov space, with ",[35,15644,15646,15730],{"className":15645},[7057],[35,15647,15649],{"className":15648},[7061],[7063,15650,15651],{"xmlns":7065},[7067,15652,15653,15727],{},[7070,15654,15655,15657,15663,15665,15667,15685,15687,15689,15691,15693,15711,15717,15719,15721],{},[7076,15656,7283],{"mathvariant":7282},[7699,15658,15659,15661],{},[7076,15660,8322],{},[7076,15662,279],{},[7223,15664,7290],{"stretchy":7289},[7223,15666,242],{},[7073,15668,15669,15671],{},[7076,15670,7817],{},[7070,15672,15673,15675,15677,15679,15681,15683],{},[7223,15674,7225],{},[7076,15676,7824],{},[7076,15678,7669],{},[7076,15680,279],{},[7076,15682,9487],{},[7076,15684,7829],{},[7076,15686,7283],{"mathvariant":7282},[7076,15688,8322],{},[7223,15690,7290],{"stretchy":7289},[7223,15692,242],{},[7073,15694,15695,15697],{},[7076,15696,7817],{},[7070,15698,15699,15701,15703,15705,15707,15709],{},[7223,15700,7225],{},[7076,15702,7824],{},[7076,15704,7669],{},[7076,15706,279],{},[7076,15708,9487],{},[7076,15710,7829],{},[7699,15712,15713,15715],{},[7076,15714,7810],{},[7076,15716,8322],{},[7076,15718,7283],{"mathvariant":7282},[7227,15720,5312],{},[7073,15722,15723,15725],{},[7223,15724,7290],{"stretchy":7289},[7076,15726,12207],{},[7083,15728,15729],{"encoding":7085},"|\\psi_k\\rangle = e^{-i H k dt } \\vert \\psi \\rangle = e^{-i H k dt } U_{\\psi} \\vert 0 \\rangle^N",[35,15731,15733,15795,15867],{"className":15732,"ariaHidden":7091},[7090],[35,15734,15736,15739,15742,15783,15786,15789,15792],{"className":15735},[7095],[35,15737],{"className":15738,"style":7303},[7099],[35,15740,7283],{"className":15741},[7104],[35,15743,15745,15748],{"className":15744},[7104],[35,15746,8322],{"className":15747,"style":7310},[7104,7145],[35,15749,15751],{"className":15750},[7113],[35,15752,15754,15775],{"className":15753},[7117,7731],[35,15755,15757,15772],{"className":15756},[7121],[35,15758,15761],{"className":15759,"style":15760},[7125],"height:0.3361em;",[35,15762,15763,15766],{"style":9100},[35,15764],{"className":15765,"style":7134},[7133],[35,15767,15769],{"className":15768},[7138,7139,7140,7141],[35,15770,279],{"className":15771,"style":8221},[7104,7145,7141],[35,15773,7756],{"className":15774},[7755],[35,15776,15778],{"className":15777},[7121],[35,15779,15781],{"className":15780,"style":7763},[7125],[35,15782],{},[35,15784,7290],{"className":15785},[7314],[35,15787],{"className":15788,"style":7450},[7248],[35,15790,242],{"className":15791},[7454],[35,15793],{"className":15794,"style":7450},[7248],[35,15796,15798,15802,15849,15852,15855,15858,15861,15864],{"className":15797},[7095],[35,15799],{"className":15800,"style":15801},[7099],"height:1.0991em;vertical-align:-0.25em;",[35,15803,15805,15808],{"className":15804},[7104],[35,15806,7817],{"className":15807},[7104,7145],[35,15809,15811],{"className":15810},[7113],[35,15812,15814],{"className":15813},[7117],[35,15815,15817],{"className":15816},[7121],[35,15818,15820],{"className":15819,"style":8191},[7125],[35,15821,15822,15825],{"style":7129},[35,15823],{"className":15824,"style":7134},[7133],[35,15826,15828],{"className":15827},[7138,7139,7140,7141],[35,15829,15831,15834,15837,15840,15843,15846],{"className":15830},[7104,7141],[35,15832,7225],{"className":15833},[7104,7141],[35,15835,7824],{"className":15836},[7104,7145,7141],[35,15838,7669],{"className":15839,"style":7684},[7104,7145,7141],[35,15841,279],{"className":15842,"style":8221},[7104,7145,7141],[35,15844,9487],{"className":15845},[7104,7145,7141],[35,15847,7829],{"className":15848},[7104,7145,7141],[35,15850,7283],{"className":15851},[7104],[35,15853,8322],{"className":15854,"style":7310},[7104,7145],[35,15856,7290],{"className":15857},[7314],[35,15859],{"className":15860,"style":7450},[7248],[35,15862,242],{"className":15863},[7454],[35,15865],{"className":15866,"style":7450},[7248],[35,15868,15870,15874,15921,15965,15968],{"className":15869},[7095],[35,15871],{"className":15872,"style":15873},[7099],"height:1.1352em;vertical-align:-0.2861em;",[35,15875,15877,15880],{"className":15876},[7104],[35,15878,7817],{"className":15879},[7104,7145],[35,15881,15883],{"className":15882},[7113],[35,15884,15886],{"className":15885},[7117],[35,15887,15889],{"className":15888},[7121],[35,15890,15892],{"className":15891,"style":8191},[7125],[35,15893,15894,15897],{"style":7129},[35,15895],{"className":15896,"style":7134},[7133],[35,15898,15900],{"className":15899},[7138,7139,7140,7141],[35,15901,15903,15906,15909,15912,15915,15918],{"className":15902},[7104,7141],[35,15904,7225],{"className":15905},[7104,7141],[35,15907,7824],{"className":15908},[7104,7145,7141],[35,15910,7669],{"className":15911,"style":7684},[7104,7145,7141],[35,15913,279],{"className":15914,"style":8221},[7104,7145,7141],[35,15916,9487],{"className":15917},[7104,7145,7141],[35,15919,7829],{"className":15920},[7104,7145,7141],[35,15922,15924,15927],{"className":15923},[7104],[35,15925,7810],{"className":15926,"style":7845},[7104,7145],[35,15928,15930],{"className":15929},[7113],[35,15931,15933,15957],{"className":15932},[7117,7731],[35,15934,15936,15954],{"className":15935},[7121],[35,15937,15939],{"className":15938,"style":15760},[7125],[35,15940,15942,15945],{"style":15941},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[35,15943],{"className":15944,"style":7134},[7133],[35,15946,15948],{"className":15947},[7138,7139,7140,7141],[35,15949,15951],{"className":15950},[7104,7141],[35,15952,8322],{"className":15953,"style":7310},[7104,7145,7141],[35,15955,7756],{"className":15956},[7755],[35,15958,15960],{"className":15959},[7121],[35,15961,15963],{"className":15962,"style":11360},[7125],[35,15964],{},[35,15966,14253],{"className":15967},[7104],[35,15969,15971,15974],{"className":15970},[7314],[35,15972,7290],{"className":15973},[7314],[35,15975,15977],{"className":15976},[7113],[35,15978,15980],{"className":15979},[7117],[35,15981,15983],{"className":15982},[7121],[35,15984,15986],{"className":15985,"style":7861},[7125],[35,15987,15988,15991],{"style":7129},[35,15989],{"className":15990,"style":7134},[7133],[35,15992,15994],{"className":15993},[7138,7139,7140,7141],[35,15995,12207],{"className":15996,"style":7845},[7104,7145,7141],". To measure ",[35,15999,16001,16014],{"className":16000},[7057],[35,16002,16004],{"className":16003},[7061],[7063,16005,16006],{"xmlns":7065},[7067,16007,16008,16012],{},[7070,16009,16010],{},[7076,16011,12257],{},[7083,16013,12257],{"encoding":7085},[35,16015,16017],{"className":16016,"ariaHidden":7091},[7090],[35,16018,16020,16023],{"className":16019},[7095],[35,16021],{"className":16022,"style":7100},[7099],[35,16024,12257],{"className":16025,"style":12402},[7104,7145],", first apply ",[35,16028,16030,16043],{"className":16029},[7057],[35,16031,16033],{"className":16032},[7061],[7063,16034,16035],{"xmlns":7065},[7067,16036,16037,16041],{},[7070,16038,16039],{},[7076,16040,7669],{},[7083,16042,7669],{"encoding":7085},[35,16044,16046],{"className":16045,"ariaHidden":7091},[7090],[35,16047,16049,16052],{"className":16048},[7095],[35,16050],{"className":16051,"style":7100},[7099],[35,16053,7669],{"className":16054,"style":7684},[7104,7145],[35,16056,16058],{"className":16057},[7318],[35,16059,16061,16167],{"className":16060},[7057],[35,16062,16064],{"className":16063},[7061],[7063,16065,16066],{"xmlns":7065,"display":7327},[7067,16067,16068,16164],{},[13970,16069,16070],{"rowspacing":13972,"columnspacing":13973},[13975,16071,16072],{},[13978,16073,16074],{},[13981,16075,16076],{"scriptlevel":5312,"displaystyle":7091},[7070,16077,16078,16080,16082,16088,16090,16092,16094,16096,16098,16104,16106,16108,16110,16112,16118,16120,16122,16124,16130,16132,16134,16136,16138,16140,16146,16148,16150,16152,16154,16160,16162],{},[7223,16079,14005],{},[7248,16081],{"width":13973},[14009,16083,16084,16086],{},[7227,16085,500],{},[7227,16087,460],{},[7076,16089,7283],{"mathvariant":7282},[7227,16091,5312],{},[7223,16093,7290],{"stretchy":7289},[7223,16095,275],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7076,16097,7283],{"mathvariant":7282},[7699,16099,16100,16102],{},[7076,16101,8322],{},[7076,16103,6920],{},[7223,16105,7290],{"stretchy":7289},[7223,16107,445],{},[7076,16109,7705],{},[7076,16111,7283],{"mathvariant":7282},[7699,16113,16114,16116],{},[7076,16115,8322],{},[7076,16117,7824],{},[7223,16119,7290],{"stretchy":7289},[7223,16121,1694],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7223,16123,445],{},[14009,16125,16126,16128],{},[7227,16127,500],{},[7227,16129,460],{},[7076,16131,7283],{"mathvariant":7282},[7227,16133,500],{},[7223,16135,7290],{"stretchy":7289},[7223,16137,275],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7076,16139,7283],{"mathvariant":7282},[7699,16141,16142,16144],{},[7076,16143,8322],{},[7076,16145,6920],{},[7223,16147,7290],{"stretchy":7289},[7223,16149,7225],{},[7076,16151,7705],{},[7076,16153,7283],{"mathvariant":7282},[7699,16155,16156,16158],{},[7076,16157,8322],{},[7076,16159,7824],{},[7223,16161,7290],{"stretchy":7289},[7223,16163,1694],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7083,16165,16166],{"encoding":7085},"\\begin{equation*}\n    \\longrightarrow\\quad\\frac{1}{2}|0\\rangle\\Big( |\\psi_j\\rangle + P|\\psi_i\\rangle\\Big) + \\frac{1}{2}|1\\rangle\\Big(|\\psi_j\\rangle - P|\\psi_i\\rangle\\Big)\n\\end{equation*}",[35,16168,16170],{"className":16169,"ariaHidden":7091},[7090],[35,16171,16173,16177],{"className":16172},[7095],[35,16174],{"className":16175,"style":16176},[7099],"height:2.0074em;vertical-align:-0.7537em;",[35,16178,16180],{"className":16179},[7104],[35,16181,16183],{"className":16182},[13970],[35,16184,16186],{"className":16185},[14229],[35,16187,16189,16596],{"className":16188},[7117,7731],[35,16190,16192,16593],{"className":16191},[7121],[35,16193,16196],{"className":16194,"style":16195},[7125],"height:1.2537em;",[35,16197,16199,16202],{"style":16198},"top:-3.2537em;",[35,16200],{"className":16201,"style":14246},[7133],[35,16203,16205,16208,16211,16214,16278,16281,16284,16290,16293,16333,16336,16339,16342,16345,16348,16351,16391,16394,16400,16403,16406,16409,16471,16474,16477,16483,16486,16526,16529,16532,16535,16538,16541,16544,16584,16587],{"className":16204},[7104],[35,16206,14005],{"className":16207},[7454],[35,16209],{"className":16210,"style":14292},[7248],[35,16212],{"className":16213,"style":7450},[7248],[35,16215,16217,16220,16275],{"className":16216},[7104],[35,16218],{"className":16219},[7472,14311],[35,16221,16223],{"className":16222},[14009],[35,16224,16226,16266],{"className":16225},[7117,7731],[35,16227,16229,16263],{"className":16228},[7121],[35,16230,16232,16244,16252],{"className":16231,"style":14324},[7125],[35,16233,16235,16238],{"style":16234},"top:-2.314em;",[35,16236],{"className":16237,"style":8839},[7133],[35,16239,16241],{"className":16240},[7104],[35,16242,460],{"className":16243},[7104],[35,16245,16246,16249],{"style":14395},[35,16247],{"className":16248,"style":8839},[7133],[35,16250],{"className":16251,"style":14403},[14402],[35,16253,16254,16257],{"style":14406},[35,16255],{"className":16256,"style":8839},[7133],[35,16258,16260],{"className":16259},[7104],[35,16261,500],{"className":16262},[7104],[35,16264,7756],{"className":16265},[7755],[35,16267,16269],{"className":16268},[7121],[35,16270,16273],{"className":16271,"style":16272},[7125],"height:0.686em;",[35,16274],{},[35,16276],{"className":16277},[7314,14311],[35,16279,14253],{"className":16280},[7104],[35,16282,7290],{"className":16283},[7314],[35,16285,16287],{"className":16286},[7104],[35,16288,275],{"className":16289},[7478,14437],[35,16291,7283],{"className":16292},[7104],[35,16294,16296,16299],{"className":16295},[7104],[35,16297,8322],{"className":16298,"style":7310},[7104,7145],[35,16300,16302],{"className":16301},[7113],[35,16303,16305,16325],{"className":16304},[7117,7731],[35,16306,16308,16322],{"className":16307},[7121],[35,16309,16311],{"className":16310,"style":11329},[7125],[35,16312,16313,16316],{"style":9100},[35,16314],{"className":16315,"style":7134},[7133],[35,16317,16319],{"className":16318},[7138,7139,7140,7141],[35,16320,6920],{"className":16321,"style":11350},[7104,7145,7141],[35,16323,7756],{"className":16324},[7755],[35,16326,16328],{"className":16327},[7121],[35,16329,16331],{"className":16330,"style":11360},[7125],[35,16332],{},[35,16334,7290],{"className":16335},[7314],[35,16337],{"className":16338,"style":7249},[7248],[35,16340,445],{"className":16341},[7253],[35,16343],{"className":16344,"style":7249},[7248],[35,16346,7705],{"className":16347,"style":7751},[7104,7145],[35,16349,7283],{"className":16350},[7104],[35,16352,16354,16357],{"className":16353},[7104],[35,16355,8322],{"className":16356,"style":7310},[7104,7145],[35,16358,16360],{"className":16359},[7113],[35,16361,16363,16383],{"className":16362},[7117,7731],[35,16364,16366,16380],{"className":16365},[7121],[35,16367,16369],{"className":16368,"style":11329},[7125],[35,16370,16371,16374],{"style":9100},[35,16372],{"className":16373,"style":7134},[7133],[35,16375,16377],{"className":16376},[7138,7139,7140,7141],[35,16378,7824],{"className":16379},[7104,7145,7141],[35,16381,7756],{"className":16382},[7755],[35,16384,16386],{"className":16385},[7121],[35,16387,16389],{"className":16388,"style":7763},[7125],[35,16390],{},[35,16392,7290],{"className":16393},[7314],[35,16395,16397],{"className":16396},[7104],[35,16398,1694],{"className":16399},[7478,14437],[35,16401],{"className":16402,"style":7249},[7248],[35,16404,445],{"className":16405},[7253],[35,16407],{"className":16408,"style":7249},[7248],[35,16410,16412,16415,16468],{"className":16411},[7104],[35,16413],{"className":16414},[7472,14311],[35,16416,16418],{"className":16417},[14009],[35,16419,16421,16460],{"className":16420},[7117,7731],[35,16422,16424,16457],{"className":16423},[7121],[35,16425,16427,16438,16446],{"className":16426,"style":14324},[7125],[35,16428,16429,16432],{"style":16234},[35,16430],{"className":16431,"style":8839},[7133],[35,16433,16435],{"className":16434},[7104],[35,16436,460],{"className":16437},[7104],[35,16439,16440,16443],{"style":14395},[35,16441],{"className":16442,"style":8839},[7133],[35,16444],{"className":16445,"style":14403},[14402],[35,16447,16448,16451],{"style":14406},[35,16449],{"className":16450,"style":8839},[7133],[35,16452,16454],{"className":16453},[7104],[35,16455,500],{"className":16456},[7104],[35,16458,7756],{"className":16459},[7755],[35,16461,16463],{"className":16462},[7121],[35,16464,16466],{"className":16465,"style":16272},[7125],[35,16467],{},[35,16469],{"className":16470},[7314,14311],[35,16472,14456],{"className":16473},[7104],[35,16475,7290],{"className":16476},[7314],[35,16478,16480],{"className":16479},[7104],[35,16481,275],{"className":16482},[7478,14437],[35,16484,7283],{"className":16485},[7104],[35,16487,16489,16492],{"className":16488},[7104],[35,16490,8322],{"className":16491,"style":7310},[7104,7145],[35,16493,16495],{"className":16494},[7113],[35,16496,16498,16518],{"className":16497},[7117,7731],[35,16499,16501,16515],{"className":16500},[7121],[35,16502,16504],{"className":16503,"style":11329},[7125],[35,16505,16506,16509],{"style":9100},[35,16507],{"className":16508,"style":7134},[7133],[35,16510,16512],{"className":16511},[7138,7139,7140,7141],[35,16513,6920],{"className":16514,"style":11350},[7104,7145,7141],[35,16516,7756],{"className":16517},[7755],[35,16519,16521],{"className":16520},[7121],[35,16522,16524],{"className":16523,"style":11360},[7125],[35,16525],{},[35,16527,7290],{"className":16528},[7314],[35,16530],{"className":16531,"style":7249},[7248],[35,16533,7225],{"className":16534},[7253],[35,16536],{"className":16537,"style":7249},[7248],[35,16539,7705],{"className":16540,"style":7751},[7104,7145],[35,16542,7283],{"className":16543},[7104],[35,16545,16547,16550],{"className":16546},[7104],[35,16548,8322],{"className":16549,"style":7310},[7104,7145],[35,16551,16553],{"className":16552},[7113],[35,16554,16556,16576],{"className":16555},[7117,7731],[35,16557,16559,16573],{"className":16558},[7121],[35,16560,16562],{"className":16561,"style":11329},[7125],[35,16563,16564,16567],{"style":9100},[35,16565],{"className":16566,"style":7134},[7133],[35,16568,16570],{"className":16569},[7138,7139,7140,7141],[35,16571,7824],{"className":16572},[7104,7145,7141],[35,16574,7756],{"className":16575},[7755],[35,16577,16579],{"className":16578},[7121],[35,16580,16582],{"className":16581,"style":7763},[7125],[35,16583],{},[35,16585,7290],{"className":16586},[7314],[35,16588,16590],{"className":16589},[7104],[35,16591,1694],{"className":16592},[7478,14437],[35,16594,7756],{"className":16595},[7755],[35,16597,16599],{"className":16598},[7121],[35,16600,16603],{"className":16601,"style":16602},[7125],"height:0.7537em;",[35,16604],{},[12,16606,16607],{},"... then measure:",[35,16609,16611],{"className":16610},[7318],[35,16612,16614,16794],{"className":16613},[7057],[35,16615,16617],{"className":16616},[7061],[7063,16618,16619],{"xmlns":7065,"display":7327},[7067,16620,16621,16791],{},[13970,16622,16623],{"rowspacing":13972,"columnspacing":13973},[13975,16624,16625],{},[13978,16626,16627],{},[13981,16628,16629],{"scriptlevel":5312,"displaystyle":7091},[13970,16630,16634,16741],{"rowspacing":16631,"columnalign":16632,"columnspacing":16633},"0.25em","right left","0em",[13975,16635,16636,16653],{},[13978,16637,16638],{},[13981,16639,16640],{"scriptlevel":5312,"displaystyle":7091},[7070,16641,16642,16645,16647,16649,16651],{},[7223,16643,16644],{},"⇒",[7248,16646],{"width":13973},[7223,16648,8953],{"stretchy":7289},[7076,16650,12257],{},[7223,16652,7290],{"stretchy":7289},[13978,16654,16655],{},[13981,16656,16657],{"scriptlevel":5312,"displaystyle":7091},[7070,16658,16659,16661,16663,16669,16672,16675,16677,16683,16685,16687,16689,16691,16697,16699,16705,16707,16709,16711,16717,16719,16721,16723,16725,16731,16733,16739],{},[7070,16660],{},[7223,16662,242],{},[14009,16664,16665,16667],{},[7227,16666,500],{},[7227,16668,528],{},[7223,16670,275],{"fence":7289,"stretchy":7091,"minsize":16671,"maxsize":16671},"3em",[7223,16673,16674],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},"∥",[7076,16676,7283],{"mathvariant":7282},[7699,16678,16679,16681],{},[7076,16680,8322],{},[7076,16682,6920],{},[7223,16684,7290],{"stretchy":7289},[7223,16686,445],{},[7076,16688,7705],{},[7076,16690,7283],{"mathvariant":7282},[7699,16692,16693,16695],{},[7076,16694,8322],{},[7076,16696,7824],{},[7223,16698,7290],{"stretchy":7289},[7073,16700,16701,16703],{},[7223,16702,16674],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7227,16704,460],{},[7223,16706,7225],{},[7223,16708,16674],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7076,16710,7283],{"mathvariant":7282},[7699,16712,16713,16715],{},[7076,16714,8322],{},[7076,16716,6920],{},[7223,16718,7290],{"stretchy":7289},[7223,16720,7225],{},[7076,16722,7705],{},[7076,16724,7283],{"mathvariant":7282},[7699,16726,16727,16729],{},[7076,16728,8322],{},[7076,16730,7824],{},[7223,16732,7290],{"stretchy":7289},[7073,16734,16735,16737],{},[7223,16736,16674],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7227,16738,460],{},[7223,16740,1694],{"fence":7289,"stretchy":7091,"minsize":16671,"maxsize":16671},[13975,16742,16743,16749],{},[13978,16744,16745],{},[13981,16746,16747],{"scriptlevel":5312,"displaystyle":7091},[7070,16748],{},[13978,16750,16751],{},[13981,16752,16753],{"scriptlevel":5312,"displaystyle":7091},[7070,16754,16755,16757,16759,16762,16764,16766,16772,16774,16776,16778,16784,16786,16789],{},[7070,16756],{},[7223,16758,242],{},[10694,16760,16761],{},"Re",[7223,16763,1254],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7223,16765,8953],{"stretchy":7289},[7699,16767,16768,16770],{},[7076,16769,8322],{},[7076,16771,6920],{},[7076,16773,7283],{"mathvariant":7282},[7076,16775,7705],{},[7076,16777,7283],{"mathvariant":7282},[7699,16779,16780,16782],{},[7076,16781,8322],{},[7076,16783,7824],{},[7223,16785,7290],{"stretchy":7289},[7223,16787,16788],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},"]",[7076,16790,5122],{"mathvariant":7282},[7083,16792,16793],{"encoding":7085},"\\begin{equation*}\n\\begin{split}\n    \\Rightarrow\\quad\\langle X\\rangle &= \\frac{1}{4}\\Bigg(\\Big\\|| \\psi_j\\rangle + P|\\psi_i\\rangle \\Big\\|^2-\\Big\\||\\psi_j\\rangle - P|\\psi_i\\rangle\\Big\\|^2\\Bigg) \\\\\n    &= \\text{Re}\\Big[\\langle\\psi_j| 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",[35,17598,17600,17687],{"className":17599},[7057],[35,17601,17603],{"className":17602},[7061],[7063,17604,17605],{"xmlns":7065},[7067,17606,17607,17684],{},[7070,17608,17609,17611,17613,17615,17618,17624,17626,17628,17630,17632,17634,17636,17638,17640,17642,17644,17646,17648,17650,17656,17658,17660,17662,17668,17670,17672,17674,17676,17678,17680,17682],{},[7076,17610,7283],{"mathvariant":7282},[7076,17612,19],{},[7223,17614,445],{},[7076,17616,17617],{},"b",[7073,17619,17620,17622],{},[7076,17621,16674],{"mathvariant":7282},[7227,17623,460],{},[7223,17625,242],{},[7223,17627,8953],{"stretchy":7289},[7076,17629,19],{},[7223,17631,445],{},[7076,17633,17617],{},[7076,17635,7283],{"mathvariant":7282},[7076,17637,19],{},[7223,17639,445],{},[7076,17641,17617],{},[7223,17643,7290],{"stretchy":7289},[7223,17645,242],{},[7076,17647,16674],{"mathvariant":7282},[7076,17649,19],{},[7073,17651,17652,17654],{},[7076,17653,16674],{"mathvariant":7282},[7227,17655,460],{},[7223,17657,445],{},[7076,17659,16674],{"mathvariant":7282},[7076,17661,17617],{},[7073,17663,17664,17666],{},[7076,17665,16674],{"mathvariant":7282},[7227,17667,460],{},[7223,17669,445],{},[7227,17671,460],{},[10694,17673,16761],{},[7223,17675,8953],{"stretchy":7289},[7076,17677,19],{},[7076,17679,7283],{"mathvariant":7282},[7076,17681,17617],{},[7223,17683,7290],{"stretchy":7289},[7083,17685,17686],{"encoding":7085},"|a + b\\|^2 = \\langle a + b | a + b \\rangle = \\|a\\|^2 + \\|b\\|^2 + 2\\text{Re}\\langle a | b \\rangle",[35,17688,17690,17711,17760,17781,17805,17826,17876,17926],{"className":17689,"ariaHidden":7091},[7090],[35,17691,17693,17696,17699,17702,17705,17708],{"className":17692},[7095],[35,17694],{"className":17695,"style":7303},[7099],[35,17697,7283],{"className":17698},[7104],[35,17700,19],{"className":17701},[7104,7145],[35,17703],{"className":17704,"style":7249},[7248],[35,17706,445],{"className":17707},[7253],[35,17709],{"className":17710,"style":7249},[7248],[35,17712,17714,17718,17721,17751,17754,17757],{"className":17713},[7095],[35,17715],{"className":17716,"style":17717},[7099],"height:1.0641em;vertical-align:-0.25em;",[35,17719,17617],{"className":17720},[7104,7145],[35,17722,17724,17727],{"className":17723},[7104],[35,17725,16674],{"className":17726},[7104],[35,17728,17730],{"className":17729},[7113],[35,17731,17733],{"className":17732},[7117],[35,17734,17736],{"className":17735},[7121],[35,17737,17740],{"className":17738,"style":17739},[7125],"height:0.8141em;",[35,17741,17742,17745],{"style":7129},[35,17743],{"className":17744,"style":7134},[7133],[35,17746,17748],{"className":17747},[7138,7139,7140,7141],[35,17749,460],{"className":17750},[7104,7141],[35,17752],{"className":17753,"style":7450},[7248],[35,17755,242],{"className":17756},[7454],[35,17758],{"className":17759,"style":7450},[7248],[35,17761,17763,17766,17769,17772,17775,17778],{"className":17762},[7095],[35,17764],{"className":17765,"style":7303},[7099],[35,17767,8953],{"className":17768},[7472],[35,17770,19],{"className":17771},[7104,7145],[35,17773],{"className":17774,"style":7249},[7248],[35,17776,445],{"className":17777},[7253],[35,17779],{"className":17780,"style":7249},[7248],[35,17782,17784,17787,17790,17793,17796,17799,17802],{"className":17783},[7095],[35,17785],{"className":17786,"style":7303},[7099],[35,17788,17617],{"className":17789},[7104,7145],[35,17791,7283],{"className":17792},[7104],[35,17794,19],{"className":17795},[7104,7145],[35,17797],{"className":17798,"style":7249},[7248],[35,17800,445],{"className":17801},[7253],[35,17803],{"className":17804,"style":7249},[7248],[35,17806,17808,17811,17814,17817,17820,17823],{"className":17807},[7095],[35,17809],{"className":17810,"style":7303},[7099],[35,17812,17617],{"className":17813},[7104,7145],[35,17815,7290],{"className":17816},[7314],[35,17818],{"className":17819,"style":7450},[7248],[35,17821,242],{"className":17822},[7454],[35,17824],{"className":17825,"style":7450},[7248],[35,17827,17829,17832,17835,17838,17867,17870,17873],{"className":17828},[7095],[35,17830],{"className":17831,"style":17717},[7099],[35,17833,16674],{"className":17834},[7104],[35,17836,19],{"className":17837},[7104,7145],[35,17839,17841,17844],{"className":17840},[7104],[35,17842,16674],{"className":17843},[7104],[35,17845,17847],{"className":17846},[7113],[35,17848,17850],{"className":17849},[7117],[35,17851,17853],{"className":17852},[7121],[35,17854,17856],{"className":17855,"style":17739},[7125],[35,17857,17858,17861],{"style":7129},[35,17859],{"className":17860,"style":7134},[7133],[35,17862,17864],{"className":17863},[7138,7139,7140,7141],[35,17865,460],{"className":17866},[7104,7141],[35,17868],{"className":17869,"style":7249},[7248],[35,17871,445],{"className":17872},[7253],[35,17874],{"className":17875,"style":7249},[7248],[35,17877,17879,17882,17885,17888,17917,17920,17923],{"className":17878},[7095],[35,17880],{"className":17881,"style":17717},[7099],[35,17883,16674],{"className":17884},[7104],[35,17886,17617],{"className":17887},[7104,7145],[35,17889,17891,17894],{"className":17890},[7104],[35,17892,16674],{"className":17893},[7104],[35,17895,17897],{"className":17896},[7113],[35,17898,17900],{"className":17899},[7117],[35,17901,17903],{"className":17902},[7121],[35,17904,17906],{"className":17905,"style":17739},[7125],[35,17907,17908,17911],{"style":7129},[35,17909],{"className":17910,"style":7134},[7133],[35,17912,17914],{"className":17913},[7138,7139,7140,7141],[35,17915,460],{"className":17916},[7104,7141],[35,17918],{"className":17919,"style":7249},[7248],[35,17921,445],{"className":17922},[7253],[35,17924],{"className":17925,"style":7249},[7248],[35,17927,17929,17932,17935,17941,17944,17947,17950,17953],{"className":17928},[7095],[35,17930],{"className":17931,"style":7303},[7099],[35,17933,460],{"className":17934},[7104],[35,17936,17938],{"className":17937},[7104,4991],[35,17939,16761],{"className":17940},[7104],[35,17942,8953],{"className":17943},[7472],[35,17945,19],{"className":17946},[7104,7145],[35,17948,7283],{"className":17949},[7104],[35,17951,17617],{"className":17952},[7104,7145],[35,17954,7290],{"className":17955},[7314],". Similarly, measuring ",[35,17958,17960,17973],{"className":17959},[7057],[35,17961,17963],{"className":17962},[7061],[7063,17964,17965],{"xmlns":7065},[7067,17966,17967,17971],{},[7070,17968,17969],{},[7076,17970,12272],{},[7083,17972,12272],{"encoding":7085},[35,17974,17976],{"className":17975,"ariaHidden":7091},[7090],[35,17977,17979,17982],{"className":17978},[7095],[35,17980],{"className":17981,"style":7100},[7099],[35,17983,12272],{"className":17984,"style":7249},[7104,7145]," yields",[35,17987,17989],{"className":17988},[7318],[35,17990,17992,18051],{"className":17991},[7057],[35,17993,17995],{"className":17994},[7061],[7063,17996,17997],{"xmlns":7065,"display":7327},[7067,17998,17999,18048],{},[13970,18000,18001],{"rowspacing":13972,"columnspacing":13973},[13975,18002,18003],{},[13978,18004,18005],{},[13981,18006,18007],{"scriptlevel":5312,"displaystyle":7091},[7070,18008,18009,18011,18013,18015,18017,18020,18022,18024,18030,18032,18034,18036,18042,18044,18046],{},[7223,18010,8953],{"stretchy":7289},[7076,18012,12272],{},[7223,18014,7290],{"stretchy":7289},[7223,18016,242],{},[10694,18018,18019],{},"Im",[7223,18021,1254],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7223,18023,8953],{"stretchy":7289},[7699,18025,18026,18028],{},[7076,18027,8322],{},[7076,18029,6920],{},[7076,18031,7283],{"mathvariant":7282},[7076,18033,7705],{},[7076,18035,7283],{"mathvariant":7282},[7699,18037,18038,18040],{},[7076,18039,8322],{},[7076,18041,7824],{},[7223,18043,7290],{"stretchy":7289},[7223,18045,16788],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7076,18047,5122],{"mathvariant":7282},[7083,18049,18050],{"encoding":7085},"\\begin{equation*}\n    \\langle Y\\rangle = \\text{Im}\\Big[\\langle\\psi_j| P|\\psi_i\\rangle\\Big].\n\\end{equation*}",[35,18052,18054],{"className":18053,"ariaHidden":7091},[7090],[35,18055,18057,18061],{"className":18056},[7095],[35,18058],{"className":18059,"style":18060},[7099],"height:1.8em;vertical-align:-0.65em;",[35,18062,18064],{"className":18063},[7104],[35,18065,18067],{"className":18066},[13970],[35,18068,18070],{"className":18069},[14229],[35,18071,18073,18225],{"className":18072},[7117,7731],[35,18074,18076,18222],{"className":18075},[7121],[35,18077,18079],{"className":18078,"style":17018},[7125],[35,18080,18081,18085],{"style":17021},[35,18082],{"className":18083,"style":18084},[7133],"height:3.15em;",[35,18086,18088,18091,18094,18097,18100,18103,18106,18112,18118,18121,18161,18164,18167,18170,18210,18213,18219],{"className":18087},[7104],[35,18089,8953],{"className":18090},[7472],[35,18092,12272],{"className":18093,"style":7249},[7104,7145],[35,18095,7290],{"className":18096},[7314],[35,18098],{"className":18099,"style":7450},[7248],[35,18101,242],{"className":18102},[7454],[35,18104],{"className":18105,"style":7450},[7248],[35,18107,18109],{"className":18108},[7104,4991],[35,18110,18019],{"className":18111},[7104],[35,18113,18115],{"className":18114},[7104],[35,18116,1254],{"className":18117},[7478,14437],[35,18119,8953],{"className":18120},[7472],[35,18122,18124,18127],{"className":18123},[7104],[35,18125,8322],{"className":18126,"style":7310},[7104,7145],[35,18128,18130],{"className":18129},[7113],[35,18131,18133,18153],{"className":18132},[7117,7731],[35,18134,18136,18150],{"className":18135},[7121],[35,18137,18139],{"className":18138,"style":11329},[7125],[35,18140,18141,18144],{"style":9100},[35,18142],{"className":18143,"style":7134},[7133],[35,18145,18147],{"className":18146},[7138,7139,7140,7141],[35,18148,6920],{"className":18149,"style":11350},[7104,7145,7141],[35,18151,7756],{"className":18152},[7755],[35,18154,18156],{"className":18155},[7121],[35,18157,18159],{"className":18158,"style":11360},[7125],[35,18160],{},[35,18162,7283],{"className":18163},[7104],[35,18165,7705],{"className":18166,"style":7751},[7104,7145],[35,18168,7283],{"className":18169},[7104],[35,18171,18173,18176],{"className":18172},[7104],[35,18174,8322],{"className":18175,"style":7310},[7104,7145],[35,18177,18179],{"className":18178},[7113],[35,18180,18182,18202],{"className":18181},[7117,7731],[35,18183,18185,18199],{"className":18184},[7121],[35,18186,18188],{"className":18187,"style":11329},[7125],[35,18189,18190,18193],{"style":9100},[35,18191],{"className":18192,"style":7134},[7133],[35,18194,18196],{"className":18195},[7138,7139,7140,7141],[35,18197,7824],{"className":18198},[7104,7145,7141],[35,18200,7756],{"className":18201},[7755],[35,18203,18205],{"className":18204},[7121],[35,18206,18208],{"className":18207,"style":7763},[7125],[35,18209],{},[35,18211,7290],{"className":18212},[7314],[35,18214,18216],{"className":18215},[7104],[35,18217,16788],{"className":18218},[7478,14437],[35,18220,5122],{"className":18221},[7104],[35,18223,7756],{"className":18224},[7755],[35,18226,18228],{"className":18227},[7121],[35,18229,18231],{"className":18230,"style":17045},[7125],[35,18232],{},[25,18234,18236],{"className":27,"code":18235,"language":29,"meta":30,"style":30},"## Create the time-evo op circuit\nevol_gate = PauliEvolutionGate(\n    H_op, time=dt, synthesis=LieTrotter(reps=num_trotter_steps)\n)\n\n## Create the time-evo op dagger circuit\nevol_gate_d = PauliEvolutionGate(\n    H_op, time=dt, synthesis=LieTrotter(reps=num_trotter_steps)\n)\nevol_gate_d = evol_gate_d.inverse()\n\n# Put pieces together\nqc_reg = QuantumRegister(n_qubits)\nqc_temp = QuantumCircuit(qc_reg)\nqc_temp.compose(qc_state_prep, inplace=True)\nfor _ in range(num_trotter_steps):\n    qc_temp.append(evol_gate, qargs=qc_reg)\nfor _ in range(num_trotter_steps):\n    qc_temp.append(evol_gate_d, qargs=qc_reg)\nqc_temp.compose(qc_state_prep.inverse(), inplace=True)\n\n# Create controlled version of the circuit\ncontrolled_U = qc_temp.to_gate().control(1)\n\n# Create hadamard test circuit for real part\nqr = QuantumRegister(n_qubits + 1)\nqc_real = QuantumCircuit(qr)\nqc_real.h(0)\nqc_real.append(controlled_U, list(range(n_qubits + 1)))\nqc_real.h(0)\n\nprint(\n    \"Circuit for calculating the real part of the overlap in S via Hadamard test\"\n)\nqc_real.draw(\"mpl\", fold=-1, scale=0.5)\n",[32,18237,18238,18242,18252,18277,18281,18285,18290,18301,18325,18329,18343,18347,18352,18363,18375,18395,18408,18424,18436,18451,18472,18476,18481,18506,18510,18515,18531,18542,18555,18580,18592,18596,18602,18607,18611],{"__ignoreMap":30},[35,18239,18240],{"class":37,"line":38},[35,18241,13846],{"class":41},[35,18243,18244,18246,18248,18250],{"class":37,"line":45},[35,18245,13851],{"class":113},[35,18247,242],{"class":241},[35,18249,13856],{"class":271},[35,18251,903],{"class":113},[35,18253,18254,18256,18258,18260,18263,18265,18267,18269,18271,18273,18275],{"class":37,"line":51},[35,18255,13863],{"class":113},[35,18257,13866],{"class":822},[35,18259,242],{"class":241},[35,18261,18262],{"class":113},"dt, ",[35,18264,13874],{"class":822},[35,18266,242],{"class":241},[35,18268,13879],{"class":271},[35,18270,275],{"class":113},[35,18272,13884],{"class":822},[35,18274,242],{"class":241},[35,18276,13889],{"class":113},[35,18278,18279],{"class":37,"line":57},[35,18280,384],{"class":113},[35,18282,18283],{"class":37,"line":63},[35,18284,103],{"emptyLinePlaceholder":102},[35,18286,18287],{"class":37,"line":69},[35,18288,18289],{"class":41},"## Create the time-evo op dagger circuit\n",[35,18291,18292,18295,18297,18299],{"class":37,"line":75},[35,18293,18294],{"class":113},"evol_gate_d ",[35,18296,242],{"class":241},[35,18298,13856],{"class":271},[35,18300,903],{"class":113},[35,18302,18303,18305,18307,18309,18311,18313,18315,18317,18319,18321,18323],{"class":37,"line":81},[35,18304,13863],{"class":113},[35,18306,13866],{"class":822},[35,18308,242],{"class":241},[35,18310,18262],{"class":113},[35,18312,13874],{"class":822},[35,18314,242],{"class":241},[35,18316,13879],{"class":271},[35,18318,275],{"class":113},[35,18320,13884],{"class":822},[35,18322,242],{"class":241},[35,18324,13889],{"class":113},[35,18326,18327],{"class":37,"line":87},[35,18328,384],{"class":113},[35,18330,18331,18333,18335,18338,18341],{"class":37,"line":93},[35,18332,18294],{"class":113},[35,18334,242],{"class":241},[35,18336,18337],{"class":113}," evol_gate_d.",[35,18339,18340],{"class":271},"inverse",[35,18342,2101],{"class":113},[35,18344,18345],{"class":37,"line":99},[35,18346,103],{"emptyLinePlaceholder":102},[35,18348,18349],{"class":37,"line":106},[35,18350,18351],{"class":41},"# Put pieces together\n",[35,18353,18354,18357,18359,18361],{"class":37,"line":117},[35,18355,18356],{"class":113},"qc_reg ",[35,18358,242],{"class":241},[35,18360,13907],{"class":271},[35,18362,13301],{"class":113},[35,18364,18365,18368,18370,18372],{"class":37,"line":125},[35,18366,18367],{"class":113},"qc_temp ",[35,18369,242],{"class":241},[35,18371,5293],{"class":271},[35,18373,18374],{"class":113},"(qc_reg)\n",[35,18376,18377,18380,18383,18386,18389,18391,18393],{"class":37,"line":139},[35,18378,18379],{"class":113},"qc_temp.",[35,18381,18382],{"class":271},"compose",[35,18384,18385],{"class":113},"(qc_state_prep, ",[35,18387,18388],{"class":822},"inplace",[35,18390,242],{"class":241},[35,18392,828],{"class":245},[35,18394,384],{"class":113},[35,18396,18397,18399,18401,18403,18405],{"class":37,"line":152},[35,18398,1874],{"class":109},[35,18400,12741],{"class":113},[35,18402,1663],{"class":109},[35,18404,12746],{"class":241},[35,18406,18407],{"class":113},"(num_trotter_steps):\n",[35,18409,18410,18413,18415,18417,18419,18421],{"class":37,"line":157},[35,18411,18412],{"class":113},"    qc_temp.",[35,18414,1927],{"class":271},[35,18416,13931],{"class":113},[35,18418,13934],{"class":822},[35,18420,242],{"class":241},[35,18422,18423],{"class":113},"qc_reg)\n",[35,18425,18426,18428,18430,18432,18434],{"class":37,"line":171},[35,18427,1874],{"class":109},[35,18429,12741],{"class":113},[35,18431,1663],{"class":109},[35,18433,12746],{"class":241},[35,18435,18407],{"class":113},[35,18437,18438,18440,18442,18445,18447,18449],{"class":37,"line":184},[35,18439,18412],{"class":113},[35,18441,1927],{"class":271},[35,18443,18444],{"class":113},"(evol_gate_d, ",[35,18446,13934],{"class":822},[35,18448,242],{"class":241},[35,18450,18423],{"class":113},[35,18452,18453,18455,18457,18460,18462,18464,18466,18468,18470],{"class":37,"line":197},[35,18454,18379],{"class":113},[35,18456,18382],{"class":271},[35,18458,18459],{"class":113},"(qc_state_prep.",[35,18461,18340],{"class":271},[35,18463,2672],{"class":113},[35,18465,18388],{"class":822},[35,18467,242],{"class":241},[35,18469,828],{"class":245},[35,18471,384],{"class":113},[35,18473,18474],{"class":37,"line":210},[35,18475,103],{"emptyLinePlaceholder":102},[35,18477,18478],{"class":37,"line":223},[35,18479,18480],{"class":41},"# Create controlled version of the circuit\n",[35,18482,18483,18486,18488,18491,18494,18497,18500,18502,18504],{"class":37,"line":228},[35,18484,18485],{"class":113},"controlled_U ",[35,18487,242],{"class":241},[35,18489,18490],{"class":113}," qc_temp.",[35,18492,18493],{"class":271},"to_gate",[35,18495,18496],{"class":113},"().",[35,18498,18499],{"class":271},"control",[35,18501,275],{"class":113},[35,18503,500],{"class":245},[35,18505,384],{"class":113},[35,18507,18508],{"class":37,"line":249},[35,18509,103],{"emptyLinePlaceholder":102},[35,18511,18512],{"class":37,"line":254},[35,18513,18514],{"class":41},"# Create hadamard test circuit for real part\n",[35,18516,18517,18519,18521,18523,18525,18527,18529],{"class":37,"line":260},[35,18518,13902],{"class":113},[35,18520,242],{"class":241},[35,18522,13907],{"class":271},[35,18524,6676],{"class":113},[35,18526,445],{"class":241},[35,18528,6079],{"class":245},[35,18530,384],{"class":113},[35,18532,18533,18536,18538,18540],{"class":37,"line":265},[35,18534,18535],{"class":113},"qc_real ",[35,18537,242],{"class":241},[35,18539,5293],{"class":271},[35,18541,13921],{"class":113},[35,18543,18544,18547,18549,18551,18553],{"class":37,"line":326},[35,18545,18546],{"class":113},"qc_real.",[35,18548,5307],{"class":271},[35,18550,275],{"class":113},[35,18552,5312],{"class":245},[35,18554,384],{"class":113},[35,18556,18557,18559,18561,18564,18567,18569,18571,18573,18575,18577],{"class":37,"line":332},[35,18558,18546],{"class":113},[35,18560,1927],{"class":271},[35,18562,18563],{"class":113},"(controlled_U, ",[35,18565,18566],{"class":241},"list",[35,18568,275],{"class":113},[35,18570,6767],{"class":241},[35,18572,6676],{"class":113},[35,18574,445],{"class":241},[35,18576,6079],{"class":245},[35,18578,18579],{"class":113},")))\n",[35,18581,18582,18584,18586,18588,18590],{"class":37,"line":338},[35,18583,18546],{"class":113},[35,18585,5307],{"class":271},[35,18587,275],{"class":113},[35,18589,5312],{"class":245},[35,18591,384],{"class":113},[35,18593,18594],{"class":37,"line":344},[35,18595,103],{"emptyLinePlaceholder":102},[35,18597,18598,18600],{"class":37,"line":349},[35,18599,1861],{"class":241},[35,18601,903],{"class":113},[35,18603,18604],{"class":37,"line":367},[35,18605,18606],{"class":234},"    \"Circuit for calculating the real part of the overlap in S via Hadamard test\"\n",[35,18608,18609],{"class":37,"line":387},[35,18610,384],{"class":113},[35,18612,18613,18615,18617,18619,18621,18623,18626,18629,18631,18633,18635,18637,18639],{"class":37,"line":404},[35,18614,18546],{"class":113},[35,18616,13682],{"class":271},[35,18618,275],{"class":113},[35,18620,13687],{"class":234},[35,18622,288],{"class":113},[35,18624,18625],{"class":822},"fold",[35,18627,18628],{"class":241},"=-",[35,18630,500],{"class":245},[35,18632,288],{"class":113},[35,18634,13692],{"class":822},[35,18636,242],{"class":241},[35,18638,13697],{"class":245},[35,18640,384],{"class":113},[25,18642,18645],{"className":18643,"code":18644,"language":4991,"meta":30},[4989],"Circuit for calculating the real part of the overlap in S via Hadamard test\n",[32,18646,18644],{"__ignoreMap":30},[4995,18648],{"alt":13702,"src":18649},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-02.avif",[12,18651,18652],{},"The Hadamard test circuit can be a deep circuit once we decompose to native gates (which will increase even more if we account for the topology of the device)",[25,18654,18656],{"className":27,"code":18655,"language":29,"meta":30,"style":30},"print(\n    \"Number of layers of 2Q operations\",\n    qc_real.decompose(reps=2).depth(lambda x: x[0].num_qubits == 2),\n)\n",[32,18657,18658,18664,18671,18716],{"__ignoreMap":30},[35,18659,18660,18662],{"class":37,"line":38},[35,18661,1861],{"class":241},[35,18663,903],{"class":113},[35,18665,18666,18669],{"class":37,"line":45},[35,18667,18668],{"class":234},"    \"Number of layers of 2Q operations\"",[35,18670,912],{"class":113},[35,18672,18673,18676,18679,18681,18683,18685,18687,18690,18693,18695,18698,18701,18704,18706,18709,18711,18713],{"class":37,"line":51},[35,18674,18675],{"class":113},"    qc_real.",[35,18677,18678],{"class":271},"decompose",[35,18680,275],{"class":113},[35,18682,13884],{"class":822},[35,18684,242],{"class":241},[35,18686,460],{"class":245},[35,18688,18689],{"class":113},").",[35,18691,18692],{"class":271},"depth",[35,18694,275],{"class":113},[35,18696,18697],{"class":109},"lambda",[35,18699,18700],{"class":278}," x",[35,18702,18703],{"class":113},": x[",[35,18705,5312],{"class":245},[35,18707,18708],{"class":113},"].num_qubits ",[35,18710,358],{"class":241},[35,18712,424],{"class":245},[35,18714,18715],{"class":113},"),\n",[35,18717,18718],{"class":37,"line":57},[35,18719,384],{"class":113},[25,18721,18724],{"className":18722,"code":18723,"language":4991,"meta":30},[4989],"Number of layers of 2Q operations 112753\n",[32,18725,18723],{"__ignoreMap":30},[5019,18727,18729],{"id":18728},"step-2-optimize-problem-for-quantum-hardware-execution","Step 2: Optimize problem for quantum hardware execution",[7047,18731,18733],{"id":18732},"efficient-hadamard-test","Efficient Hadamard test",[12,18735,18736],{},"We can optimize the deep circuits for the Hadamard test that we have obtained by introducing some approximations and relying on some assumption about the model Hamiltonian. For example, consider the following circuit for the Hadamard test:",[12,18738,18739],{},[11235,18740],{"alt":18741,"src":18742},"fig3.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-08.avif",[12,18744,18745,18746,18817,18818,18884,18885,18913,18914,18979,18980,19022,19023,19302,19303,19344,19345,19386],{},"Assume we can classically calculate ",[35,18747,18749,18767],{"className":18748},[7057],[35,18750,18752],{"className":18751},[7061],[7063,18753,18754],{"xmlns":7065},[7067,18755,18756,18764],{},[7070,18757,18758],{},[7699,18759,18760,18762],{},[7076,18761,10709],{},[7227,18763,5312],{},[7083,18765,18766],{"encoding":7085},"E_0",[35,18768,18770],{"className":18769,"ariaHidden":7091},[7090],[35,18771,18773,18776],{"className":18772},[7095],[35,18774],{"className":18775,"style":7718},[7099],[35,18777,18779,18782],{"className":18778},[7104],[35,18780,10709],{"className":18781,"style":10419},[7104,7145],[35,18783,18785],{"className":18784},[7113],[35,18786,18788,18809],{"className":18787},[7117,7731],[35,18789,18791,18806],{"className":18790},[7121],[35,18792,18795],{"className":18793,"style":18794},[7125],"height:0.3011em;",[35,18796,18797,18800],{"style":10541},[35,18798],{"className":18799,"style":7134},[7133],[35,18801,18803],{"className":18802},[7138,7139,7140,7141],[35,18804,5312],{"className":18805},[7104,7141],[35,18807,7756],{"className":18808},[7755],[35,18810,18812],{"className":18811},[7121],[35,18813,18815],{"className":18814,"style":7763},[7125],[35,18816],{},", the eigenvalue of ",[35,18819,18821,18843],{"className":18820},[7057],[35,18822,18824],{"className":18823},[7061],[7063,18825,18826],{"xmlns":7065},[7067,18827,18828,18840],{},[7070,18829,18830,18832,18834],{},[7076,18831,7283],{"mathvariant":7282},[7227,18833,5312],{},[7073,18835,18836,18838],{},[7223,18837,7290],{"stretchy":7289},[7076,18839,12207],{},[7083,18841,18842],{"encoding":7085},"|0\\rangle^N",[35,18844,18846],{"className":18845,"ariaHidden":7091},[7090],[35,18847,18849,18852,18855],{"className":18848},[7095],[35,18850],{"className":18851,"style":9778},[7099],[35,18853,14253],{"className":18854},[7104],[35,18856,18858,18861],{"className":18857},[7314],[35,18859,7290],{"className":18860},[7314],[35,18862,18864],{"className":18863},[7113],[35,18865,18867],{"className":18866},[7117],[35,18868,18870],{"className":18869},[7121],[35,18871,18873],{"className":18872,"style":7861},[7125],[35,18874,18875,18878],{"style":7129},[35,18876],{"className":18877,"style":7134},[7133],[35,18879,18881],{"className":18880},[7138,7139,7140,7141],[35,18882,12207],{"className":18883,"style":7845},[7104,7145,7141]," under the Hamiltonian ",[35,18886,18888,18901],{"className":18887},[7057],[35,18889,18891],{"className":18890},[7061],[7063,18892,18893],{"xmlns":7065},[7067,18894,18895,18899],{},[7070,18896,18897],{},[7076,18898,7669],{},[7083,18900,7669],{"encoding":7085},[35,18902,18904],{"className":18903,"ariaHidden":7091},[7090],[35,18905,18907,18910],{"className":18906},[7095],[35,18908],{"className":18909,"style":7100},[7099],[35,18911,7669],{"className":18912,"style":7684},[7104,7145],".\nThis is satisfied when the Hamiltonian preserves the U(1) symmetry. Although this may seem like a strong assumption, there are many cases where it is safe to assume that there is a vacuum state (in this case it maps to the ",[35,18915,18917,18938],{"className":18916},[7057],[35,18918,18920],{"className":18919},[7061],[7063,18921,18922],{"xmlns":7065},[7067,18923,18924,18936],{},[7070,18925,18926,18928,18930],{},[7076,18927,7283],{"mathvariant":7282},[7227,18929,5312],{},[7073,18931,18932,18934],{},[7223,18933,7290],{"stretchy":7289},[7076,18935,12207],{},[7083,18937,18842],{"encoding":7085},[35,18939,18941],{"className":18940,"ariaHidden":7091},[7090],[35,18942,18944,18947,18950],{"className":18943},[7095],[35,18945],{"className":18946,"style":9778},[7099],[35,18948,14253],{"className":18949},[7104],[35,18951,18953,18956],{"className":18952},[7314],[35,18954,7290],{"className":18955},[7314],[35,18957,18959],{"className":18958},[7113],[35,18960,18962],{"className":18961},[7117],[35,18963,18965],{"className":18964},[7121],[35,18966,18968],{"className":18967,"style":7861},[7125],[35,18969,18970,18973],{"style":7129},[35,18971],{"className":18972,"style":7134},[7133],[35,18974,18976],{"className":18975},[7138,7139,7140,7141],[35,18977,12207],{"className":18978,"style":7845},[7104,7145,7141]," state) which is unaffected by the action of the Hamiltonian. This is true for example for chemistry Hamiltonians that describe stable molecule (where the number of electrons is conserved).\nGiven that the gate ",[35,18981,18983,19001],{"className":18982},[7057],[35,18984,18986],{"className":18985},[7061],[7063,18987,18988],{"xmlns":7065},[7067,18989,18990,18998],{},[7070,18991,18992,18994,18996],{},[10694,18993,11540],{},[10694,18995,11543],{},[7076,18997,8322],{},[7083,18999,19000],{"encoding":7085},"\\text{Prep} \\; \\psi",[35,19002,19004],{"className":19003,"ariaHidden":7091},[7090],[35,19005,19007,19010,19016,19019],{"className":19006},[7095],[35,19008],{"className":19009,"style":11562},[7099],[35,19011,19013],{"className":19012},[7104,4991],[35,19014,11540],{"className":19015},[7104],[35,19017],{"className":19018,"style":7450},[7248],[35,19020,8322],{"className":19021,"style":7310},[7104,7145],", prepares the desired reference state ",[35,19024,19026,19104],{"className":19025},[7057],[35,19027,19029],{"className":19028},[7061],[7063,19030,19031],{"xmlns":7065},[7067,19032,19033,19101],{},[7070,19034,19035,19051,19053,19055,19057,19059,19067,19069,19087,19093],{},[19036,19037,19038,19040,19049],"mpadded",{},[7076,19039,7283],{"mathvariant":7282},[7070,19041,19042,19044,19047],{},[7076,19043,12],{},[7076,19045,19046],{},"s",[7076,19048,7824],{},[7223,19050,7290],{"stretchy":7289},[7223,19052,242],{},[10694,19054,11540],{},[10694,19056,11543],{},[7076,19058,8322],{},[19036,19060,19061,19063,19065],{},[7076,19062,7283],{"mathvariant":7282},[7227,19064,5312],{},[7223,19066,7290],{"stretchy":7289},[7223,19068,242],{},[7073,19070,19071,19073],{},[7076,19072,7817],{},[7070,19074,19075,19077,19079,19081,19083,19085],{},[7223,19076,7225],{},[7076,19078,7824],{},[7076,19080,7669],{},[7227,19082,5312],{},[7076,19084,9487],{},[7076,19086,7829],{},[7699,19088,19089,19091],{},[7076,19090,7810],{},[7076,19092,8322],{},[19036,19094,19095,19097,19099],{},[7076,19096,7283],{"mathvariant":7282},[7227,19098,5312],{},[7223,19100,7290],{"stretchy":7289},[7083,19102,19103],{"encoding":7085},"\\ket{psi} = \\text{Prep} \\; \\psi \\ket{0} = e^{-i H 0 dt} U_{\\psi} \\ket{0}",[35,19105,19107,19143,19188],{"className":19106,"ariaHidden":7091},[7090],[35,19108,19110,19113,19134,19137,19140],{"className":19109},[7095],[35,19111],{"className":19112,"style":7303},[7099],[35,19114,19116,19119,19131],{"className":19115},[7468],[35,19117,7283],{"className":19118},[7104],[35,19120,19122,19125,19128],{"className":19121},[7104],[35,19123,12],{"className":19124},[7104,7145],[35,19126,19046],{"className":19127},[7104,7145],[35,19129,7824],{"className":19130},[7104,7145],[35,19132,7290],{"className":19133},[7314],[35,19135],{"className":19136,"style":7450},[7248],[35,19138,242],{"className":19139},[7454],[35,19141],{"className":19142,"style":7450},[7248],[35,19144,19146,19149,19155,19158,19161,19164,19179,19182,19185],{"className":19145},[7095],[35,19147],{"className":19148,"style":7303},[7099],[35,19150,19152],{"className":19151},[7104,4991],[35,19153,11540],{"className":19154},[7104],[35,19156],{"className":19157,"style":7450},[7248],[35,19159,8322],{"className":19160,"style":7310},[7104,7145],[35,19162],{"className":19163,"style":7496},[7248],[35,19165,19167,19170,19176],{"className":19166},[7468],[35,19168,7283],{"className":19169},[7104],[35,19171,19173],{"className":19172},[7104],[35,19174,5312],{"className":19175},[7104],[35,19177,7290],{"className":19178},[7314],[35,19180],{"className":19181,"style":7450},[7248],[35,19183,242],{"className":19184},[7454],[35,19186],{"className":19187,"style":7450},[7248],[35,19189,19191,19194,19241,19284,19287],{"className":19190},[7095],[35,19192],{"className":19193,"style":15873},[7099],[35,19195,19197,19200],{"className":19196},[7104],[35,19198,7817],{"className":19199},[7104,7145],[35,19201,19203],{"className":19202},[7113],[35,19204,19206],{"className":19205},[7117],[35,19207,19209],{"className":19208},[7121],[35,19210,19212],{"className":19211,"style":8191},[7125],[35,19213,19214,19217],{"style":7129},[35,19215],{"className":19216,"style":7134},[7133],[35,19218,19220],{"className":19219},[7138,7139,7140,7141],[35,19221,19223,19226,19229,19232,19235,19238],{"className":19222},[7104,7141],[35,19224,7225],{"className":19225},[7104,7141],[35,19227,7824],{"className":19228},[7104,7145,7141],[35,19230,7669],{"className":19231,"style":7684},[7104,7145,7141],[35,19233,5312],{"className":19234},[7104,7141],[35,19236,9487],{"className":19237},[7104,7145,7141],[35,19239,7829],{"className":19240},[7104,7145,7141],[35,19242,19244,19247],{"className":19243},[7104],[35,19245,7810],{"className":19246,"style":7845},[7104,7145],[35,19248,19250],{"className":19249},[7113],[35,19251,19253,19276],{"className":19252},[7117,7731],[35,19254,19256,19273],{"className":19255},[7121],[35,19257,19259],{"className":19258,"style":15760},[7125],[35,19260,19261,19264],{"style":15941},[35,19262],{"className":19263,"style":7134},[7133],[35,19265,19267],{"className":19266},[7138,7139,7140,7141],[35,19268,19270],{"className":19269},[7104,7141],[35,19271,8322],{"className":19272,"style":7310},[7104,7145,7141],[35,19274,7756],{"className":19275},[7755],[35,19277,19279],{"className":19278},[7121],[35,19280,19282],{"className":19281,"style":11360},[7125],[35,19283],{},[35,19285],{"className":19286,"style":7496},[7248],[35,19288,19290,19293,19299],{"className":19289},[7468],[35,19291,7283],{"className":19292},[7104],[35,19294,19296],{"className":19295},[7104],[35,19297,5312],{"className":19298},[7104],[35,19300,7290],{"className":19301},[7314],", for example, to prepare the HF state for chemistry ",[35,19304,19306,19323],{"className":19305},[7057],[35,19307,19309],{"className":19308},[7061],[7063,19310,19311],{"xmlns":7065},[7067,19312,19313,19321],{},[7070,19314,19315,19317,19319],{},[10694,19316,11540],{},[10694,19318,11543],{},[7076,19320,8322],{},[7083,19322,19000],{"encoding":7085},[35,19324,19326],{"className":19325,"ariaHidden":7091},[7090],[35,19327,19329,19332,19338,19341],{"className":19328},[7095],[35,19330],{"className":19331,"style":11562},[7099],[35,19333,19335],{"className":19334},[7104,4991],[35,19336,11540],{"className":19337},[7104],[35,19339],{"className":19340,"style":7450},[7248],[35,19342,8322],{"className":19343,"style":7310},[7104,7145]," would be a product of single-qubit NOTs, so controlled-",[35,19346,19348,19365],{"className":19347},[7057],[35,19349,19351],{"className":19350},[7061],[7063,19352,19353],{"xmlns":7065},[7067,19354,19355,19363],{},[7070,19356,19357,19359,19361],{},[10694,19358,11540],{},[10694,19360,11543],{},[7076,19362,8322],{},[7083,19364,19000],{"encoding":7085},[35,19366,19368],{"className":19367,"ariaHidden":7091},[7090],[35,19369,19371,19374,19380,19383],{"className":19370},[7095],[35,19372],{"className":19373,"style":11562},[7099],[35,19375,19377],{"className":19376},[7104,4991],[35,19378,11540],{"className":19379},[7104],[35,19381],{"className":19382,"style":7450},[7248],[35,19384,8322],{"className":19385,"style":7310},[7104,7145]," is just a product of CNOTs.\nThen the circuit above implements the following state prior to measurement:",[35,19388,19390],{"className":19389},[7318],[35,19391,19393,20013],{"className":19392},[7057],[35,19394,19396],{"className":19395},[7061],[7063,19397,19398],{"xmlns":7065,"display":7327},[7067,19399,19400,20010],{},[13970,19401,19402],{"rowspacing":13972,"columnspacing":13973},[13975,19403,19404,19408,20003,20006],{},[13978,19405],{"className":19406},[19407],"mtr-glue",[13978,19409,19410],{},[13981,19411,19412],{"scriptlevel":5312,"displaystyle":7091},[13970,19413,19414,19518,19585,19672,19755,19873],{"rowspacing":16631,"columnalign":16632,"columnspacing":16633},[13975,19415,19416,19454],{},[13978,19417,19418],{},[13981,19419,19420],{"scriptlevel":5312,"displaystyle":7091},[7070,19421,19422,19430,19442],{},[19036,19423,19424,19426,19428],{},[7076,19425,7283],{"mathvariant":7282},[7227,19427,5312],{},[7223,19429,7290],{"stretchy":7289},[7073,19431,19432,19440],{},[19036,19433,19434,19436,19438],{},[7076,19435,7283],{"mathvariant":7282},[7227,19437,5312],{},[7223,19439,7290],{"stretchy":7289},[7076,19441,12207],{},[8800,19443,19444,19448],{},[7223,19445,19447],{"stretchy":7091,"minsize":19446},"3.0em","→",[19036,19449,19452],{"width":19450,"lspace":19451},"+0.6em","0.3em",[7076,19453,7669],{},[13978,19455,19456],{},[13981,19457,19458],{"scriptlevel":5312,"displaystyle":7091},[7070,19459,19460,19462,19470],{},[7070,19461],{},[14009,19463,19464,19466],{},[7227,19465,500],{},[14014,19467,19468],{},[7227,19469,460],{},[7070,19471,19472,19474,19482,19494,19496,19504,19516],{},[7223,19473,275],{"fence":7091},[19036,19475,19476,19478,19480],{},[7076,19477,7283],{"mathvariant":7282},[7227,19479,5312],{},[7223,19481,7290],{"stretchy":7289},[7073,19483,19484,19492],{},[19036,19485,19486,19488,19490],{},[7076,19487,7283],{"mathvariant":7282},[7227,19489,5312],{},[7223,19491,7290],{"stretchy":7289},[7076,19493,12207],{},[7223,19495,445],{},[19036,19497,19498,19500,19502],{},[7076,19499,7283],{"mathvariant":7282},[7227,19501,500],{},[7223,19503,7290],{"stretchy":7289},[7073,19505,19506,19514],{},[19036,19507,19508,19510,19512],{},[7076,19509,7283],{"mathvariant":7282},[7227,19511,5312],{},[7223,19513,7290],{"stretchy":7289},[7076,19515,12207],{},[7223,19517,1694],{"fence":7091},[13975,19519,19520,19533],{},[13978,19521,19522],{},[13981,19523,19524],{"scriptlevel":5312,"displaystyle":7091},[8800,19525,19526,19528],{},[7223,19527,19447],{"stretchy":7091,"minsize":19446},[19036,19529,19530],{"width":19450,"lspace":19451},[10694,19531,19532],{},"1-ctrl-init",[13978,19534,19535],{},[13981,19536,19537],{"scriptlevel":5312,"displaystyle":7091},[7070,19538,19539,19541,19549],{},[7070,19540],{},[14009,19542,19543,19545],{},[7227,19544,500],{},[14014,19546,19547],{},[7227,19548,460],{},[7070,19550,19551,19553,19555,19557,19559,19561,19563,19569,19571,19573,19575,19577,19579,19581,19583],{},[7223,19552,275],{"fence":7091},[7076,19554,7283],{"mathvariant":7282},[7227,19556,5312],{},[7223,19558,7290],{"stretchy":7289},[7076,19560,7283],{"mathvariant":7282},[7227,19562,5312],{},[7073,19564,19565,19567],{},[7223,19566,7290],{"stretchy":7289},[7076,19568,12207],{},[7223,19570,445],{},[7076,19572,7283],{"mathvariant":7282},[7227,19574,500],{},[7223,19576,7290],{"stretchy":7289},[7076,19578,7283],{"mathvariant":7282},[7076,19580,8322],{},[7223,19582,7290],{"stretchy":7289},[7223,19584,1694],{"fence":7091},[13975,19586,19587,19599],{},[13978,19588,19589],{},[13981,19590,19591],{"scriptlevel":5312,"displaystyle":7091},[8800,19592,19593,19595],{},[7223,19594,19447],{"stretchy":7091,"minsize":19446},[19036,19596,19597],{"width":19450,"lspace":19451},[7076,19598,7810],{},[13978,19600,19601],{},[13981,19602,19603],{"scriptlevel":5312,"displaystyle":7091},[7070,19604,19605,19607,19615],{},[7070,19606],{},[14009,19608,19609,19611],{},[7227,19610,500],{},[14014,19612,19613],{},[7227,19614,460],{},[7070,19616,19617,19619,19630,19638,19650,19652,19660,19662,19670],{},[7223,19618,275],{"fence":7091},[7073,19620,19621,19623],{},[7076,19622,7817],{},[7070,19624,19625,19627],{},[7076,19626,7824],{},[7076,19628,19629],{},"ϕ",[19036,19631,19632,19634,19636],{},[7076,19633,7283],{"mathvariant":7282},[7227,19635,5312],{},[7223,19637,7290],{"stretchy":7289},[7073,19639,19640,19648],{},[19036,19641,19642,19644,19646],{},[7076,19643,7283],{"mathvariant":7282},[7227,19645,5312],{},[7223,19647,7290],{"stretchy":7289},[7076,19649,12207],{},[7223,19651,445],{},[19036,19653,19654,19656,19658],{},[7076,19655,7283],{"mathvariant":7282},[7227,19657,500],{},[7223,19659,7290],{"stretchy":7289},[7076,19661,7810],{},[19036,19663,19664,19666,19668],{},[7076,19665,7283],{"mathvariant":7282},[7076,19667,8322],{},[7223,19669,7290],{"stretchy":7289},[7223,19671,1694],{"fence":7091},[13975,19673,19674,19687],{},[13978,19675,19676],{},[13981,19677,19678],{"scriptlevel":5312,"displaystyle":7091},[8800,19679,19680,19682],{},[7223,19681,19447],{"stretchy":7091,"minsize":19446},[19036,19683,19684],{"width":19450,"lspace":19451},[10694,19685,19686],{},"0-ctrl-init",[13978,19688,19689],{},[13981,19690,19691],{"scriptlevel":5312,"displaystyle":7091},[7070,19692,19693,19695,19703],{},[7070,19694],{},[14009,19696,19697,19699],{},[7227,19698,500],{},[14014,19700,19701],{},[7227,19702,460],{},[7070,19704,19705,19707,19717,19725,19733,19735,19743,19745,19753],{},[7223,19706,275],{"fence":7091},[7073,19708,19709,19711],{},[7076,19710,7817],{},[7070,19712,19713,19715],{},[7076,19714,7824],{},[7076,19716,19629],{},[19036,19718,19719,19721,19723],{},[7076,19720,7283],{"mathvariant":7282},[7227,19722,5312],{},[7223,19724,7290],{"stretchy":7289},[19036,19726,19727,19729,19731],{},[7076,19728,7283],{"mathvariant":7282},[7076,19730,8322],{},[7223,19732,7290],{"stretchy":7289},[7223,19734,445],{},[19036,19736,19737,19739,19741],{},[7076,19738,7283],{"mathvariant":7282},[7227,19740,500],{},[7223,19742,7290],{"stretchy":7289},[7076,19744,7810],{},[19036,19746,19747,19749,19751],{},[7076,19748,7283],{"mathvariant":7282},[7076,19750,8322],{},[7223,19752,7290],{"stretchy":7289},[7223,19754,1694],{"fence":7091},[13975,19756,19757,19763],{},[13978,19758,19759],{},[13981,19760,19761],{"scriptlevel":5312,"displaystyle":7091},[7223,19762,242],{"lspace":16633,"rspace":16633},[13978,19764,19765],{},[13981,19766,19767],{"scriptlevel":5312,"displaystyle":7091},[7070,19768,19769,19771,19777],{},[7070,19770],{},[14009,19772,19773,19775],{},[7227,19774,500],{},[7227,19776,460],{},[7070,19778,19779,19781,19789,19825,19827,19835,19871],{},[7223,19780,275],{"fence":7091},[19036,19782,19783,19785,19787],{},[7076,19784,7283],{"mathvariant":7282},[7223,19786,445],{"lspace":16633,"rspace":16633},[7223,19788,7290],{"stretchy":7289},[7070,19790,19791,19793,19803,19811,19813,19815,19823],{},[7223,19792,275],{"fence":7091},[7073,19794,19795,19797],{},[7076,19796,7817],{},[7070,19798,19799,19801],{},[7076,19800,7824],{},[7076,19802,19629],{},[19036,19804,19805,19807,19809],{},[7076,19806,7283],{"mathvariant":7282},[7076,19808,8322],{},[7223,19810,7290],{"stretchy":7289},[7223,19812,445],{},[7076,19814,7810],{},[19036,19816,19817,19819,19821],{},[7076,19818,7283],{"mathvariant":7282},[7076,19820,8322],{},[7223,19822,7290],{"stretchy":7289},[7223,19824,1694],{"fence":7091},[7223,19826,445],{},[19036,19828,19829,19831,19833],{},[7076,19830,7283],{"mathvariant":7282},[7223,19832,7225],{"lspace":16633,"rspace":16633},[7223,19834,7290],{"stretchy":7289},[7070,19836,19837,19839,19849,19857,19859,19861,19869],{},[7223,19838,275],{"fence":7091},[7073,19840,19841,19843],{},[7076,19842,7817],{},[7070,19844,19845,19847],{},[7076,19846,7824],{},[7076,19848,19629],{},[19036,19850,19851,19853,19855],{},[7076,19852,7283],{"mathvariant":7282},[7076,19854,8322],{},[7223,19856,7290],{"stretchy":7289},[7223,19858,7225],{},[7076,19860,7810],{},[19036,19862,19863,19865,19867],{},[7076,19864,7283],{"mathvariant":7282},[7076,19866,8322],{},[7223,19868,7290],{"stretchy":7289},[7223,19870,1694],{"fence":7091},[7223,19872,1694],{"fence":7091},[13975,19874,19875,19881],{},[13978,19876,19877],{},[13981,19878,19879],{"scriptlevel":5312,"displaystyle":7091},[7223,19880,242],{"lspace":16633,"rspace":16633},[13978,19882,19883],{},[13981,19884,19885],{"scriptlevel":5312,"displaystyle":7091},[7070,19886,19887,19889,19895],{},[7070,19888],{},[14009,19890,19891,19893],{},[7227,19892,500],{},[7227,19894,460],{},[7070,19896,19897,19899,19911,19949,19951,19963,20001],{},[7223,19898,275],{"fence":7091},[19036,19900,19901,19903,19909],{},[7076,19902,7283],{"mathvariant":7282},[7070,19904,19905,19907],{},[7223,19906,445],{},[7076,19908,7824],{},[7223,19910,7290],{"stretchy":7289},[7070,19912,19913,19915,19925,19933,19935,19937,19939,19947],{},[7223,19914,275],{"fence":7091},[7073,19916,19917,19919],{},[7076,19918,7817],{},[7070,19920,19921,19923],{},[7076,19922,7824],{},[7076,19924,19629],{},[19036,19926,19927,19929,19931],{},[7076,19928,7283],{"mathvariant":7282},[7076,19930,8322],{},[7223,19932,7290],{"stretchy":7289},[7223,19934,7225],{},[7076,19936,7824],{},[7076,19938,7810],{},[19036,19940,19941,19943,19945],{},[7076,19942,7283],{"mathvariant":7282},[7076,19944,8322],{},[7223,19946,7290],{"stretchy":7289},[7223,19948,1694],{"fence":7091},[7223,19950,445],{},[19036,19952,19953,19955,19961],{},[7076,19954,7283],{"mathvariant":7282},[7070,19956,19957,19959],{},[7223,19958,7225],{},[7076,19960,7824],{},[7223,19962,7290],{"stretchy":7289},[7070,19964,19965,19967,19977,19985,19987,19989,19991,19999],{},[7223,19966,275],{"fence":7091},[7073,19968,19969,19971],{},[7076,19970,7817],{},[7070,19972,19973,19975],{},[7076,19974,7824],{},[7076,19976,19629],{},[19036,19978,19979,19981,19983],{},[7076,19980,7283],{"mathvariant":7282},[7076,19982,8322],{},[7223,19984,7290],{"stretchy":7289},[7223,19986,445],{},[7076,19988,7824],{},[7076,19990,7810],{},[19036,19992,19993,19995,19997],{},[7076,19994,7283],{"mathvariant":7282},[7076,19996,8322],{},[7223,19998,7290],{"stretchy":7289},[7223,20000,1694],{"fence":7091},[7223,20002,1694],{"fence":7091},[13978,20004],{"className":20005},[19407],[13978,20007],{"className":20008},[20009],"mml-eqn-num",[7083,20011,20012],{"encoding":7085},"\\begin{equation}\n\\begin{split}\n    \\ket{0} \\ket{0}^N\\xrightarrow{H}&\\frac{1}{\\sqrt{2}}\n    \\left(\n    \\ket{0}\\ket{0}^N+ \\ket{1} \\ket{0}^N\n    \\right)\\\\\n    \\xrightarrow{\\text{1-ctrl-init}}&\\frac{1}{\\sqrt{2}}\\left(|0\\rangle|0\\rangle^N+|1\\rangle|\\psi\\rangle\\right)\\\\\n    \\xrightarrow{U}&\\frac{1}{\\sqrt{2}}\\left(e^{i\\phi}\\ket{0}\\ket{0}^N+\\ket{1} U\\ket{\\psi}\\right)\\\\\n    \\xrightarrow{\\text{0-ctrl-init}}&\\frac{1}{\\sqrt{2}}\n    \\left(\n    e^{i\\phi}\\ket{0} \\ket{\\psi}\n    +\\ket{1} U\\ket{\\psi}\n    \\right)\\\\\n    =&\\frac{1}{2}\n    \\left(\n    \\ket{+}\\left(e^{i\\phi}\\ket{\\psi}+U\\ket{\\psi}\\right)\n    +\\ket{-}\\left(e^{i\\phi}\\ket{\\psi}-U\\ket{\\psi}\\right)\n    \\right)\\\\\n    =&\\frac{1}{2}\n    \\left(\n    \\ket{+i}\\left(e^{i\\phi}\\ket{\\psi}-iU\\ket{\\psi}\\right)\n    +\\ket{-i}\\left(e^{i\\phi}\\ket{\\psi}+iU\\ket{\\psi}\\right)\n    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1,21914,21920],{"className":21910},[7468],[35,21912,7283],{"className":21913},[7104],[35,21915,21917],{"className":21916},[7104],[35,21918,8322],{"className":21919,"style":7310},[7104,7145],[35,21921,7290],{"className":21922},[7314],[35,21924],{"className":21925,"style":7249},[7248],[35,21927,7225],{"className":21928},[7253],[35,21930],{"className":21931,"style":7249},[7248],[35,21933,7824],{"className":21934},[7104,7145],[35,21936,7810],{"className":21937,"style":7845},[7104,7145],[35,21939],{"className":21940,"style":7496},[7248],[35,21942,21944,21947,21953],{"className":21943},[7468],[35,21945,7283],{"className":21946},[7104],[35,21948,21950],{"className":21949},[7104],[35,21951,8322],{"className":21952,"style":7310},[7104,7145],[35,21954,7290],{"className":21955},[7314],[35,21957,21959],{"className":21958,"style":7474},[7314,7473],[35,21960,1694],{"className":21961},[7478,7479],[35,21963],{"className":21964,"style":7249},[7248],[35,21966,445],{"className":21967},[7253],[35,21969],{"className":21970,"style":7249},[7248],[35,21972,21974,21977,21986],{"className":21973},[7468],[35,21975,7283],{"className":21976},[7104],[35,21978,21980,21983],{"className":21979},[7104],[35,21981,7225],{"className":21982},[7104],[35,21984,7824],{"className":21985},[7104,7145],[35,21987,7290],{"className":21988},[7314],[35,21990],{"className":21991,"style":7496},[7248],[35,21993,21995,22001,22036,22039,22054,22057,22060,22063,22066,22069,22072,22087],{"className":21994},[7468],[35,21996,21998],{"className":21997,"style":7474},[7472,7473],[35,21999,275],{"className":22000},[7478,7479],[35,22002,22004,22007],{"className":22003},[7104],[35,22005,7817],{"className":22006},[7104,7145],[35,22008,22010],{"className":22009},[7113],[35,22011,22013],{"className":22012},[7117],[35,22014,22016],{"className":22015},[7121],[35,22017,22019],{"className":22018,"style":9568},[7125],[35,22020,22021,22024],{"style":7437},[35,22022],{"className":22023,"style":7134},[7133],[35,22025,22027],{"className":22026},[7138,7139,7140,7141],[35,22028,22030,22033],{"className":22029},[7104,7141],[35,22031,7824],{"className":22032},[7104,7145,7141],[35,22034,19629],{"className":22035},[7104,7145,7141],[35,22037],{"className":22038,"style":7496},[7248],[35,22040,22042,22045,22051],{"className":22041},[7468],[35,22043,7283],{"className":22044},[7104],[35,22046,22048],{"className":22047},[7104],[35,22049,8322],{"className":22050,"style":7310},[7104,7145],[35,22052,7290],{"className":22053},[7314],[35,22055],{"className":22056,"style":7249},[7248],[35,22058,445],{"className":22059},[7253],[35,22061],{"className":22062,"style":7249},[7248],[35,22064,7824],{"className":22065},[7104,7145],[35,22067,7810],{"className":22068,"style":7845},[7104,7145],[35,22070],{"className":22071,"style":7496},[7248],[35,22073,22075,22078,22084],{"className":22074},[7468],[35,22076,7283],{"className":22077},[7104],[35,22079,22081],{"className":22080},[7104],[35,22082,8322],{"className":22083,"style":7310},[7104,7145],[35,22085,7290],{"className":22086},[7314],[35,22088,22090],{"className":22089,"style":7474},[7314,7473],[35,22091,1694],{"className":22092},[7478,7479],[35,22094,22096],{"className":22095,"style":7474},[7314,7473],[35,22097,1694],{"className":22098},[7478,7479],[35,22100,7756],{"className":22101},[7755],[35,22103,22105],{"className":22104},[7121],[35,22106,22108],{"className":22107,"style":20415},[7125],[35,22109],{},[35,22111,7756],{"className":22112},[7755],[35,22114,22116],{"className":22115},[7121],[35,22117,22119],{"className":22118,"style":20415},[7125],[35,22120],{},[35,22122,22125],{"className":22123},[22124],"tag",[35,22126,22128,22146],{"className":22127},[7117,7731],[35,22129,22131,22143],{"className":22130},[7121],[35,22132,22134],{"className":22133,"style":20038},[7125],[35,22135,22136,22139],{"style":20041},[35,22137],{"className":22138,"style":20045},[7133],[35,22140],{"className":22141},[22142],"eqn-num",[35,22144,7756],{"className":22145},[7755],[35,22147,22149],{"className":22148},[7121],[35,22150,22152],{"className":22151,"style":20415},[7125],[35,22153],{},[12,22155,22156,22157,22361],{},"where we have used the classical simulable phase shift ",[35,22158,22160,22210],{"className":22159},[7057],[35,22161,22163],{"className":22162},[7061],[7063,22164,22165],{"xmlns":7065},[7067,22166,22167,22207],{},[7070,22168,22169,22171,22183,22185,22195],{},[7076,22170,7810],{},[7073,22172,22173,22181],{},[19036,22174,22175,22177,22179],{},[7076,22176,7283],{"mathvariant":7282},[7227,22178,5312],{},[7223,22180,7290],{"stretchy":7289},[7076,22182,12207],{},[7223,22184,242],{},[7073,22186,22187,22189],{},[7076,22188,7817],{},[7070,22190,22191,22193],{},[7076,22192,7824],{},[7076,22194,19629],{},[7073,22196,22197,22205],{},[19036,22198,22199,22201,22203],{},[7076,22200,7283],{"mathvariant":7282},[7227,22202,5312],{},[7223,22204,7290],{"stretchy":7289},[7076,22206,12207],{},[7083,22208,22209],{"encoding":7085}," U\\ket{0}^N = e^{i\\phi}\\ket{0}^N",[35,22211,22213,22276],{"className":22212,"ariaHidden":7091},[7090],[35,22214,22216,22220,22223,22226,22267,22270,22273],{"className":22215},[7095],[35,22217],{"className":22218,"style":22219},[7099],"height:1.2312em;vertical-align:-0.25em;",[35,22221,7810],{"className":22222,"style":7845},[7104,7145],[35,22224],{"className":22225,"style":7496},[7248],[35,22227,22229,22244],{"className":22228},[7468],[35,22230,22232,22235,22241],{"className":22231},[7468],[35,22233,7283],{"className":22234},[7104],[35,22236,22238],{"className":22237},[7104],[35,22239,5312],{"className":22240},[7104],[35,22242,7290],{"className":22243},[7314],[35,22245,22247],{"className":22246},[7113],[35,22248,22250],{"className":22249},[7117],[35,22251,22253],{"className":22252},[7121],[35,22254,22256],{"className":22255,"style":12348},[7125],[35,22257,22258,22261],{"style":12371},[35,22259],{"className":22260,"style":7134},[7133],[35,22262,22264],{"className":22263},[7138,7139,7140,7141],[35,22265,12207],{"className":22266,"style":7845},[7104,7145,7141],[35,22268],{"className":22269,"style":7450},[7248],[35,22271,242],{"className":22272},[7454],[35,22274],{"className":22275,"style":7450},[7248],[35,22277,22279,22282,22317,22320],{"className":22278},[7095],[35,22280],{"className":22281,"style":22219},[7099],[35,22283,22285,22288],{"className":22284},[7104],[35,22286,7817],{"className":22287},[7104,7145],[35,22289,22291],{"className":22290},[7113],[35,22292,22294],{"className":22293},[7117],[35,22295,22297],{"className":22296},[7121],[35,22298,22300],{"className":22299,"style":8191},[7125],[35,22301,22302,22305],{"style":7129},[35,22303],{"className":22304,"style":7134},[7133],[35,22306,22308],{"className":22307},[7138,7139,7140,7141],[35,22309,22311,22314],{"className":22310},[7104,7141],[35,22312,7824],{"className":22313},[7104,7145,7141],[35,22315,19629],{"className":22316},[7104,7145,7141],[35,22318],{"className":22319,"style":7496},[7248],[35,22321,22323,22338],{"className":22322},[7468],[35,22324,22326,22329,22335],{"className":22325},[7468],[35,22327,7283],{"className":22328},[7104],[35,22330,22332],{"className":22331},[7104],[35,22333,5312],{"className":22334},[7104],[35,22336,7290],{"className":22337},[7314],[35,22339,22341],{"className":22340},[7113],[35,22342,22344],{"className":22343},[7117],[35,22345,22347],{"className":22346},[7121],[35,22348,22350],{"className":22349,"style":12348},[7125],[35,22351,22352,22355],{"style":12371},[35,22353],{"className":22354,"style":7134},[7133],[35,22356,22358],{"className":22357},[7138,7139,7140,7141],[35,22359,12207],{"className":22360,"style":7845},[7104,7145,7141]," in the third line. Therefore the expectation values are obtained as",[35,22363,22365],{"className":22364},[7318],[35,22366,22368,22678],{"className":22367},[7057],[35,22369,22371],{"className":22370},[7061],[7063,22372,22373],{"xmlns":7065,"display":7327},[7067,22374,22375,22675],{},[13970,22376,22377],{"rowspacing":13972,"columnspacing":13973},[13975,22378,22379,22382,22669,22672],{},[13978,22380],{"className":22381},[19407],[13978,22383,22384],{},[13981,22385,22386],{"scriptlevel":5312,"displaystyle":7091},[13970,22387,22388,22506,22609],{"rowspacing":16631,"columnalign":16632,"columnspacing":16633},[13975,22389,22390,22407],{},[13978,22391,22392],{},[13981,22393,22394],{"scriptlevel":5312,"displaystyle":7091},[7070,22395,22396,22398,22400,22403,22405],{},[7223,22397,8953],{"stretchy":7289},[7076,22399,12257],{},[7223,22401,22402],{},"⊗",[7076,22404,7705],{},[7223,22406,7290],{"stretchy":7289},[13978,22408,22409],{},[13981,22410,22411],{"scriptlevel":5312,"displaystyle":7091},[7070,22412,22413,22415,22417,22423,22425,22468,22470],{},[7070,22414],{},[7223,22416,242],{},[14009,22418,22419,22421],{},[7227,22420,500],{},[7227,22422,528],{},[7223,22424,275],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[7070,22426,22427,22429,22441,22449,22451,22459,22466],{},[7223,22428,275],{"fence":7091},[7073,22430,22431,22433],{},[7076,22432,7817],{},[7070,22434,22435,22437,22439],{},[7223,22436,7225],{},[7076,22438,7824],{},[7076,22440,19629],{},[19036,22442,22443,22445,22447],{},[7223,22444,8953],{"stretchy":7289},[7076,22446,8322],{},[7076,22448,7283],{"mathvariant":7282},[7223,22450,445],{},[19036,22452,22453,22455,22457],{},[7223,22454,8953],{"stretchy":7289},[7076,22456,8322],{},[7076,22458,7283],{"mathvariant":7282},[7073,22460,22461,22463],{},[7076,22462,7810],{},[7223,22464,22465],{},"†",[7223,22467,1694],{"fence":7091},[7076,22469,7705],{},[7070,22471,22472,22474,22484,22492,22494,22496,22504],{},[7223,22473,275],{"fence":7091},[7073,22475,22476,22478],{},[7076,22477,7817],{},[7070,22479,22480,22482],{},[7076,22481,7824],{},[7076,22483,19629],{},[19036,22485,22486,22488,22490],{},[7076,22487,7283],{"mathvariant":7282},[7076,22489,8322],{},[7223,22491,7290],{"stretchy":7289},[7223,22493,445],{},[7076,22495,7810],{},[19036,22497,22498,22500,22502],{},[7076,22499,7283],{"mathvariant":7282},[7076,22501,8322],{},[7223,22503,7290],{"stretchy":7289},[7223,22505,1694],{"fence":7091},[13975,22507,22508,22514],{},[13978,22509,22510],{},[13981,22511,22512],{"scriptlevel":5312,"displaystyle":7091},[7070,22513],{},[13978,22515,22516],{},[13981,22517,22518],{"scriptlevel":5312,"displaystyle":7091},[7070,22519,22520,22522,22525,22527,22569,22571,22607],{},[7070,22521],{},[7248,22523],{"width":22524},"2em",[7223,22526,7225],{},[7070,22528,22529,22531,22543,22551,22553,22561,22567],{},[7223,22530,275],{"fence":7091},[7073,22532,22533,22535],{},[7076,22534,7817],{},[7070,22536,22537,22539,22541],{},[7223,22538,7225],{},[7076,22540,7824],{},[7076,22542,19629],{},[19036,22544,22545,22547,22549],{},[7223,22546,8953],{"stretchy":7289},[7076,22548,8322],{},[7076,22550,7283],{"mathvariant":7282},[7223,22552,7225],{},[19036,22554,22555,22557,22559],{},[7223,22556,8953],{"stretchy":7289},[7076,22558,8322],{},[7076,22560,7283],{"mathvariant":7282},[7073,22562,22563,22565],{},[7076,22564,7810],{},[7223,22566,22465],{},[7223,22568,1694],{"fence":7091},[7076,22570,7705],{},[7070,22572,22573,22575,22585,22593,22595,22597,22605],{},[7223,22574,275],{"fence":7091},[7073,22576,22577,22579],{},[7076,22578,7817],{},[7070,22580,22581,22583],{},[7076,22582,7824],{},[7076,22584,19629],{},[19036,22586,22587,22589,22591],{},[7076,22588,7283],{"mathvariant":7282},[7076,22590,8322],{},[7223,22592,7290],{"stretchy":7289},[7223,22594,7225],{},[7076,22596,7810],{},[19036,22598,22599,22601,22603],{},[7076,22600,7283],{"mathvariant":7282},[7076,22602,8322],{},[7223,22604,7290],{"stretchy":7289},[7223,22606,1694],{"fence":7091},[7223,22608,1694],{"fence":7289,"stretchy":7091,"minsize":14020,"maxsize":14020},[13975,22610,22611,22617],{},[13978,22612,22613],{},[13981,22614,22615],{"scriptlevel":5312,"displaystyle":7091},[7070,22616],{},[13978,22618,22619],{},[13981,22620,22621],{"scriptlevel":5312,"displaystyle":7091},[7070,22622,22623,22625,22627,22629,22667],{},[7070,22624],{},[7223,22626,242],{},[10694,22628,16761],{},[7070,22630,22631,22633,22645,22653,22655,22657,22665],{},[7223,22632,1254],{"fence":7091},[7073,22634,22635,22637],{},[7076,22636,7817],{},[7070,22638,22639,22641,22643],{},[7223,22640,7225],{},[7076,22642,7824],{},[7076,22644,19629],{},[19036,22646,22647,22649,22651],{},[7223,22648,8953],{"stretchy":7289},[7076,22650,8322],{},[7076,22652,7283],{"mathvariant":7282},[7076,22654,7705],{},[7076,22656,7810],{},[19036,22658,22659,22661,22663],{},[7076,22660,7283],{"mathvariant":7282},[7076,22662,8322],{},[7223,22664,7290],{"stretchy":7289},[7223,22666,16788],{"fence":7091},[7223,22668,2877],{"separator":7091},[13978,22670],{"className":22671},[19407],[13978,22673],{"className":22674},[20009],[7083,22676,22677],{"encoding":7085},"\\begin{equation}\n\\begin{split}\n    \\langle X\\otimes P\\rangle&=\\frac{1}{4}\n    \\Big(\n    \\left(e^{-i\\phi}\\bra{\\psi}+\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}+U\\ket{\\psi}\\right)\n    \\\\\n    &\\qquad-\\left(e^{-i\\phi}\\bra{\\psi}-\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}-U\\ket{\\psi}\\right)\n    \\Big)\\\\\n    &=\\text{Re}\\left[e^{-i\\phi}\\bra{\\psi}PU\\ket{\\psi}\\right],\n\\end{split}\n\\end{equation}",[35,22679,22681,23550],{"className":22680,"ariaHidden":7091},[7090],[35,22682,22684,22688],{"className":22683},[7095],[35,22685],{"className":22686,"style":22687},[7099],"height:5.6666em;vertical-align:-2.5833em;",[35,22689,22691],{"className":22690},[13970],[35,22692,22694],{"className":22693},[14229],[35,22695,22697,23542],{"className":22696},[7117,7731],[35,22698,22700,23539],{"className":22699},[7121],[35,22701,22704],{"className":22702,"style":22703},[7125],"height:3.0833em;",[35,22705,22707,22711],{"style":22706},"top:-5.0833em;",[35,22708],{"className":22709,"style":22710},[7133],"height:5.0833em;",[35,22712,22714],{"className":22713},[7104],[35,22715,22717],{"className":22716},[7104],[35,22718,22720,22791],{"className":22719},[13970],[35,22721,22723],{"className":22722},[16842],[35,22724,22726,22782],{"className":22725},[7117,7731],[35,22727,22729,22779],{"className":22728},[7121],[35,22730,22732,22761,22770],{"className":22731,"style":22703},[7125],[35,22733,22734,22737],{"style":22706},[35,22735],{"className":22736,"style":14246},[7133],[35,22738,22740,22743,22746,22749,22752,22755,22758],{"className":22739},[7104],[35,22741,8953],{"className":22742},[7472],[35,22744,12257],{"className":22745,"style":12402},[7104,7145],[35,22747],{"className":22748,"style":7249},[7248],[35,22750,22402],{"className":22751},[7253],[35,22753],{"className":22754,"style":7249},[7248],[35,22756,7705],{"className":22757,"style":7751},[7104,7145],[35,22759,7290],{"className":22760},[7314],[35,22762,22764,22767],{"style":22763},"top:-2.9473em;",[35,22765],{"className":22766,"style":14246},[7133],[35,22768],{"className":22769},[7104],[35,22771,22773,22776],{"style":22772},"top:-1.0982em;",[35,22774],{"className":22775,"style":14246},[7133],[35,22777],{"className":22778},[7104],[35,22780,7756],{"className":22781},[7755],[35,22783,22785],{"className":22784},[7121],[35,22786,22789],{"className":22787,"style":22788},[7125],"height:2.5833em;",[35,22790],{},[35,22792,22794],{"className":22793},[16903],[35,22795,22797,23531],{"className":22796},[7117,7731],[35,22798,22800,23528],{"className":22799},[7121],[35,22801,22803,23128,23395],{"className":22802,"style":22703},[7125],[35,22804,22805,22808],{"style":22706},[35,22806],{"className":22807,"style":14246},[7133],[35,22809,22811,22814,22817,22820,22823,22885,22891,22894,23021,23024,23027,23030],{"className":22810},[7104],[35,22812],{"className":22813},[7104],[35,22815],{"className":22816,"style":7450},[7248],[35,22818,242],{"className":22819},[7454],[35,22821],{"className":22822,"style":7450},[7248],[35,22824,22826,22829,22882],{"className":22825},[7104],[35,22827],{"className":22828},[7472,14311],[35,22830,22832],{"className":22831},[14009],[35,22833,22835,22874],{"className":22834},[7117,7731],[35,22836,22838,22871],{"className":22837},[7121],[35,22839,22841,22852,22860],{"className":22840,"style":14324},[7125],[35,22842,22843,22846],{"style":16234},[35,22844],{"className":22845,"style":8839},[7133],[35,22847,22849],{"className":22848},[7104],[35,22850,528],{"className":22851},[7104],[35,22853,22854,22857],{"style":14395},[35,22855],{"className":22856,"style":8839},[7133],[35,22858],{"className":22859,"style":14403},[14402],[35,22861,22862,22865],{"style":14406},[35,22863],{"className":22864,"style":8839},[7133],[35,22866,22868],{"className":22867},[7104],[35,22869,500],{"className":22870},[7104],[35,22872,7756],{"className":22873},[7755],[35,22875,22877],{"className":22876},[7121],[35,22878,22880],{"className":22879,"style":16272},[7125],[35,22881],{},[35,22883],{"className":22884},[7314,14311],[35,22886,22888],{"className":22887},[7104],[35,22889,275],{"className":22890},[7478,14437],[35,22892],{"className":22893,"style":7496},[7248],[35,22895,22897,22903,22941,22944,22959,22962,22965,22968,22983,22986,23015],{"className":22896},[7468],[35,22898,22900],{"className":22899,"style":7474},[7472,7473],[35,22901,275],{"className":22902},[7478,7479],[35,22904,22906,22909],{"className":22905},[7104],[35,22907,7817],{"className":22908},[7104,7145],[35,22910,22912],{"className":22911},[7113],[35,22913,22915],{"className":22914},[7117],[35,22916,22918],{"className":22917},[7121],[35,22919,22921],{"className":22920,"style":9568},[7125],[35,22922,22923,22926],{"style":7437},[35,22924],{"className":22925,"style":7134},[7133],[35,22927,22929],{"className":22928},[7138,7139,7140,7141],[35,22930,22932,22935,22938],{"className":22931},[7104,7141],[35,22933,7225],{"className":22934},[7104,7141],[35,22936,7824],{"className":22937},[7104,7145,7141],[35,22939,19629],{"className":22940},[7104,7145,7141],[35,22942],{"className":22943,"style":7496},[7248],[35,22945,22947,22950,22956],{"className":22946},[7468],[35,22948,8953],{"className":22949},[7472],[35,22951,22953],{"className":22952},[7104],[35,22954,8322],{"className":22955,"style":7310},[7104,7145],[35,22957,7283],{"className":22958},[7104],[35,229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{},[13981,23844,23845],{"scriptlevel":5312,"displaystyle":7091},[7070,23846,23847,23849,23851,23853,23891],{},[7070,23848],{},[7223,23850,242],{},[10694,23852,18019],{},[7070,23854,23855,23857,23869,23877,23879,23881,23889],{},[7223,23856,1254],{"fence":7091},[7073,23858,23859,23861],{},[7076,23860,7817],{},[7070,23862,23863,23865,23867],{},[7223,23864,7225],{},[7076,23866,7824],{},[7076,23868,19629],{},[19036,23870,23871,23873,23875],{},[7223,23872,8953],{"stretchy":7289},[7076,23874,8322],{},[7076,23876,7283],{"mathvariant":7282},[7076,23878,7705],{},[7076,23880,7810],{},[19036,23882,23883,23885,23887],{},[7076,23884,7283],{"mathvariant":7282},[7076,23886,8322],{},[7223,23888,7290],{"stretchy":7289},[7223,23890,16788],{"fence":7091},[7076,23892,5122],{"mathvariant":7282},[13978,23894],{"className":23895},[19407],[13978,23897],{"className":23898},[20009],[7083,23900,23901],{"encoding":7085},"\\begin{equation}\n\\begin{split}\n    \\langle Y\\otimes P\\rangle&=\\frac{1}{4}\n    \\Big(\n    \\left(e^{-i\\phi}\\bra{\\psi}+i\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}-iU\\ket{\\psi}\\right)\n    \\\\\n    &\\qquad-\\left(e^{-i\\phi}\\bra{\\psi}-i\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}+iU\\ket{\\psi}\\right)\n    \\Big)\\\\\n    &=\\text{Im}\\left[e^{-i\\phi}\\bra{\\psi}PU\\ket{\\psi}\\right].\n\\end{split}\n\\end{equation}",[35,23903,23905,24784],{"className":23904,"ariaHidden":7091},[7090],[35,23906,23908,23911],{"className":23907},[7095],[35,23909],{"className":23910,"style":22687},[7099],[35,23912,23914],{"className":23913},[13970],[35,23915,23917],{"className":23916},[14229],[35,23918,23920,24776],{"className":23919},[7117,7731],[35,23921,23923,24773],{"className":23922},[7121],[35,23924,23926],{"className":23925,"style":22703},[7125],[35,23927,23928,23931],{"style":22706},[35,23929],{"className":23930,"style":22710},[7133],[35,23932,23934],{"className":23933},[7104],[35,23935,23937],{"className":23936},[7104],[35,23938,23940,24008],{"className":23939},[13970],[35,23941,23943],{"className":23942},[16842],[35,23944,23946,24000],{"className":23945},[7117,7731],[35,23947,23949,23997],{"className":23948},[7121],[35,23950,23952,23981,23989],{"className":23951,"style":22703},[7125],[35,23953,23954,23957],{"style":22706},[35,23955],{"className":23956,"style":14246},[7133],[35,23958,23960,23963,23966,23969,23972,23975,23978],{"className":23959},[7104],[35,23961,8953],{"className":23962},[7472],[35,23964,12272],{"className":23965,"style":7249},[7104,7145],[35,23967],{"className":23968,"style":7249},[7248],[35,23970,22402],{"className":23971},[7253],[35,23973],{"className":23974,"style":7249},[7248],[35,23976,7705],{"className":23977,"style":7751},[7104,7145],[35,23979,7290],{"className":23980},[7314],[35,23982,23983,23986],{"style":22763},[35,23984],{"className":23985,"style":14246},[7133],[35,23987],{"className":23988},[7104],[35,23990,23991,23994],{"style":22772},[35,23992],{"className":23993,"style":14246},[7133],[35,23995],{"className":23996},[7104],[35,23998,7756],{"className":23999},[7755],[35,24001,24003],{"className":24002},[7121],[35,24004,24006],{"className":24005,"style":22788},[7125],[35,24007],{},[35,24009,24011],{"className":24010},[16903],[35,24012,24014,24765],{"className":24013},[7117,7731],[35,24015,24017,24762],{"className":24016},[7121],[35,24018,24020,24354,24629],{"className":24019,"style":22703},[7125],[35,24021,24022,24025],{"style":22706},[35,24023],{"className":24024,"style":14246},[7133],[35,24026,24028,24031,24034,24037,24040,24102,24108,24111,24244,24247,24250,24253],{"className":24027},[7104],[35,24029],{"className":24030},[7104],[35,24032],{"className":24033,"style":7450},[7248],[35,24035,242],{"className":24036},[7454],[35,24038],{"className":24039,"style":7450},[7248],[35,24041,24043,24046,24099],{"className":24042},[7104],[35,24044],{"className":24045},[7472,14311],[35,24047,24049],{"className":24048},[14009],[35,24050,24052,24091],{"className":24051},[7117,7731],[35,24053,24055,24088],{"className":24054},[7121],[35,24056,24058,24069,24077],{"className":24057,"style":14324},[7125],[35,24059,24060,24063],{"style":16234},[35,24061],{"className":24062,"style":8839},[7133],[35,24064,24066],{"className":24065},[7104],[35,24067,528],{"className":24068},[7104],[35,24070,24071,24074],{"style":14395},[35,24072],{"className":24073,"style":8839},[7133],[35,24075],{"className":24076,"style":14403},[14402],[35,24078,24079,24082],{"style":14406},[35,24080],{"className":24081,"style":8839},[7133],[35,24083,24085],{"className":24084},[7104],[35,24086,500],{"className":24087},[7104],[35,24089,7756],{"className":24090},[7755],[35,24092,24094],{"className":24093},[7121],[35,24095,24097],{"className":24096,"style":16272},[7125],[35,24098],{},[35,24100],{"className":24101},[7314,14311],[35,24103,24105],{"className":24104},[7104],[35,24106,275],{"className":24107},[7478,14437],[35,24109],{"className":24110,"style":7496},[7248],[35,24112,24114,24120,24158,24161,24176,24179,24182,24185,24188,24191,24206,24209,24238],{"className":24113},[7468],[35,24115,24117],{"className":24116,"style":7474},[7472,7473],[35,24118,275],{"className":24119},[7478,7479],[35,24121,24123,24126],{"className":24122},[7104],[35,24124,7817],{"className":24125},[7104,7145],[35,24127,24129],{"className":24128},[7113],[35,24130,24132],{"className":24131},[7117],[35,24133,24135],{"className":24134},[7121],[35,24136,24138],{"className":24137,"style":9568},[7125],[35,24139,24140,24143],{"style":7437},[35,24141],{"className":24142,"style":7134},[7133],[35,24144,24146],{"className":24145},[7138,7139,7140,7141],[35,24147,24149,24152,24155],{"className":24148},[7104,7141],[35,24150,7225],{"className":24151},[7104,7141],[35,24153,7824],{"className":24154},[7104,7145,7141],[35,24156,19629],{"className":24157},[7104,7145,7141],[35,24159],{"className":24160,"style":7496},[7248],[35,24162,24164,24167,24173],{"className":24163},[7468],[35,24165,8953],{"className":24166},[7472],[35,24168,24170],{"className":24169},[7104],[35,24171,8322],{"className":24172,"style":7310},[7104,7145],[35,24174,7283],{"className":24175},[7104],[35,24177],{"className":24178,"style":7249},[7248],[35,24180,445],{"className":24181},[7253],[35,24183],{"className":24184,"style":7249},[7248],[35,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assName":24755},[7478,7479],[35,24757],{"className":24758,"style":7496},[7248],[35,24760,5122],{"className":24761},[7104],[35,24763,7756],{"className":24764},[7755],[35,24766,24768],{"className":24767},[7121],[35,24769,24771],{"className":24770,"style":22788},[7125],[35,24772],{},[35,24774,7756],{"className":24775},[7755],[35,24777,24779],{"className":24778},[7121],[35,24780,24782],{"className":24781,"style":22788},[7125],[35,24783],{},[35,24785,24787],{"className":24786},[22124],[35,24788,24790,24807],{"className":24789},[7117,7731],[35,24791,24793,24804],{"className":24792},[7121],[35,24794,24796],{"className":24795,"style":22703},[7125],[35,24797,24798,24801],{"style":22706},[35,24799],{"className":24800,"style":22710},[7133],[35,24802],{"className":24803},[22142],[35,24805,7756],{"className":24806},[7755],[35,24808,24810],{"className":24809},[7121],[35,24811,24813],{"className":24812,"style":22788},[7125],[35,24814],{},[12,24816,24817,24818,24859],{},"Using these assumptions we were able to write the expectation values of operators of interest with fewer controlled operations. In fact, we only need to implement the controlled state preparation ",[35,24819,24821,24838],{"className":24820},[7057],[35,24822,24824],{"className":24823},[7061],[7063,24825,24826],{"xmlns":7065},[7067,24827,24828,24836],{},[7070,24829,24830,24832,24834],{},[10694,24831,11540],{},[10694,24833,11543],{},[7076,24835,8322],{},[7083,24837,19000],{"encoding":7085},[35,24839,24841],{"className":24840,"ariaHidden":7091},[7090],[35,24842,24844,24847,24853,24856],{"className":24843},[7095],[35,24845],{"className":24846,"style":11562},[7099],[35,24848,24850],{"className":24849},[7104,4991],[35,24851,11540],{"className":24852},[7104],[35,24854],{"className":24855,"style":7450},[7248],[35,24857,8322],{"className":24858,"style":7310},[7104,7145]," and not controlled time evolutions. Reframing our calculation as above will allow us to greatly reduce the depth of the resulting circuits.",[7047,24861,24863],{"id":24862},"decompose-time-evolution-operator-with-trotter-decomposition","Decompose time-evolution operator with Trotter decomposition",[12,24865,24866,24867,288,24948,288,25029,25110,25111,25140,25141,25265,25266,25323,25324,25357,25358,25407,25408,25411],{},"Instead of implementing the time-evolution operator exactly we can use the Trotter decomposition to implement an approximation of it. Repeating several times a certain order Trotter decomposition gives us further reduction of the error introduced from the approximation. In the following, we directly build the Trotter implementation in the most efficient way for the interaction graph of the Hamiltonian we are considering (nearest neighbor interactions only). In practice we insert Pauli rotations ",[35,24868,24870,24893],{"className":24869},[7057],[35,24871,24873],{"className":24872},[7061],[7063,24874,24875],{"xmlns":7065},[7067,24876,24877,24890],{},[7070,24878,24879],{},[7699,24880,24881,24884],{},[7076,24882,24883],{},"R",[7070,24885,24886,24888],{},[7076,24887,13657],{},[7076,24889,13657],{},[7083,24891,24892],{"encoding":7085},"R_{xx}",[35,24894,24896],{"className":24895,"ariaHidden":7091},[7090],[35,24897,24899,24902],{"className":24898},[7095],[35,24900],{"className":24901,"style":7718},[7099],[35,24903,24905,24909],{"className":24904},[7104],[35,24906,24883],{"className":24907,"style":24908},[7104,7145],"margin-right:0.0077em;",[35,24910,24912],{"className":24911},[7113],[35,24913,24915,24940],{"className":24914},[7117,7731],[35,24916,24918,24937],{"className":24917},[7121],[35,24919,24921],{"className":24920,"style":9035},[7125],[35,24922,24924,24927],{"style":24923},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[35,24925],{"className":24926,"style":7134},[7133],[35,24928,24930],{"className":24929},[7138,7139,7140,7141],[35,24931,24933],{"className":24932},[7104,7141],[35,24934,24936],{"className":24935},[7104,7145,7141],"xx",[35,24938,7756],{"className":24939},[7755],[35,24941,24943],{"className":24942},[7121],[35,24944,24946],{"className":24945,"style":7763},[7125],[35,24947],{},[35,24949,24951,24974],{"className":24950},[7057],[35,24952,24954],{"className":24953},[7061],[7063,24955,24956],{"xmlns":7065},[7067,24957,24958,24971],{},[7070,24959,24960],{},[7699,24961,24962,24964],{},[7076,24963,24883],{},[7070,24965,24966,24969],{},[7076,24967,24968],{},"y",[7076,24970,24968],{},[7083,24972,24973],{"encoding":7085},"R_{yy}",[35,24975,24977],{"className":24976,"ariaHidden":7091},[7090],[35,24978,24980,24983],{"className":24979},[7095],[35,24981],{"className":24982,"style":12493},[7099],[35,24984,24986,24989],{"className":24985},[7104],[35,24987,24883],{"className":24988,"style":24908},[7104,7145],[35,24990,24992],{"className":24991},[7113],[35,24993,24995,25021],{"className":24994},[7117,7731],[35,24996,24998,25018],{"className":24997},[7121],[35,24999,25001],{"className":25000,"style":9035},[7125],[35,25002,25003,25006],{"style":24923},[35,25004],{"className":25005,"style":7134},[7133],[35,25007,25009],{"className":25008},[7138,7139,7140,7141],[35,25010,25012,25015],{"className":25011},[7104,7141],[35,25013,24968],{"className":25014,"style":7310},[7104,7145,7141],[35,25016,24968],{"className":25017,"style":7310},[7104,7145,7141],[35,25019,7756],{"className":25020},[7755],[35,25022,25024],{"className":25023},[7121],[35,25025,25027],{"className":25026,"style":11360},[7125],[35,25028],{},[35,25030,25032,25054],{"className":25031},[7057],[35,25033,25035],{"className":25034},[7061],[7063,25036,25037],{"xmlns":7065},[7067,25038,25039,25051],{},[7070,25040,25041],{},[7699,25042,25043,25045],{},[7076,25044,24883],{},[7070,25046,25047,25049],{},[7076,25048,300],{},[7076,25050,300],{},[7083,25052,25053],{"encoding":7085},"R_{zz}",[35,25055,25057],{"className":25056,"ariaHidden":7091},[7090],[35,25058,25060,25063],{"className":25059},[7095],[35,25061],{"className":25062,"style":7718},[7099],[35,25064,25066,25069],{"className":25065},[7104],[35,25067,24883],{"className":25068,"style":24908},[7104,7145],[35,25070,25072],{"className":25071},[7113],[35,25073,25075,25102],{"className":25074},[7117,7731],[35,25076,25078,25099],{"className":25077},[7121],[35,25079,25081],{"className":25080,"style":9035},[7125],[35,25082,25083,25086],{"style":24923},[35,25084],{"className":25085,"style":7134},[7133],[35,25087,25089],{"className":25088},[7138,7139,7140,7141],[35,25090,25092,25096],{"className":25091},[7104,7141],[35,25093,300],{"className":25094,"style":25095},[7104,7145,7141],"margin-right:0.044em;",[35,25097,300],{"className":25098,"style":25095},[7104,7145,7141],[35,25100,7756],{"className":25101},[7755],[35,25103,25105],{"className":25104},[7121],[35,25106,25108],{"className":25107,"style":7763},[7125],[35,25109],{}," with a parametrized angle ",[35,25112,25114,25127],{"className":25113},[7057],[35,25115,25117],{"className":25116},[7061],[7063,25118,25119],{"xmlns":7065},[7067,25120,25121,25125],{},[7070,25122,25123],{},[7076,25124,7829],{},[7083,25126,7829],{"encoding":7085},[35,25128,25130],{"className":25129,"ariaHidden":7091},[7090],[35,25131,25133,25137],{"className":25132},[7095],[35,25134],{"className":25135,"style":25136},[7099],"height:0.6151em;",[35,25138,7829],{"className":25139},[7104,7145]," which correspond to the approximate implementation of ",[35,25142,25144,25188],{"className":25143},[7057],[35,25145,25147],{"className":25146},[7061],[7063,25148,25149],{"xmlns":7065},[7067,25150,25151,25185],{},[7070,25152,25153],{},[7073,25154,25155,25157],{},[7076,25156,7817],{},[7070,25158,25159,25161,25163,25165,25167,25169,25171,25173,25175,25177,25179,25181,25183],{},[7223,25160,7225],{},[7076,25162,7824],{},[7223,25164,275],{"stretchy":7289},[7076,25166,12257],{},[7076,25168,12257],{},[7223,25170,445],{},[7076,25172,12272],{},[7076,25174,12272],{},[7223,25176,445],{},[7076,25178,12290],{},[7076,25180,12290],{},[7223,25182,1694],{"stretchy":7289},[7076,25184,7829],{},[7083,25186,25187],{"encoding":7085},"e^{-i (XX + YY + ZZ) t}",[35,25189,25191],{"className":25190,"ariaHidden":7091},[7090],[35,25192,25194,25197],{"className":25193},[7095],[35,25195],{"className":25196,"style":8237},[7099],[35,25198,25200,25203],{"className":25199},[7104],[35,25201,7817],{"className":25202},[7104,7145],[35,25204,25206],{"className":25205},[7113],[35,25207,25209],{"className":25208},[7117],[35,25210,25212],{"className":25211},[7121],[35,25213,25215],{"className":25214,"style":8237},[7125],[35,25216,25217,25220],{"style":7129},[35,25218],{"className":25219,"style":7134},[7133],[35,25221,25223],{"className":25222},[7138,7139,7140,7141],[35,25224,25226,25229,25232,25235,25238,25241,25244,25247,25250,25253,25256,25259,25262],{"className":25225},[7104,7141],[35,25227,7225],{"className":25228},[7104,7141],[35,25230,7824],{"className":25231},[7104,7145,7141],[35,25233,275],{"className":25234},[7472,7141],[35,25236,12257],{"className":25237,"style":12402},[7104,7145,7141],[35,25239,12257],{"className":25240,"style":12402},[7104,7145,7141],[35,25242,445],{"className":25243},[7253,7141],[35,25245,12272],{"className":25246,"style":7249},[7104,7145,7141],[35,25248,12272],{"className":25249,"style":7249},[7104,7145,7141],[35,25251,445],{"className":25252},[7253,7141],[35,25254,12290],{"className":25255,"style":12600},[7104,7145,7141],[35,25257,12290],{"className":25258,"style":12600},[7104,7145,7141],[35,25260,1694],{"className":25261},[7314,7141],[35,25263,7829],{"className":25264},[7104,7145,7141],". Given the difference in definition of the Pauli rotations and the time-evolution that we are trying to implement, we'll have to use the parameter ",[35,25267,25269,25290],{"className":25268},[7057],[35,25270,25272],{"className":25271},[7061],[7063,25273,25274],{"xmlns":7065},[7067,25275,25276,25287],{},[7070,25277,25278,25280,25283,25285],{},[7227,25279,460],{},[7223,25281,25282],{},"∗",[7076,25284,9487],{},[7076,25286,7829],{},[7083,25288,25289],{"encoding":7085},"2*dt",[35,25291,25293,25311],{"className":25292,"ariaHidden":7091},[7090],[35,25294,25296,25299,25302,25305,25308],{"className":25295},[7095],[35,25297],{"className":25298,"style":7263},[7099],[35,25300,460],{"className":25301},[7104],[35,25303],{"className":25304,"style":7249},[7248],[35,25306,25282],{"className":25307},[7253],[35,25309],{"className":25310,"style":7249},[7248],[35,25312,25314,25317,25320],{"className":25313},[7095],[35,25315],{"className":25316,"style":9962},[7099],[35,25318,9487],{"className":25319},[7104,7145],[35,25321,7829],{"className":25322},[7104,7145]," to achieve a time-evolution of ",[35,25325,25327,25342],{"className":25326},[7057],[35,25328,25330],{"className":25329},[7061],[7063,25331,25332],{"xmlns":7065},[7067,25333,25334,25340],{},[7070,25335,25336,25338],{},[7076,25337,9487],{},[7076,25339,7829],{},[7083,25341,9991],{"encoding":7085},[35,25343,25345],{"className":25344,"ariaHidden":7091},[7090],[35,25346,25348,25351,25354],{"className":25347},[7095],[35,25349],{"className":25350,"style":9962},[7099],[35,25352,9487],{"className":25353},[7104,7145],[35,25355,7829],{"className":25356},[7104,7145],". Furthermore, we reverse the order of the operations for odd number of repetitions of the Trotter steps, which is functionally equivalent but allows for synthesizing adjacent operations in a single ",[35,25359,25361,25383],{"className":25360},[7057],[35,25362,25364],{"className":25363},[7061],[7063,25365,25366],{"xmlns":7065},[7067,25367,25368,25380],{},[7070,25369,25370,25372,25374,25376,25378],{},[7076,25371,10384],{},[7076,25373,7810],{},[7223,25375,275],{"stretchy":7289},[7227,25377,460],{},[7223,25379,1694],{"stretchy":7289},[7083,25381,25382],{"encoding":7085},"SU(2)",[35,25384,25386],{"className":25385,"ariaHidden":7091},[7090],[35,25387,25389,25392,25395,25398,25401,25404],{"className":25388},[7095],[35,25390],{"className":25391,"style":7303},[7099],[35,25393,10384],{"className":25394,"style":10419},[7104,7145],[35,25396,7810],{"className":25397,"style":7845},[7104,7145],[35,25399,275],{"className":25400},[7472],[35,25402,460],{"className":25403},[7104],[35,25405,1694],{"className":25406},[7314]," unitary. This gives a much shallower circuit than what is obtained using the generic ",[32,25409,25410],{},"PauliEvolutionGate()"," functionality.",[25,25413,25415],{"className":27,"code":25414,"language":29,"meta":30,"style":30},"t = Parameter(\"t\")\n\n# Create instruction for rotation about XX+YY-ZZ:\nRxyz_circ = QuantumCircuit(2)\nRxyz_circ.rxx(t, 0, 1)\nRxyz_circ.ryy(t, 0, 1)\nRxyz_circ.rzz(t, 0, 1)\nRxyz_instr = Rxyz_circ.to_instruction(label=\"RXX+YY+ZZ\")\n\ninteraction_list = [\n    [[i, i + 1] for i in range(0, n_qubits - 1, 2)],\n    [[i, i + 1] for i in range(1, n_qubits - 1, 2)],\n]  # linear chain\n\nqr = QuantumRegister(n_qubits)\ntrotter_step_circ = QuantumCircuit(qr)\nfor i, color in enumerate(interaction_list):\n    for interaction in color:\n        trotter_step_circ.append(Rxyz_instr, interaction)\n    if i \u003C len(interaction_list) - 1:\n        trotter_step_circ.barrier()\nreverse_trotter_step_circ = trotter_step_circ.reverse_ops()\n\nqc_evol = QuantumCircuit(qr)\nfor step in range(num_trotter_steps):\n    if step % 2 == 0:\n        qc_evol = qc_evol.compose(trotter_step_circ)\n    else:\n        qc_evol = qc_evol.compose(reverse_trotter_step_circ)\n\nqc_evol.decompose().draw(\"mpl\", fold=-1, scale=0.5)\n",[32,25416,25417,25431,25435,25440,25455,25474,25491,25508,25532,25536,25545,25582,25616,25624,25628,25638,25649,25663,25675,25685,25704,25713,25728,25732,25742,25755,25772,25787,25793,25806,25810],{"__ignoreMap":30},[35,25418,25419,25421,25423,25425,25427,25429],{"class":37,"line":38},[35,25420,13825],{"class":113},[35,25422,242],{"class":241},[35,25424,13830],{"class":271},[35,25426,275],{"class":113},[35,25428,13835],{"class":234},[35,25430,384],{"class":113},[35,25432,25433],{"class":37,"line":45},[35,25434,103],{"emptyLinePlaceholder":102},[35,25436,25437],{"class":37,"line":51},[35,25438,25439],{"class":41},"# Create instruction for rotation about XX+YY-ZZ:\n",[35,25441,25442,25445,25447,25449,25451,25453],{"class":37,"line":57},[35,25443,25444],{"class":113},"Rxyz_circ ",[35,25446,242],{"class":241},[35,25448,5293],{"class":271},[35,25450,275],{"class":113},[35,25452,460],{"class":245},[35,25454,384],{"class":113},[35,25456,25457,25460,25463,25466,25468,25470,25472],{"class":37,"line":63},[35,25458,25459],{"class":113},"Rxyz_circ.",[35,25461,25462],{"class":271},"rxx",[35,25464,25465],{"class":113},"(t, ",[35,25467,5312],{"class":245},[35,25469,288],{"class":113},[35,25471,500],{"class":245},[35,25473,384],{"class":113},[35,25475,25476,25478,25481,25483,25485,25487,25489],{"class":37,"line":69},[35,25477,25459],{"class":113},[35,25479,25480],{"class":271},"ryy",[35,25482,25465],{"class":113},[35,25484,5312],{"class":245},[35,25486,288],{"class":113},[35,25488,500],{"class":245},[35,25490,384],{"class":113},[35,25492,25493,25495,25498,25500,25502,25504,25506],{"class":37,"line":75},[35,25494,25459],{"class":113},[35,25496,25497],{"class":271},"rzz",[35,25499,25465],{"class":113},[35,25501,5312],{"class":245},[35,25503,288],{"class":113},[35,25505,500],{"class":245},[35,25507,384],{"class":113},[35,25509,25510,25513,25515,25518,25521,25523,25525,25527,25530],{"class":37,"line":81},[35,25511,25512],{"class":113},"Rxyz_instr ",[35,25514,242],{"class":241},[35,25516,25517],{"class":113}," Rxyz_circ.",[35,25519,25520],{"class":271},"to_instruction",[35,25522,275],{"class":113},[35,25524,2966],{"class":822},[35,25526,242],{"class":241},[35,25528,25529],{"class":234},"\"RXX+YY+ZZ\"",[35,25531,384],{"class":113},[35,25533,25534],{"class":37,"line":87},[35,25535,103],{"emptyLinePlaceholder":102},[35,25537,25538,25541,25543],{"class":37,"line":93},[35,25539,25540],{"class":113},"interaction_list ",[35,25542,242],{"class":241},[35,25544,894],{"class":113},[35,25546,25547,25550,25552,25554,25556,25558,25560,25562,25564,25566,25568,25571,25573,25575,25577,25579],{"class":37,"line":99},[35,25548,25549],{"class":113},"    [[i, i ",[35,25551,445],{"class":241},[35,25553,6079],{"class":245},[35,25555,238],{"class":113},[35,25557,1874],{"class":109},[35,25559,2958],{"class":113},[35,25561,1663],{"class":109},[35,25563,12746],{"class":241},[35,25565,275],{"class":113},[35,25567,5312],{"class":245},[35,25569,25570],{"class":113},", n_qubits ",[35,25572,564],{"class":241},[35,25574,6079],{"class":245},[35,25576,288],{"class":113},[35,25578,460],{"class":245},[35,25580,25581],{"class":113},")],\n",[35,25583,25584,25586,25588,25590,25592,25594,25596,25598,25600,25602,25604,25606,25608,25610,25612,25614],{"class":37,"line":106},[35,25585,25549],{"class":113},[35,25587,445],{"class":241},[35,25589,6079],{"class":245},[35,25591,238],{"class":113},[35,25593,1874],{"class":109},[35,25595,2958],{"class":113},[35,25597,1663],{"class":109},[35,25599,12746],{"class":241},[35,25601,275],{"class":113},[35,25603,500],{"class":245},[35,25605,25570],{"class":113},[35,25607,564],{"class":241},[35,25609,6079],{"class":245},[35,25611,288],{"class":113},[35,25613,460],{"class":245},[35,25615,25581],{"class":113},[35,25617,25618,25621],{"class":37,"line":117},[35,25619,25620],{"class":113},"]  ",[35,25622,25623],{"class":41},"# linear chain\n",[35,25625,25626],{"class":37,"line":125},[35,25627,103],{"emptyLinePlaceholder":102},[35,25629,25630,25632,25634,25636],{"class":37,"line":139},[35,25631,13902],{"class":113},[35,25633,242],{"class":241},[35,25635,13907],{"class":271},[35,25637,13301],{"class":113},[35,25639,25640,25643,25645,25647],{"class":37,"line":152},[35,25641,25642],{"class":113},"trotter_step_circ ",[35,25644,242],{"class":241},[35,25646,5293],{"class":271},[35,25648,13921],{"class":113},[35,25650,25651,25653,25656,25658,25660],{"class":37,"line":157},[35,25652,1874],{"class":109},[35,25654,25655],{"class":113}," i, color ",[35,25657,1663],{"class":109},[35,25659,2905],{"class":241},[35,25661,25662],{"class":113},"(interaction_list):\n",[35,25664,25665,25667,25670,25672],{"class":37,"line":171},[35,25666,1657],{"class":109},[35,25668,25669],{"class":113}," interaction ",[35,25671,1663],{"class":109},[35,25673,25674],{"class":113}," color:\n",[35,25676,25677,25680,25682],{"class":37,"line":184},[35,25678,25679],{"class":113},"        trotter_step_circ.",[35,25681,1927],{"class":271},[35,25683,25684],{"class":113},"(Rxyz_instr, interaction)\n",[35,25686,25687,25689,25691,25693,25695,25698,25700,25702],{"class":37,"line":197},[35,25688,352],{"class":109},[35,25690,2958],{"class":113},[35,25692,10205],{"class":241},[35,25694,6854],{"class":241},[35,25696,25697],{"class":113},"(interaction_list) ",[35,25699,564],{"class":241},[35,25701,6079],{"class":245},[35,25703,364],{"class":113},[35,25705,25706,25708,25711],{"class":37,"line":210},[35,25707,25679],{"class":113},[35,25709,25710],{"class":271},"barrier",[35,25712,2101],{"class":113},[35,25714,25715,25718,25720,25723,25726],{"class":37,"line":223},[35,25716,25717],{"class":113},"reverse_trotter_step_circ ",[35,25719,242],{"class":241},[35,25721,25722],{"class":113}," trotter_step_circ.",[35,25724,25725],{"class":271},"reverse_ops",[35,25727,2101],{"class":113},[35,25729,25730],{"class":37,"line":228},[35,25731,103],{"emptyLinePlaceholder":102},[35,25733,25734,25736,25738,25740],{"class":37,"line":249},[35,25735,13914],{"class":113},[35,25737,242],{"class":241},[35,25739,5293],{"class":271},[35,25741,13921],{"class":113},[35,25743,25744,25746,25749,25751,25753],{"class":37,"line":254},[35,25745,1874],{"class":109},[35,25747,25748],{"class":113}," step ",[35,25750,1663],{"class":109},[35,25752,12746],{"class":241},[35,25754,18407],{"class":113},[35,25756,25757,25759,25761,25763,25765,25768,25770],{"class":37,"line":260},[35,25758,352],{"class":109},[35,25760,25748],{"class":113},[35,25762,13287],{"class":241},[35,25764,424],{"class":245},[35,25766,25767],{"class":241}," ==",[35,25769,361],{"class":245},[35,25771,364],{"class":113},[35,25773,25774,25777,25779,25782,25784],{"class":37,"line":265},[35,25775,25776],{"class":113},"        qc_evol ",[35,25778,242],{"class":241},[35,25780,25781],{"class":113}," qc_evol.",[35,25783,18382],{"class":271},[35,25785,25786],{"class":113},"(trotter_step_circ)\n",[35,25788,25789,25791],{"class":37,"line":326},[35,25790,6403],{"class":109},[35,25792,364],{"class":113},[35,25794,25795,25797,25799,25801,25803],{"class":37,"line":332},[35,25796,25776],{"class":113},[35,25798,242],{"class":241},[35,25800,25781],{"class":113},[35,25802,18382],{"class":271},[35,25804,25805],{"class":113},"(reverse_trotter_step_circ)\n",[35,25807,25808],{"class":37,"line":338},[35,25809,103],{"emptyLinePlaceholder":102},[35,25811,25812,25814,25816,25818,25820,25822,25824,25826,25828,25830,25832,25834,25836,25838,25840],{"class":37,"line":344},[35,25813,13926],{"class":113},[35,25815,18678],{"class":271},[35,25817,18496],{"class":113},[35,25819,13682],{"class":271},[35,25821,275],{"class":113},[35,25823,13687],{"class":234},[35,25825,288],{"class":113},[35,25827,18625],{"class":822},[35,25829,18628],{"class":241},[35,25831,500],{"class":245},[35,25833,288],{"class":113},[35,25835,13692],{"class":822},[35,25837,242],{"class":241},[35,25839,13697],{"class":245},[35,25841,384],{"class":113},[4995,25843],{"alt":13702,"src":25844},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-03.avif",[7047,25846,25848],{"id":25847},"use-an-optimized-circuit-for-state-preparation","Use an optimized circuit for state preparation",[25,25850,25852],{"className":27,"code":25851,"language":29,"meta":30,"style":30},"control = 0\nexcitation = int(n_qubits \u002F 2) + 1\ncontrolled_state_prep = QuantumCircuit(n_qubits + 1)\ncontrolled_state_prep.cx(control, excitation)\ncontrolled_state_prep.draw(\"mpl\", fold=-1, scale=0.5)\n",[32,25853,25854,25863,25884,25901,25911],{"__ignoreMap":30},[35,25855,25856,25859,25861],{"class":37,"line":38},[35,25857,25858],{"class":113},"control ",[35,25860,242],{"class":241},[35,25862,6805],{"class":245},[35,25864,25865,25868,25870,25872,25874,25876,25878,25880,25882],{"class":37,"line":45},[35,25866,25867],{"class":113},"excitation ",[35,25869,242],{"class":241},[35,25871,1434],{"class":241},[35,25873,6676],{"class":113},[35,25875,398],{"class":241},[35,25877,424],{"class":245},[35,25879,427],{"class":113},[35,25881,445],{"class":241},[35,25883,734],{"class":245},[35,25885,25886,25889,25891,25893,25895,25897,25899],{"class":37,"line":51},[35,25887,25888],{"class":113},"controlled_state_prep ",[35,25890,242],{"class":241},[35,25892,5293],{"class":271},[35,25894,6676],{"class":113},[35,25896,445],{"class":241},[35,25898,6079],{"class":245},[35,25900,384],{"class":113},[35,25902,25903,25906,25908],{"class":37,"line":57},[35,25904,25905],{"class":113},"controlled_state_prep.",[35,25907,5325],{"class":271},[35,25909,25910],{"class":113},"(control, excitation)\n",[35,25912,25913,25915,25917,25919,25921,25923,25925,25927,25929,25931,25933,25935,25937],{"class":37,"line":63},[35,25914,25905],{"class":113},[35,25916,13682],{"class":271},[35,25918,275],{"class":113},[35,25920,13687],{"class":234},[35,25922,288],{"class":113},[35,25924,18625],{"class":822},[35,25926,18628],{"class":241},[35,25928,500],{"class":245},[35,25930,288],{"class":113},[35,25932,13692],{"class":822},[35,25934,242],{"class":241},[35,25936,13697],{"class":245},[35,25938,384],{"class":113},[4995,25940],{"alt":13702,"src":25941},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-04.avif",[7047,25943,25945,25946,15316,26006,26066],{"id":25944},"template-circuits-for-calculating-matrix-elements-of-stildess-and-htildehh-via-hadamard-test","Template circuits for calculating matrix elements of ",[35,25947,25949,25966],{"className":25948},[7057],[35,25950,25952],{"className":25951},[7061],[7063,25953,25954],{"xmlns":7065},[7067,25955,25956,25964],{},[7070,25957,25958],{},[8800,25959,25960,25962],{"accent":7091},[7076,25961,10384],{},[7223,25963,8806],{},[7083,25965,10389],{"encoding":7085},[35,25967,25969],{"className":25968,"ariaHidden":7091},[7090],[35,25970,25972,25975],{"className":25971},[7095],[35,25973],{"className":25974,"style":8819},[7099],[35,25976,25978],{"className":25977},[7104,8823],[35,25979,25981],{"className":25980},[7117],[35,25982,25984],{"className":25983},[7121],[35,25985,25987,25995],{"className":25986,"style":8819},[7125],[35,25988,25989,25992],{"style":8835},[35,25990],{"className":25991,"style":8839},[7133],[35,25993,10384],{"className":25994,"style":10419},[7104,7145],[35,25996,25997,26000],{"style":8845},[35,25998],{"className":25999,"style":8839},[7133],[35,26001,26003],{"className":26002,"style":10428},[8852],[35,26004,8806],{"className":26005},[7104],[35,26007,26009,26026],{"className":26008},[7057],[35,26010,26012],{"className":26011},[7061],[7063,26013,26014],{"xmlns":7065},[7067,26015,26016,26024],{},[7070,26017,26018],{},[8800,26019,26020,26022],{"accent":7091},[7076,26021,7669],{},[7223,26023,8806],{},[7083,26025,8809],{"encoding":7085},[35,26027,26029],{"className":26028,"ariaHidden":7091},[7090],[35,26030,26032,26035],{"className":26031},[7095],[35,26033],{"className":26034,"style":8819},[7099],[35,26036,26038],{"className":26037},[7104,8823],[35,26039,26041],{"className":26040},[7117],[35,26042,26044],{"className":26043},[7121],[35,26045,26047,26055],{"className":26046,"style":8819},[7125],[35,26048,26049,26052],{"style":8835},[35,26050],{"className":26051,"style":8839},[7133],[35,26053,7669],{"className":26054,"style":7684},[7104,7145],[35,26056,26057,26060],{"style":8845},[35,26058],{"className":26059,"style":8839},[7133],[35,26061,26063],{"className":26062,"style":8853},[8852],[35,26064,8806],{"className":26065},[7104]," via Hadamard test",[12,26068,26069],{},"The only difference between the circuits used in the Hadamard test will be the phase in the time-evolution operator and the observables measured. Therefore we can prepare a template circuit which represent the generic circuit for the Hadamard test, with placeholders for the gates that depend on the time-evolution operator.",[25,26071,26073],{"className":27,"code":26072,"language":29,"meta":30,"style":30},"# Parameters for the template circuits\nparameters = []\nfor idx in range(1, krylov_dim):\n    parameters.append(2 * dt_circ * (idx))\n",[32,26074,26075,26080,26089,26107],{"__ignoreMap":30},[35,26076,26077],{"class":37,"line":38},[35,26078,26079],{"class":41},"# Parameters for the template circuits\n",[35,26081,26082,26085,26087],{"class":37,"line":45},[35,26083,26084],{"class":113},"parameters ",[35,26086,242],{"class":241},[35,26088,1850],{"class":113},[35,26090,26091,26093,26096,26098,26100,26102,26104],{"class":37,"line":51},[35,26092,1874],{"class":109},[35,26094,26095],{"class":113}," idx ",[35,26097,1663],{"class":109},[35,26099,12746],{"class":241},[35,26101,275],{"class":113},[35,26103,500],{"class":245},[35,26105,26106],{"class":113},", krylov_dim):\n",[35,26108,26109,26112,26114,26116,26118,26120,26123,26125],{"class":37,"line":57},[35,26110,26111],{"class":113},"    parameters.",[35,26113,1927],{"class":271},[35,26115,275],{"class":113},[35,26117,460],{"class":245},[35,26119,463],{"class":241},[35,26121,26122],{"class":113}," dt_circ ",[35,26124,484],{"class":241},[35,26126,26127],{"class":113}," (idx))\n",[25,26129,26131],{"className":27,"code":26130,"language":29,"meta":30,"style":30},"# Create modified hadamard test circuit\nqr = QuantumRegister(n_qubits + 1)\nqc = QuantumCircuit(qr)\nqc.h(0)\nqc.compose(controlled_state_prep, list(range(n_qubits + 1)), inplace=True)\nqc.barrier()\nqc.compose(qc_evol, list(range(1, n_qubits + 1)), inplace=True)\nqc.barrier()\nqc.x(0)\nqc.compose(\n    controlled_state_prep.inverse(), list(range(n_qubits + 1)), inplace=True\n)\nqc.x(0)\n\nqc.decompose().draw(\"mpl\", fold=-1)\n",[32,26132,26133,26138,26154,26164,26176,26208,26216,26251,26259,26271,26279,26309,26313,26325,26329],{"__ignoreMap":30},[35,26134,26135],{"class":37,"line":38},[35,26136,26137],{"class":41},"# Create modified hadamard test circuit\n",[35,26139,26140,26142,26144,26146,26148,26150,26152],{"class":37,"line":45},[35,26141,13902],{"class":113},[35,26143,242],{"class":241},[35,26145,13907],{"class":271},[35,26147,6676],{"class":113},[35,26149,445],{"class":241},[35,26151,6079],{"class":245},[35,26153,384],{"class":113},[35,26155,26156,26158,26160,26162],{"class":37,"line":51},[35,26157,5288],{"class":113},[35,26159,242],{"class":241},[35,26161,5293],{"class":271},[35,26163,13921],{"class":113},[35,26165,26166,26168,26170,26172,26174],{"class":37,"line":57},[35,26167,5304],{"class":113},[35,26169,5307],{"class":271},[35,26171,275],{"class":113},[35,26173,5312],{"class":245},[35,26175,384],{"class":113},[35,26177,26178,26180,26182,26185,26187,26189,26191,26193,26195,26197,26200,26202,26204,26206],{"class":37,"line":63},[35,26179,5304],{"class":113},[35,26181,18382],{"class":271},[35,26183,26184],{"class":113},"(controlled_state_prep, ",[35,26186,18566],{"class":241},[35,26188,275],{"class":113},[35,26190,6767],{"class":241},[35,26192,6676],{"class":113},[35,26194,445],{"class":241},[35,26196,6079],{"class":245},[35,26198,26199],{"class":113},")), ",[35,26201,18388],{"class":822},[35,26203,242],{"class":241},[35,26205,828],{"class":245},[35,26207,384],{"class":113},[35,26209,26210,26212,26214],{"class":37,"line":69},[35,26211,5304],{"class":113},[35,26213,25710],{"class":271},[35,26215,2101],{"class":113},[35,26217,26218,26220,26222,26225,26227,26229,26231,26233,26235,26237,26239,26241,26243,26245,26247,26249],{"class":37,"line":75},[35,26219,5304],{"class":113},[35,26221,18382],{"class":271},[35,26223,26224],{"class":113},"(qc_evol, ",[35,26226,18566],{"class":241},[35,26228,275],{"class":113},[35,26230,6767],{"class":241},[35,26232,275],{"class":113},[35,26234,500],{"class":245},[35,26236,25570],{"class":113},[35,26238,445],{"class":241},[35,26240,6079],{"class":245},[35,26242,26199],{"class":113},[35,26244,18388],{"class":822},[35,26246,242],{"class":241},[35,26248,828],{"class":245},[35,26250,384],{"class":113},[35,26252,26253,26255,26257],{"class":37,"line":81},[35,26254,5304],{"class":113},[35,26256,25710],{"class":271},[35,26258,2101],{"class":113},[35,26260,26261,26263,26265,26267,26269],{"class":37,"line":87},[35,26262,5304],{"class":113},[35,26264,13657],{"class":271},[35,26266,275],{"class":113},[35,26268,5312],{"class":245},[35,26270,384],{"class":113},[35,26272,26273,26275,26277],{"class":37,"line":93},[35,26274,5304],{"class":113},[35,26276,18382],{"class":271},[35,26278,903],{"class":113},[35,26280,26281,26284,26286,26288,26290,26292,26294,26296,26298,26300,26302,26304,26306],{"class":37,"line":99},[35,26282,26283],{"class":113},"    controlled_state_prep.",[35,26285,18340],{"class":271},[35,26287,2672],{"class":113},[35,26289,18566],{"class":241},[35,26291,275],{"class":113},[35,26293,6767],{"class":241},[35,26295,6676],{"class":113},[35,26297,445],{"class":241},[35,26299,6079],{"class":245},[35,26301,26199],{"class":113},[35,26303,18388],{"class":822},[35,26305,242],{"class":241},[35,26307,26308],{"class":245},"True\n",[35,26310,26311],{"class":37,"line":106},[35,26312,384],{"class":113},[35,26314,26315,26317,26319,26321,26323],{"class":37,"line":117},[35,26316,5304],{"class":113},[35,26318,13657],{"class":271},[35,26320,275],{"class":113},[35,26322,5312],{"class":245},[35,26324,384],{"class":113},[35,26326,26327],{"class":37,"line":125},[35,26328,103],{"emptyLinePlaceholder":102},[35,26330,26331,26333,26335,26337,26339,26341,26343,26345,26347,26349,26351],{"class":37,"line":139},[35,26332,5304],{"class":113},[35,26334,18678],{"class":271},[35,26336,18496],{"class":113},[35,26338,13682],{"class":271},[35,26340,275],{"class":113},[35,26342,13687],{"class":234},[35,26344,288],{"class":113},[35,26346,18625],{"class":822},[35,26348,18628],{"class":241},[35,26350,500],{"class":245},[35,26352,384],{"class":113},[4995,26354],{"alt":13702,"src":26355},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-05.avif",[25,26357,26359],{"className":27,"code":26358,"language":29,"meta":30,"style":30},"print(\n    \"The optimized circuit has 2Q gates depth: \",\n    qc.decompose().decompose().depth(lambda x: x[0].num_qubits == 2),\n)\n",[32,26360,26361,26367,26374,26407],{"__ignoreMap":30},[35,26362,26363,26365],{"class":37,"line":38},[35,26364,1861],{"class":241},[35,26366,903],{"class":113},[35,26368,26369,26372],{"class":37,"line":45},[35,26370,26371],{"class":234},"    \"The optimized circuit has 2Q gates depth: \"",[35,26373,912],{"class":113},[35,26375,26376,26379,26381,26383,26385,26387,26389,26391,26393,26395,26397,26399,26401,26403,26405],{"class":37,"line":51},[35,26377,26378],{"class":113},"    qc.",[35,26380,18678],{"class":271},[35,26382,18496],{"class":113},[35,26384,18678],{"class":271},[35,26386,18496],{"class":113},[35,26388,18692],{"class":271},[35,26390,275],{"class":113},[35,26392,18697],{"class":109},[35,26394,18700],{"class":278},[35,26396,18703],{"class":113},[35,26398,5312],{"class":245},[35,26400,18708],{"class":113},[35,26402,358],{"class":241},[35,26404,424],{"class":245},[35,26406,18715],{"class":113},[35,26408,26409],{"class":37,"line":57},[35,26410,384],{"class":113},[25,26412,26415],{"className":26413,"code":26414,"language":4991,"meta":30},[4989],"The optimized circuit has 2Q gates depth:  74\n",[32,26416,26414],{"__ignoreMap":30},[12,26418,26419],{},"We have considerably reduced the depth of the Hadamard test with a combination of Trotter approximation and uncontrolled unitaries",[5019,26421,26423],{"id":26422},"step-3-execute-using-qiskit-primitives","Step 3: Execute using Qiskit primitives",[12,26425,26426],{},"Instantiate the backend and set runtime parameters",[25,26428,26430],{"className":27,"code":26429,"language":29,"meta":30,"style":30},"service = QiskitRuntimeService()\nbackend = service.least_busy(operational=True, simulator=False)\nif (\n    \"if_else\" not in backend.target.operation_names\n):  # Needed as \"op_name\" could be \"if_else\"\n    backend.target.add_instruction(IfElseOp, name=\"if_else\")\nprint(backend.name)\n",[32,26431,26432,26444,26477,26483,26497,26505,26526],{"__ignoreMap":30},[35,26433,26434,26437,26439,26442],{"class":37,"line":38},[35,26435,26436],{"class":113},"service ",[35,26438,242],{"class":241},[35,26440,26441],{"class":271}," QiskitRuntimeService",[35,26443,2101],{"class":113},[35,26445,26446,26449,26451,26454,26457,26459,26462,26464,26466,26468,26471,26473,26475],{"class":37,"line":45},[35,26447,26448],{"class":113},"backend ",[35,26450,242],{"class":241},[35,26452,26453],{"class":113}," service.",[35,26455,26456],{"class":271},"least_busy",[35,26458,275],{"class":113},[35,26460,26461],{"class":822},"operational",[35,26463,242],{"class":241},[35,26465,828],{"class":245},[35,26467,288],{"class":113},[35,26469,26470],{"class":822},"simulator",[35,26472,242],{"class":241},[35,26474,2218],{"class":245},[35,26476,384],{"class":113},[35,26478,26479,26481],{"class":37,"line":51},[35,26480,1492],{"class":109},[35,26482,6003],{"class":113},[35,26484,26485,26488,26491,26494],{"class":37,"line":57},[35,26486,26487],{"class":234},"    \"if_else\"",[35,26489,26490],{"class":109}," not",[35,26492,26493],{"class":109}," in",[35,26495,26496],{"class":113}," backend.target.operation_names\n",[35,26498,26499,26502],{"class":37,"line":63},[35,26500,26501],{"class":113},"):  ",[35,26503,26504],{"class":41},"# Needed as \"op_name\" could be \"if_else\"\n",[35,26506,26507,26510,26513,26516,26519,26521,26524],{"class":37,"line":69},[35,26508,26509],{"class":113},"    backend.target.",[35,26511,26512],{"class":271},"add_instruction",[35,26514,26515],{"class":113},"(IfElseOp, ",[35,26517,26518],{"class":822},"name",[35,26520,242],{"class":241},[35,26522,26523],{"class":234},"\"if_else\"",[35,26525,384],{"class":113},[35,26527,26528,26530],{"class":37,"line":75},[35,26529,1861],{"class":241},[35,26531,26532],{"class":113},"(backend.name)\n",[7047,26534,26536],{"id":26535},"transpiling-to-a-qpu","Transpiling to a QPU",[12,26538,26539],{},"First, let's pick subsets of the coupling map with \"good\" performing qubits (where \"good\" is pretty arbitrary here, we mostly want to avoid really poor performing qubits) and create a new target for transpilation",[25,26541,26543],{"className":27,"code":26542,"language":29,"meta":30,"style":30},"target = backend.target\ncmap = target.build_coupling_map(filter_idle_qubits=True)\ncmap_list = list(cmap.get_edges())\n\ncust_cmap_list = copy.deepcopy(cmap_list)\nfor q in range(target.num_qubits):\n    meas_err = target[\"measure\"][(q,)].error\n    t2 = target.qubit_properties[q].t2 * 1e6\n    if meas_err > 0.02 or t2 \u003C 100:\n        for q_pair in cmap_list:\n            if q in q_pair:\n                try:\n                    cust_cmap_list.remove(q_pair)\n                except:\n                    continue\n\nfor q in cmap_list:\n    op_name = list(target.operation_names_for_qargs(q))[0]\n    twoq_gate_err = target[f\"{op_name}\"][q].error\n    if twoq_gate_err > 0.005:\n        for q_pair in cmap_list:\n            if q == q_pair:\n                try:\n                    cust_cmap_list.remove(q)\n                except:\n                    continue\n\n\ncust_cmap = CouplingMap(cust_cmap_list)\ncust_target = Target.from_configuration(\n    basis_gates=backend.configuration().basis_gates,\n    coupling_map=cust_cmap,\n)\n",[32,26544,26545,26555,26579,26597,26601,26617,26631,26647,26662,26686,26698,26709,26716,26727,26734,26739,26743,26753,26775,26800,26814,26824,26834,26840,26849,26855,26859,26863,26867,26880,26895,26911,26921],{"__ignoreMap":30},[35,26546,26547,26550,26552],{"class":37,"line":38},[35,26548,26549],{"class":113},"target ",[35,26551,242],{"class":241},[35,26553,26554],{"class":113}," backend.target\n",[35,26556,26557,26560,26562,26565,26568,26570,26573,26575,26577],{"class":37,"line":45},[35,26558,26559],{"class":113},"cmap ",[35,26561,242],{"class":241},[35,26563,26564],{"class":113}," target.",[35,26566,26567],{"class":271},"build_coupling_map",[35,26569,275],{"class":113},[35,26571,26572],{"class":822},"filter_idle_qubits",[35,26574,242],{"class":241},[35,26576,828],{"class":245},[35,26578,384],{"class":113},[35,26580,26581,26584,26586,26589,26592,26595],{"class":37,"line":51},[35,26582,26583],{"class":113},"cmap_list ",[35,26585,242],{"class":241},[35,26587,26588],{"class":241}," list",[35,26590,26591],{"class":113},"(cmap.",[35,26593,26594],{"class":271},"get_edges",[35,26596,1456],{"class":113},[35,26598,26599],{"class":37,"line":57},[35,26600,103],{"emptyLinePlaceholder":102},[35,26602,26603,26606,26608,26611,26614],{"class":37,"line":63},[35,26604,26605],{"class":113},"cust_cmap_list ",[35,26607,242],{"class":241},[35,26609,26610],{"class":113}," copy.",[35,26612,26613],{"class":271},"deepcopy",[35,26615,26616],{"class":113},"(cmap_list)\n",[35,26618,26619,26621,26624,26626,26628],{"class":37,"line":69},[35,26620,1874],{"class":109},[35,26622,26623],{"class":113}," q ",[35,26625,1663],{"class":109},[35,26627,12746],{"class":241},[35,26629,26630],{"class":113},"(target.num_qubits):\n",[35,26632,26633,26636,26638,26641,26644],{"class":37,"line":75},[35,26634,26635],{"class":113},"    meas_err ",[35,26637,242],{"class":241},[35,26639,26640],{"class":113}," target[",[35,26642,26643],{"class":234},"\"measure\"",[35,26645,26646],{"class":113},"][(q,)].error\n",[35,26648,26649,26652,26654,26657,26659],{"class":37,"line":81},[35,26650,26651],{"class":113},"    t2 ",[35,26653,242],{"class":241},[35,26655,26656],{"class":113}," target.qubit_properties[q].t2 ",[35,26658,484],{"class":241},[35,26660,26661],{"class":245}," 1e6\n",[35,26663,26664,26666,26669,26671,26674,26676,26679,26681,26684],{"class":37,"line":87},[35,26665,352],{"class":109},[35,26667,26668],{"class":113}," meas_err ",[35,26670,1497],{"class":241},[35,26672,26673],{"class":245}," 0.02",[35,26675,2038],{"class":109},[35,26677,26678],{"class":113}," t2 ",[35,26680,10205],{"class":241},[35,26682,26683],{"class":245}," 100",[35,26685,364],{"class":113},[35,26687,26688,26690,26693,26695],{"class":37,"line":93},[35,26689,2087],{"class":109},[35,26691,26692],{"class":113}," q_pair ",[35,26694,1663],{"class":109},[35,26696,26697],{"class":113}," cmap_list:\n",[35,26699,26700,26702,26704,26706],{"class":37,"line":99},[35,26701,6851],{"class":109},[35,26703,26623],{"class":113},[35,26705,1663],{"class":109},[35,26707,26708],{"class":113}," q_pair:\n",[35,26710,26711,26714],{"class":37,"line":106},[35,26712,26713],{"class":109},"                try",[35,26715,364],{"class":113},[35,26717,26718,26721,26724],{"class":37,"line":117},[35,26719,26720],{"class":113},"                    cust_cmap_list.",[35,26722,26723],{"class":271},"remove",[35,26725,26726],{"class":113},"(q_pair)\n",[35,26728,26729,26732],{"class":37,"line":125},[35,26730,26731],{"class":109},"                except",[35,26733,364],{"class":113},[35,26735,26736],{"class":37,"line":139},[35,26737,26738],{"class":109},"                    continue\n",[35,26740,26741],{"class":37,"line":152},[35,26742,103],{"emptyLinePlaceholder":102},[35,26744,26745,26747,26749,26751],{"class":37,"line":157},[35,26746,1874],{"class":109},[35,26748,26623],{"class":113},[35,26750,1663],{"class":109},[35,26752,26697],{"class":113},[35,26754,26755,26758,26760,26762,26765,26768,26771,26773],{"class":37,"line":171},[35,26756,26757],{"class":113},"    op_name ",[35,26759,242],{"class":241},[35,26761,26588],{"class":241},[35,26763,26764],{"class":113},"(target.",[35,26766,26767],{"class":271},"operation_names_for_qargs",[35,26769,26770],{"class":113},"(q))[",[35,26772,5312],{"class":245},[35,26774,1074],{"class":113},[35,26776,26777,26780,26782,26784,26786,26788,26790,26793,26795,26797],{"class":37,"line":184},[35,26778,26779],{"class":113},"    twoq_gate_err ",[35,26781,242],{"class":241},[35,26783,26640],{"class":113},[35,26785,1675],{"class":109},[35,26787,2986],{"class":234},[35,26789,1681],{"class":245},[35,26791,26792],{"class":113},"op_name",[35,26794,1697],{"class":245},[35,26796,2986],{"class":234},[35,26798,26799],{"class":113},"][q].error\n",[35,26801,26802,26804,26807,26809,26812],{"class":37,"line":197},[35,26803,352],{"class":109},[35,26805,26806],{"class":113}," twoq_gate_err ",[35,26808,1497],{"class":241},[35,26810,26811],{"class":245}," 0.005",[35,26813,364],{"class":113},[35,26815,26816,26818,26820,26822],{"class":37,"line":210},[35,26817,2087],{"class":109},[35,26819,26692],{"class":113},[35,26821,1663],{"class":109},[35,26823,26697],{"class":113},[35,26825,26826,26828,26830,26832],{"class":37,"line":223},[35,26827,6851],{"class":109},[35,26829,26623],{"class":113},[35,26831,358],{"class":241},[35,26833,26708],{"class":113},[35,26835,26836,26838],{"class":37,"line":228},[35,26837,26713],{"class":109},[35,26839,364],{"class":113},[35,26841,26842,26844,26846],{"class":37,"line":249},[35,26843,26720],{"class":113},[35,26845,26723],{"class":271},[35,26847,26848],{"class":113},"(q)\n",[35,26850,26851,26853],{"class":37,"line":254},[35,26852,26731],{"class":109},[35,26854,364],{"class":113},[35,26856,26857],{"class":37,"line":260},[35,26858,26738],{"class":109},[35,26860,26861],{"class":37,"line":265},[35,26862,103],{"emptyLinePlaceholder":102},[35,26864,26865],{"class":37,"line":326},[35,26866,103],{"emptyLinePlaceholder":102},[35,26868,26869,26872,26874,26877],{"class":37,"line":332},[35,26870,26871],{"class":113},"cust_cmap ",[35,26873,242],{"class":241},[35,26875,26876],{"class":271}," CouplingMap",[35,26878,26879],{"class":113},"(cust_cmap_list)\n",[35,26881,26882,26885,26887,26890,26893],{"class":37,"line":338},[35,26883,26884],{"class":113},"cust_target ",[35,26886,242],{"class":241},[35,26888,26889],{"class":113}," Target.",[35,26891,26892],{"class":271},"from_configuration",[35,26894,903],{"class":113},[35,26896,26897,26900,26902,26905,26908],{"class":37,"line":344},[35,26898,26899],{"class":822},"    basis_gates",[35,26901,242],{"class":241},[35,26903,26904],{"class":113},"backend.",[35,26906,26907],{"class":271},"configuration",[35,26909,26910],{"class":113},"().basis_gates,\n",[35,26912,26913,26916,26918],{"class":37,"line":349},[35,26914,26915],{"class":822},"    coupling_map",[35,26917,242],{"class":241},[35,26919,26920],{"class":113},"cust_cmap,\n",[35,26922,26923],{"class":37,"line":367},[35,26924,384],{"class":113},[12,26926,26927],{},"Then transpile the virtual circuit to the best physical layout in this new target",[25,26929,26931],{"className":27,"code":26930,"language":29,"meta":30,"style":30},"basis_gates = list(target.operation_names)\npm = generate_preset_pass_manager(\n    optimization_level=3,\n    target=cust_target,\n    basis_gates=basis_gates,\n)\n\nqc_trans = pm.run(qc)\n\nprint(\"depth\", qc_trans.depth(lambda x: x[0].num_qubits == 2))\nprint(\"num 2q ops\", qc_trans.count_ops())\nprint(\n    \"physical qubits\",\n    sorted(\n        [\n            idx\n            for idx, qb in qc_trans.layout.initial_layout.get_physical_bits().items()\n            if qb._register.name != \"ancilla\"\n        ]\n    ),\n)\n",[32,26932,26933,26945,26957,26968,26978,26987,26991,26995,27010,27014,27046,27062,27068,27075,27082,27087,27092,27114,27127,27132,27136],{"__ignoreMap":30},[35,26934,26935,26938,26940,26942],{"class":37,"line":38},[35,26936,26937],{"class":113},"basis_gates ",[35,26939,242],{"class":241},[35,26941,26588],{"class":241},[35,26943,26944],{"class":113},"(target.operation_names)\n",[35,26946,26947,26950,26952,26955],{"class":37,"line":45},[35,26948,26949],{"class":113},"pm ",[35,26951,242],{"class":241},[35,26953,26954],{"class":271}," generate_preset_pass_manager",[35,26956,903],{"class":113},[35,26958,26959,26962,26964,26966],{"class":37,"line":51},[35,26960,26961],{"class":822},"    optimization_level",[35,26963,242],{"class":241},[35,26965,12751],{"class":245},[35,26967,912],{"class":113},[35,26969,26970,26973,26975],{"class":37,"line":57},[35,26971,26972],{"class":822},"    target",[35,26974,242],{"class":241},[35,26976,26977],{"class":113},"cust_target,\n",[35,26979,26980,26982,26984],{"class":37,"line":63},[35,26981,26899],{"class":822},[35,26983,242],{"class":241},[35,26985,26986],{"class":113},"basis_gates,\n",[35,26988,26989],{"class":37,"line":69},[35,26990,384],{"class":113},[35,26992,26993],{"class":37,"line":75},[35,26994,103],{"emptyLinePlaceholder":102},[35,26996,26997,27000,27002,27005,27008],{"class":37,"line":81},[35,26998,26999],{"class":113},"qc_trans ",[35,27001,242],{"class":241},[35,27003,27004],{"class":113}," pm.",[35,27006,27007],{"class":271},"run",[35,27009,5395],{"class":113},[35,27011,27012],{"class":37,"line":87},[35,27013,103],{"emptyLinePlaceholder":102},[35,27015,27016,27018,27020,27023,27026,27028,27030,27032,27034,27036,27038,27040,27042,27044],{"class":37,"line":93},[35,27017,1861],{"class":241},[35,27019,275],{"class":113},[35,27021,27022],{"class":234},"\"depth\"",[35,27024,27025],{"class":113},", qc_trans.",[35,27027,18692],{"class":271},[35,27029,275],{"class":113},[35,27031,18697],{"class":109},[35,27033,18700],{"class":278},[35,27035,18703],{"class":113},[35,27037,5312],{"class":245},[35,27039,18708],{"class":113},[35,27041,358],{"class":241},[35,27043,424],{"class":245},[35,27045,2401],{"class":113},[35,27047,27048,27050,27052,27055,27057,27060],{"class":37,"line":99},[35,27049,1861],{"class":241},[35,27051,275],{"class":113},[35,27053,27054],{"class":234},"\"num 2q ops\"",[35,27056,27025],{"class":113},[35,27058,27059],{"class":271},"count_ops",[35,27061,1456],{"class":113},[35,27063,27064,27066],{"class":37,"line":106},[35,27065,1861],{"class":241},[35,27067,903],{"class":113},[35,27069,27070,27073],{"class":37,"line":117},[35,27071,27072],{"class":234},"    \"physical qubits\"",[35,27074,912],{"class":113},[35,27076,27077,27080],{"class":37,"line":125},[35,27078,27079],{"class":241},"    sorted",[35,27081,903],{"class":113},[35,27083,27084],{"class":37,"line":139},[35,27085,27086],{"class":113},"        [\n",[35,27088,27089],{"class":37,"line":152},[35,27090,27091],{"class":113},"            idx\n",[35,27093,27094,27097,27100,27102,27105,27108,27110,27112],{"class":37,"line":157},[35,27095,27096],{"class":109},"            for",[35,27098,27099],{"class":113}," idx, qb ",[35,27101,1663],{"class":109},[35,27103,27104],{"class":113}," qc_trans.layout.initial_layout.",[35,27106,27107],{"class":271},"get_physical_bits",[35,27109,18496],{"class":113},[35,27111,2145],{"class":271},[35,27113,2101],{"class":113},[35,27115,27116,27118,27121,27124],{"class":37,"line":171},[35,27117,6851],{"class":109},[35,27119,27120],{"class":113}," qb._register.name ",[35,27122,27123],{"class":241},"!=",[35,27125,27126],{"class":234}," \"ancilla\"\n",[35,27128,27129],{"class":37,"line":184},[35,27130,27131],{"class":113},"        ]\n",[35,27133,27134],{"class":37,"line":197},[35,27135,932],{"class":113},[35,27137,27138],{"class":37,"line":210},[35,27139,384],{"class":113},[25,27141,27144],{"className":27142,"code":27143,"language":4991,"meta":30},[4989],"depth 52\nnum 2q ops OrderedDict([('rz', 2058), ('sx', 1703), ('cz', 728), ('x', 84), ('barrier', 8)])\nphysical qubits [91, 92, 93, 94, 95, 98, 99, 108, 109, 110, 111, 113, 114, 115, 119, 127, 132, 133, 134, 135, 137, 139, 147, 148, 149, 150, 151, 152, 153, 154, 155]\n",[32,27145,27143],{"__ignoreMap":30},[7047,27147,27149],{"id":27148},"create-pubs-for-execution-with-estimator","Create PUBs for execution with Estimator",[25,27151,27153],{"className":27,"code":27152,"language":29,"meta":30,"style":30},"# Define observables to measure for S\nobservable_S_real = \"I\" * (n_qubits) + \"X\"\nobservable_S_imag = \"I\" * (n_qubits) + \"Y\"\n\nobservable_op_real = SparsePauliOp(\n    observable_S_real\n)  # define a sparse pauli operator for the observable\nobservable_op_imag = SparsePauliOp(observable_S_imag)\n\nlayout = qc_trans.layout  # get layout of transpiled circuit\nobservable_op_real = observable_op_real.apply_layout(\n    layout\n)  # apply physical layout to the observable\nobservable_op_imag = observable_op_imag.apply_layout(layout)\nobservable_S_real = (\n    observable_op_real.paulis.to_labels()\n)  # get the label of the physical observable\nobservable_S_imag = observable_op_imag.paulis.to_labels()\n\nobservables_S = [[observable_S_real], [observable_S_imag]]\n\n\n# Define observables to measure for H\n# Hamiltonian terms to measure\nobservable_list = []\nfor pauli, coeff in zip(H_op.paulis, H_op.coeffs):\n    # print(pauli)\n    observable_H_real = pauli[::-1].to_label() + \"X\"\n    observable_H_imag = pauli[::-1].to_label() + \"Y\"\n    observable_list.append([observable_H_real])\n    observable_list.append([observable_H_imag])\n\nlayout = qc_trans.layout\n\nobservable_trans_list = []\nfor observable in observable_list:\n    observable_op = SparsePauliOp(observable)\n    observable_op = observable_op.apply_layout(layout)\n    observable_trans_list.append([observable_op.paulis.to_labels()])\n\nobservables_H = observable_trans_list\n\n\n# Define a sweep over parameter values\nparams = np.vstack(parameters).T\n\n\n# Estimate the expectation value for all combinations of\n# observables and parameter values, where the pub result will have\n# shape (# observables, # parameter values).\npub = (qc_trans, observables_S + observables_H, params)\n",[32,27154,27155,27160,27179,27196,27200,27212,27217,27225,27237,27241,27254,27268,27273,27280,27294,27302,27312,27319,27332,27336,27346,27350,27354,27359,27364,27373,27387,27392,27417,27440,27450,27459,27463,27472,27476,27485,27497,27509,27522,27537,27541,27551,27555,27559,27564,27578,27582,27586,27591,27596,27601],{"__ignoreMap":30},[35,27156,27157],{"class":37,"line":38},[35,27158,27159],{"class":41},"# Define observables to measure for S\n",[35,27161,27162,27165,27167,27170,27172,27175,27177],{"class":37,"line":45},[35,27163,27164],{"class":113},"observable_S_real ",[35,27166,242],{"class":241},[35,27168,27169],{"class":234}," \"I\"",[35,27171,463],{"class":241},[35,27173,27174],{"class":113}," (n_qubits) ",[35,27176,445],{"class":241},[35,27178,12843],{"class":234},[35,27180,27181,27184,27186,27188,27190,27192,27194],{"class":37,"line":51},[35,27182,27183],{"class":113},"observable_S_imag ",[35,27185,242],{"class":241},[35,27187,27169],{"class":234},[35,27189,463],{"class":241},[35,27191,27174],{"class":113},[35,27193,445],{"class":241},[35,27195,12910],{"class":234},[35,27197,27198],{"class":37,"line":57},[35,27199,103],{"emptyLinePlaceholder":102},[35,27201,27202,27205,27207,27210],{"class":37,"line":63},[35,27203,27204],{"class":113},"observable_op_real ",[35,27206,242],{"class":241},[35,27208,27209],{"class":271}," SparsePauliOp",[35,27211,903],{"class":113},[35,27213,27214],{"class":37,"line":69},[35,27215,27216],{"class":113},"    observable_S_real\n",[35,27218,27219,27222],{"class":37,"line":75},[35,27220,27221],{"class":113},")  ",[35,27223,27224],{"class":41},"# define a sparse pauli operator for the observable\n",[35,27226,27227,27230,27232,27234],{"class":37,"line":81},[35,27228,27229],{"class":113},"observable_op_imag ",[35,27231,242],{"class":241},[35,27233,27209],{"class":271},[35,27235,27236],{"class":113},"(observable_S_imag)\n",[35,27238,27239],{"class":37,"line":87},[35,27240,103],{"emptyLinePlaceholder":102},[35,27242,27243,27246,27248,27251],{"class":37,"line":93},[35,27244,27245],{"class":113},"layout ",[35,27247,242],{"class":241},[35,27249,27250],{"class":113}," qc_trans.layout  ",[35,27252,27253],{"class":41},"# get layout of transpiled circuit\n",[35,27255,27256,27258,27260,27263,27266],{"class":37,"line":99},[35,27257,27204],{"class":113},[35,27259,242],{"class":241},[35,27261,27262],{"class":113}," observable_op_real.",[35,27264,27265],{"class":271},"apply_layout",[35,27267,903],{"class":113},[35,27269,27270],{"class":37,"line":106},[35,27271,27272],{"class":113},"    layout\n",[35,27274,27275,27277],{"class":37,"line":117},[35,27276,27221],{"class":113},[35,27278,27279],{"class":41},"# apply physical layout to the observable\n",[35,27281,27282,27284,27286,27289,27291],{"class":37,"line":125},[35,27283,27229],{"class":113},[35,27285,242],{"class":241},[35,27287,27288],{"class":113}," observable_op_imag.",[35,27290,27265],{"class":271},[35,27292,27293],{"class":113},"(layout)\n",[35,27295,27296,27298,27300],{"class":37,"line":139},[35,27297,27164],{"class":113},[35,27299,242],{"class":241},[35,27301,6003],{"class":113},[35,27303,27304,27307,27310],{"class":37,"line":152},[35,27305,27306],{"class":113},"    observable_op_real.paulis.",[35,27308,27309],{"class":271},"to_labels",[35,27311,2101],{"class":113},[35,27313,27314,27316],{"class":37,"line":157},[35,27315,27221],{"class":113},[35,27317,27318],{"class":41},"# get the label of the physical observable\n",[35,27320,27321,27323,27325,27328,27330],{"class":37,"line":171},[35,27322,27183],{"class":113},[35,27324,242],{"class":241},[35,27326,27327],{"class":113}," observable_op_imag.paulis.",[35,27329,27309],{"class":271},[35,27331,2101],{"class":113},[35,27333,27334],{"class":37,"line":184},[35,27335,103],{"emptyLinePlaceholder":102},[35,27337,27338,27341,27343],{"class":37,"line":197},[35,27339,27340],{"class":113},"observables_S ",[35,27342,242],{"class":241},[35,27344,27345],{"class":113}," [[observable_S_real], [observable_S_imag]]\n",[35,27347,27348],{"class":37,"line":210},[35,27349,103],{"emptyLinePlaceholder":102},[35,27351,27352],{"class":37,"line":223},[35,27353,103],{"emptyLinePlaceholder":102},[35,27355,27356],{"class":37,"line":228},[35,27357,27358],{"class":41},"# Define observables to measure for H\n",[35,27360,27361],{"class":37,"line":249},[35,27362,27363],{"class":41},"# Hamiltonian terms to measure\n",[35,27365,27366,27369,27371],{"class":37,"line":254},[35,27367,27368],{"class":113},"observable_list ",[35,27370,242],{"class":241},[35,27372,1850],{"class":113},[35,27374,27375,27377,27380,27382,27384],{"class":37,"line":260},[35,27376,1874],{"class":109},[35,27378,27379],{"class":113}," pauli, coeff ",[35,27381,1663],{"class":109},[35,27383,6262],{"class":241},[35,27385,27386],{"class":113},"(H_op.paulis, H_op.coeffs):\n",[35,27388,27389],{"class":37,"line":265},[35,27390,27391],{"class":41},"    # print(pauli)\n",[35,27393,27394,27397,27399,27402,27404,27406,27408,27411,27413,27415],{"class":37,"line":326},[35,27395,27396],{"class":113},"    observable_H_real ",[35,27398,242],{"class":241},[35,27400,27401],{"class":113}," pauli[::",[35,27403,564],{"class":241},[35,27405,500],{"class":245},[35,27407,2440],{"class":113},[35,27409,27410],{"class":271},"to_label",[35,27412,2519],{"class":113},[35,27414,445],{"class":241},[35,27416,12843],{"class":234},[35,27418,27419,27422,27424,27426,27428,27430,27432,27434,27436,27438],{"class":37,"line":332},[35,27420,27421],{"class":113},"    observable_H_imag ",[35,27423,242],{"class":241},[35,27425,27401],{"class":113},[35,27427,564],{"class":241},[35,27429,500],{"class":245},[35,27431,2440],{"class":113},[35,27433,27410],{"class":271},[35,27435,2519],{"class":113},[35,27437,445],{"class":241},[35,27439,12910],{"class":234},[35,27441,27442,27445,27447],{"class":37,"line":338},[35,27443,27444],{"class":113},"    observable_list.",[35,27446,1927],{"class":271},[35,27448,27449],{"class":113},"([observable_H_real])\n",[35,27451,27452,27454,27456],{"class":37,"line":344},[35,27453,27444],{"class":113},[35,27455,1927],{"class":271},[35,27457,27458],{"class":113},"([observable_H_imag])\n",[35,27460,27461],{"class":37,"line":349},[35,27462,103],{"emptyLinePlaceholder":102},[35,27464,27465,27467,27469],{"class":37,"line":367},[35,27466,27245],{"class":113},[35,27468,242],{"class":241},[35,27470,27471],{"class":113}," qc_trans.layout\n",[35,27473,27474],{"class":37,"line":387},[35,27475,103],{"emptyLinePlaceholder":102},[35,27477,27478,27481,27483],{"class":37,"line":404},[35,27479,27480],{"class":113},"observable_trans_list ",[35,27482,242],{"class":241},[35,27484,1850],{"class":113},[35,27486,27487,27489,27492,27494],{"class":37,"line":434},[35,27488,1874],{"class":109},[35,27490,27491],{"class":113}," observable ",[35,27493,1663],{"class":109},[35,27495,27496],{"class":113}," observable_list:\n",[35,27498,27499,27502,27504,27506],{"class":37,"line":474},[35,27500,27501],{"class":113},"    observable_op ",[35,27503,242],{"class":241},[35,27505,27209],{"class":271},[35,27507,27508],{"class":113},"(observable)\n",[35,27510,27511,27513,27515,27518,27520],{"class":37,"line":546},[35,27512,27501],{"class":113},[35,27514,242],{"class":241},[35,27516,27517],{"class":113}," observable_op.",[35,27519,27265],{"class":271},[35,27521,27293],{"class":113},[35,27523,27524,27527,27529,27532,27534],{"class":37,"line":584},[35,27525,27526],{"class":113},"    observable_trans_list.",[35,27528,1927],{"class":271},[35,27530,27531],{"class":113},"([observable_op.paulis.",[35,27533,27309],{"class":271},[35,27535,27536],{"class":113},"()])\n",[35,27538,27539],{"class":37,"line":589},[35,27540,103],{"emptyLinePlaceholder":102},[35,27542,27543,27546,27548],{"class":37,"line":594},[35,27544,27545],{"class":113},"observables_H ",[35,27547,242],{"class":241},[35,27549,27550],{"class":113}," observable_trans_list\n",[35,27552,27553],{"class":37,"line":643},[35,27554,103],{"emptyLinePlaceholder":102},[35,27556,27557],{"class":37,"line":648},[35,27558,103],{"emptyLinePlaceholder":102},[35,27560,27561],{"class":37,"line":654},[35,27562,27563],{"class":41},"# Define a sweep over parameter values\n",[35,27565,27566,27569,27571,27573,27575],{"class":37,"line":660},[35,27567,27568],{"class":113},"params ",[35,27570,242],{"class":241},[35,27572,2412],{"class":113},[35,27574,2537],{"class":271},[35,27576,27577],{"class":113},"(parameters).T\n",[35,27579,27580],{"class":37,"line":665},[35,27581,103],{"emptyLinePlaceholder":102},[35,27583,27584],{"class":37,"line":692},[35,27585,103],{"emptyLinePlaceholder":102},[35,27587,27588],{"class":37,"line":714},[35,27589,27590],{"class":41},"# Estimate the expectation value for all combinations of\n",[35,27592,27593],{"class":37,"line":729},[35,27594,27595],{"class":41},"# observables and parameter values, where the pub result will have\n",[35,27597,27598],{"class":37,"line":737},[35,27599,27600],{"class":41},"# shape (# observables, # parameter values).\n",[35,27602,27603,27606,27608,27611,27613],{"class":37,"line":768},[35,27604,27605],{"class":113},"pub ",[35,27607,242],{"class":241},[35,27609,27610],{"class":113}," (qc_trans, observables_S ",[35,27612,445],{"class":241},[35,27614,27615],{"class":113}," observables_H, params)\n",[7047,27617,27619],{"id":27618},"run-circuits","Run circuits",[12,27621,27622,27623,27674],{},"Circuits for ",[35,27624,27626,27644],{"className":27625},[7057],[35,27627,27629],{"className":27628},[7061],[7063,27630,27631],{"xmlns":7065},[7067,27632,27633,27641],{},[7070,27634,27635,27637,27639],{},[7076,27636,7829],{},[7223,27638,242],{},[7227,27640,5312],{},[7083,27642,27643],{"encoding":7085},"t=0",[35,27645,27647,27665],{"className":27646,"ariaHidden":7091},[7090],[35,27648,27650,27653,27656,27659,27662],{"className":27649},[7095],[35,27651],{"className":27652,"style":25136},[7099],[35,27654,7829],{"className":27655},[7104,7145],[35,27657],{"className":27658,"style":7450},[7248],[35,27660,242],{"className":27661},[7454],[35,27663],{"className":27664,"style":7450},[7248],[35,27666,27668,27671],{"className":27667},[7095],[35,27669],{"className":27670,"style":7263},[7099],[35,27672,5312],{"className":27673},[7104]," are classically calculable",[25,27676,27678],{"className":27,"code":27677,"language":29,"meta":30,"style":30},"qc_cliff = qc.assign_parameters({t: 0})\n\n\n# Get expectation values from experiment\nS_expval_real = StabilizerState(qc_cliff).expectation_value(\n    Pauli(\"I\" * (n_qubits) + \"X\")\n)\nS_expval_imag = StabilizerState(qc_cliff).expectation_value(\n    Pauli(\"I\" * (n_qubits) + \"Y\")\n)\n\n# Get expectation values\nS_expval = S_expval_real + 1j * S_expval_imag\n\nH_expval = 0\nfor obs_idx, (pauli, coeff) in enumerate(zip(H_op.paulis, H_op.coeffs)):\n    # Get expectation values from experiment\n    expval_real = StabilizerState(qc_cliff).expectation_value(\n        Pauli(pauli[::-1].to_label() + \"X\")\n    )\n    expval_imag = StabilizerState(qc_cliff).expectation_value(\n        Pauli(pauli[::-1].to_label() + \"Y\")\n    )\n    expval = expval_real + 1j * expval_imag\n\n    # Fill-in matrix elements\n    H_expval += coeff * expval\n\n\nprint(H_expval)\n",[32,27679,27680,27701,27705,27709,27714,27732,27752,27756,27771,27790,27794,27798,27803,27824,27828,27837,27856,27861,27876,27900,27904,27919,27941,27945,27966,27970,27975,27990,27994,27998],{"__ignoreMap":30},[35,27681,27682,27685,27687,27690,27693,27696,27698],{"class":37,"line":38},[35,27683,27684],{"class":113},"qc_cliff ",[35,27686,242],{"class":241},[35,27688,27689],{"class":113}," qc.",[35,27691,27692],{"class":271},"assign_parameters",[35,27694,27695],{"class":113},"({t: ",[35,27697,5312],{"class":245},[35,27699,27700],{"class":113},"})\n",[35,27702,27703],{"class":37,"line":45},[35,27704,103],{"emptyLinePlaceholder":102},[35,27706,27707],{"class":37,"line":51},[35,27708,103],{"emptyLinePlaceholder":102},[35,27710,27711],{"class":37,"line":57},[35,27712,27713],{"class":41},"# Get expectation values from experiment\n",[35,27715,27716,27719,27721,27724,27727,27730],{"class":37,"line":63},[35,27717,27718],{"class":113},"S_expval_real ",[35,27720,242],{"class":241},[35,27722,27723],{"class":271}," StabilizerState",[35,27725,27726],{"class":113},"(qc_cliff).",[35,27728,27729],{"class":271},"expectation_value",[35,27731,903],{"class":113},[35,27733,27734,27737,27739,27741,27743,27745,27747,27750],{"class":37,"line":69},[35,27735,27736],{"class":271},"    Pauli",[35,27738,275],{"class":113},[35,27740,12729],{"class":234},[35,27742,463],{"class":241},[35,27744,27174],{"class":113},[35,27746,445],{"class":241},[35,27748,27749],{"class":234}," \"X\"",[35,27751,384],{"class":113},[35,27753,27754],{"class":37,"line":75},[35,27755,384],{"class":113},[35,27757,27758,27761,27763,27765,27767,27769],{"class":37,"line":81},[35,27759,27760],{"class":113},"S_expval_imag ",[35,27762,242],{"class":241},[35,27764,27723],{"class":271},[35,27766,27726],{"class":113},[35,27768,27729],{"class":271},[35,27770,903],{"class":113},[35,27772,27773,27775,27777,27779,27781,27783,27785,27788],{"class":37,"line":87},[35,27774,27736],{"class":271},[35,27776,275],{"class":113},[35,27778,12729],{"class":234},[35,27780,463],{"class":241},[35,27782,27174],{"class":113},[35,27784,445],{"class":241},[35,27786,27787],{"class":234}," \"Y\"",[35,27789,384],{"class":113},[35,27791,27792],{"class":37,"line":93},[35,27793,384],{"class":113},[35,27795,27796],{"class":37,"line":99},[35,27797,103],{"emptyLinePlaceholder":102},[35,27799,27800],{"class":37,"line":106},[35,27801,27802],{"class":41},"# Get expectation values\n",[35,27804,27805,27808,27810,27813,27815,27817,27819,27821],{"class":37,"line":117},[35,27806,27807],{"class":113},"S_expval ",[35,27809,242],{"class":241},[35,27811,27812],{"class":113}," S_expval_real ",[35,27814,445],{"class":241},[35,27816,6079],{"class":245},[35,27818,6920],{"class":109},[35,27820,463],{"class":241},[35,27822,27823],{"class":113}," S_expval_imag\n",[35,27825,27826],{"class":37,"line":125},[35,27827,103],{"emptyLinePlaceholder":102},[35,27829,27830,27833,27835],{"class":37,"line":139},[35,27831,27832],{"class":113},"H_expval ",[35,27834,242],{"class":241},[35,27836,6805],{"class":245},[35,27838,27839,27841,27844,27846,27848,27850,27853],{"class":37,"line":152},[35,27840,1874],{"class":109},[35,27842,27843],{"class":113}," obs_idx, (pauli, coeff) ",[35,27845,1663],{"class":109},[35,27847,2905],{"class":241},[35,27849,275],{"class":113},[35,27851,27852],{"class":241},"zip",[35,27854,27855],{"class":113},"(H_op.paulis, H_op.coeffs)):\n",[35,27857,27858],{"class":37,"line":157},[35,27859,27860],{"class":41},"    # Get expectation values from experiment\n",[35,27862,27863,27866,27868,27870,27872,27874],{"class":37,"line":171},[35,27864,27865],{"class":113},"    expval_real ",[35,27867,242],{"class":241},[35,27869,27723],{"class":271},[35,27871,27726],{"class":113},[35,27873,27729],{"class":271},[35,27875,903],{"class":113},[35,27877,27878,27881,27884,27886,27888,27890,27892,27894,27896,27898],{"class":37,"line":184},[35,27879,27880],{"class":271},"        Pauli",[35,27882,27883],{"class":113},"(pauli[::",[35,27885,564],{"class":241},[35,27887,500],{"class":245},[35,27889,2440],{"class":113},[35,27891,27410],{"class":271},[35,27893,2519],{"class":113},[35,27895,445],{"class":241},[35,27897,27749],{"class":234},[35,27899,384],{"class":113},[35,27901,27902],{"class":37,"line":197},[35,27903,6280],{"class":113},[35,27905,27906,27909,27911,27913,27915,27917],{"class":37,"line":210},[35,27907,27908],{"class":113},"    expval_imag ",[35,27910,242],{"class":241},[35,27912,27723],{"class":271},[35,27914,27726],{"class":113},[35,27916,27729],{"class":271},[35,27918,903],{"class":113},[35,27920,27921,27923,27925,27927,27929,27931,27933,27935,27937,27939],{"class":37,"line":223},[35,27922,27880],{"class":271},[35,27924,27883],{"class":113},[35,27926,564],{"class":241},[35,27928,500],{"class":245},[35,27930,2440],{"class":113},[35,27932,27410],{"class":271},[35,27934,2519],{"class":113},[35,27936,445],{"class":241},[35,27938,27787],{"class":234},[35,27940,384],{"class":113},[35,27942,27943],{"class":37,"line":228},[35,27944,6280],{"class":113},[35,27946,27947,27950,27952,27955,27957,27959,27961,27963],{"class":37,"line":249},[35,27948,27949],{"class":113},"    expval ",[35,27951,242],{"class":241},[35,27953,27954],{"class":113}," expval_real ",[35,27956,445],{"class":241},[35,27958,6079],{"class":245},[35,27960,6920],{"class":109},[35,27962,463],{"class":241},[35,27964,27965],{"class":113}," expval_imag\n",[35,27967,27968],{"class":37,"line":254},[35,27969,103],{"emptyLinePlaceholder":102},[35,27971,27972],{"class":37,"line":260},[35,27973,27974],{"class":41},"    # Fill-in matrix elements\n",[35,27976,27977,27980,27982,27985,27987],{"class":37,"line":265},[35,27978,27979],{"class":113},"    H_expval ",[35,27981,6840],{"class":241},[35,27983,27984],{"class":113}," coeff ",[35,27986,484],{"class":241},[35,27988,27989],{"class":113}," expval\n",[35,27991,27992],{"class":37,"line":326},[35,27993,103],{"emptyLinePlaceholder":102},[35,27995,27996],{"class":37,"line":332},[35,27997,103],{"emptyLinePlaceholder":102},[35,27999,28000,28002],{"class":37,"line":338},[35,28001,1861],{"class":241},[35,28003,28004],{"class":113},"(H_expval)\n",[25,28006,28009],{"className":28007,"code":28008,"language":4991,"meta":30},[4989],"(25+0j)\n",[32,28010,28008],{"__ignoreMap":30},[12,28012,28013,28014,15316,28042,28102],{},"Execute circuits for ",[35,28015,28017,28030],{"className":28016},[7057],[35,28018,28020],{"className":28019},[7061],[7063,28021,28022],{"xmlns":7065},[7067,28023,28024,28028],{},[7070,28025,28026],{},[7076,28027,10384],{},[7083,28029,10384],{"encoding":7085},[35,28031,28033],{"className":28032,"ariaHidden":7091},[7090],[35,28034,28036,28039],{"className":28035},[7095],[35,28037],{"className":28038,"style":7100},[7099],[35,28040,10384],{"className":28041,"style":10419},[7104,7145],[35,28043,28045,28062],{"className":28044},[7057],[35,28046,28048],{"className":28047},[7061],[7063,28049,28050],{"xmlns":7065},[7067,28051,28052,28060],{},[7070,28053,28054],{},[8800,28055,28056,28058],{"accent":7091},[7076,28057,7669],{},[7223,28059,8806],{},[7083,28061,8809],{"encoding":7085},[35,28063,28065],{"className":28064,"ariaHidden":7091},[7090],[35,28066,28068,28071],{"className":28067},[7095],[35,28069],{"className":28070,"style":8819},[7099],[35,28072,28074],{"className":28073},[7104,8823],[35,28075,28077],{"className":28076},[7117],[35,28078,28080],{"className":28079},[7121],[35,28081,28083,28091],{"className":28082,"style":8819},[7125],[35,28084,28085,28088],{"style":8835},[35,28086],{"className":28087,"style":8839},[7133],[35,28089,7669],{"className":28090,"style":7684},[7104,7145],[35,28092,28093,28096],{"style":8845},[35,28094],{"className":28095,"style":8839},[7133],[35,28097,28099],{"className":28098,"style":8853},[8852],[35,28100,8806],{"className":28101},[7104]," with Estimator",[25,28104,28106],{"className":27,"code":28105,"language":29,"meta":30,"style":30},"# Experiment options\nnum_randomizations = 300\nnum_randomizations_learning = 30\nshots_per_randomization = 100\nnoise_factors = [1, 1.2, 1.4]\nlearning_pair_depths = [0, 4, 24, 48]\n\n\nexperimental_opts = {}\nexperimental_opts[\"resilience\"] = {\n    \"measure_mitigation\": True,\n    \"measure_noise_learning\": {\n        \"num_randomizations\": num_randomizations_learning,\n        \"shots_per_randomization\": shots_per_randomization,\n    },\n    \"zne_mitigation\": True,\n    \"zne\": {\"noise_factors\": noise_factors},\n    \"layer_noise_learning\": {\n        \"max_layers_to_learn\": 10,\n        \"layer_pair_depths\": learning_pair_depths,\n        \"shots_per_randomization\": shots_per_randomization,\n        \"num_randomizations\": num_randomizations_learning,\n    },\n    \"zne\": {\n        \"amplifier\": \"pea\",\n        \"extrapolated_noise_factors\": [0] + noise_factors,\n    },\n}\nexperimental_opts[\"twirling\"] = {\n    \"num_randomizations\": num_randomizations,\n    \"shots_per_randomization\": shots_per_randomization,\n    \"strategy\": \"all\",\n}\n\nestimator = Estimator(mode=backend, options=experimental_opts)\n\n\njob = estimator.run([pub])\n",[32,28107,28108,28113,28123,28132,28142,28165,28192,28196,28200,28209,28223,28234,28241,28249,28257,28261,28272,28286,28293,28304,28312,28318,28324,28328,28334,28346,28362,28366,28370,28383,28391,28398,28410,28414,28418,28446,28450,28454],{"__ignoreMap":30},[35,28109,28110],{"class":37,"line":38},[35,28111,28112],{"class":41},"# Experiment options\n",[35,28114,28115,28118,28120],{"class":37,"line":45},[35,28116,28117],{"class":113},"num_randomizations ",[35,28119,242],{"class":241},[35,28121,28122],{"class":245}," 300\n",[35,28124,28125,28128,28130],{"class":37,"line":51},[35,28126,28127],{"class":113},"num_randomizations_learning ",[35,28129,242],{"class":241},[35,28131,12695],{"class":245},[35,28133,28134,28137,28139],{"class":37,"line":57},[35,28135,28136],{"class":113},"shots_per_randomization ",[35,28138,242],{"class":241},[35,28140,28141],{"class":245}," 100\n",[35,28143,28144,28147,28149,28151,28153,28155,28158,28160,28163],{"class":37,"line":63},[35,28145,28146],{"class":113},"noise_factors ",[35,28148,242],{"class":241},[35,28150,2871],{"class":113},[35,28152,500],{"class":245},[35,28154,288],{"class":113},[35,28156,28157],{"class":245},"1.2",[35,28159,288],{"class":113},[35,28161,28162],{"class":245},"1.4",[35,28164,1074],{"class":113},[35,28166,28167,28170,28172,28174,28176,28178,28180,28182,28185,28187,28190],{"class":37,"line":69},[35,28168,28169],{"class":113},"learning_pair_depths ",[35,28171,242],{"class":241},[35,28173,2871],{"class":113},[35,28175,5312],{"class":245},[35,28177,288],{"class":113},[35,28179,528],{"class":245},[35,28181,288],{"class":113},[35,28183,28184],{"class":245},"24",[35,28186,288],{"class":113},[35,28188,28189],{"class":245},"48",[35,28191,1074],{"class":113},[35,28193,28194],{"class":37,"line":75},[35,28195,103],{"emptyLinePlaceholder":102},[35,28197,28198],{"class":37,"line":81},[35,28199,103],{"emptyLinePlaceholder":102},[35,28201,28202,28205,28207],{"class":37,"line":87},[35,28203,28204],{"class":113},"experimental_opts ",[35,28206,242],{"class":241},[35,28208,1839],{"class":113},[35,28210,28211,28214,28217,28219,28221],{"class":37,"line":93},[35,28212,28213],{"class":113},"experimental_opts[",[35,28215,28216],{"class":234},"\"resilience\"",[35,28218,238],{"class":113},[35,28220,242],{"class":241},[35,28222,1104],{"class":113},[35,28224,28225,28228,28230,28232],{"class":37,"line":99},[35,28226,28227],{"class":234},"    \"measure_mitigation\"",[35,28229,282],{"class":113},[35,28231,828],{"class":245},[35,28233,912],{"class":113},[35,28235,28236,28239],{"class":37,"line":106},[35,28237,28238],{"class":234},"    \"measure_noise_learning\"",[35,28240,1167],{"class":113},[35,28242,28243,28246],{"class":37,"line":117},[35,28244,28245],{"class":234},"        \"num_randomizations\"",[35,28247,28248],{"class":113},": num_randomizations_learning,\n",[35,28250,28251,28254],{"class":37,"line":125},[35,28252,28253],{"class":234},"        \"shots_per_randomization\"",[35,28255,28256],{"class":113},": shots_per_randomization,\n",[35,28258,28259],{"class":37,"line":139},[35,28260,1197],{"class":113},[35,28262,28263,28266,28268,28270],{"class":37,"line":152},[35,28264,28265],{"class":234},"    \"zne_mitigation\"",[35,28267,282],{"class":113},[35,28269,828],{"class":245},[35,28271,912],{"class":113},[35,28273,28274,28277,28280,28283],{"class":37,"line":157},[35,28275,28276],{"class":234},"    \"zne\"",[35,28278,28279],{"class":113},": {",[35,28281,28282],{"class":234},"\"noise_factors\"",[35,28284,28285],{"class":113},": noise_factors},\n",[35,28287,28288,28291],{"class":37,"line":171},[35,28289,28290],{"class":234},"    \"layer_noise_learning\"",[35,28292,1167],{"class":113},[35,28294,28295,28298,28300,28302],{"class":37,"line":184},[35,28296,28297],{"class":234},"        \"max_layers_to_learn\"",[35,28299,282],{"class":113},[35,28301,2393],{"class":245},[35,28303,912],{"class":113},[35,28305,28306,28309],{"class":37,"line":197},[35,28307,28308],{"class":234},"        \"layer_pair_depths\"",[35,28310,28311],{"class":113},": learning_pair_depths,\n",[35,28313,28314,28316],{"class":37,"line":210},[35,28315,28253],{"class":234},[35,28317,28256],{"class":113},[35,28319,28320,28322],{"class":37,"line":223},[35,28321,28245],{"class":234},[35,28323,28248],{"class":113},[35,28325,28326],{"class":37,"line":228},[35,28327,1197],{"class":113},[35,28329,28330,28332],{"class":37,"line":249},[35,28331,28276],{"class":234},[35,28333,1167],{"class":113},[35,28335,28336,28339,28341,28344],{"class":37,"line":254},[35,28337,28338],{"class":234},"        \"amplifier\"",[35,28340,282],{"class":113},[35,28342,28343],{"class":234},"\"pea\"",[35,28345,912],{"class":113},[35,28347,28348,28351,28353,28355,28357,28359],{"class":37,"line":260},[35,28349,28350],{"class":234},"        \"extrapolated_noise_factors\"",[35,28352,1139],{"class":113},[35,28354,5312],{"class":245},[35,28356,238],{"class":113},[35,28358,445],{"class":241},[35,28360,28361],{"class":113}," noise_factors,\n",[35,28363,28364],{"class":37,"line":265},[35,28365,1197],{"class":113},[35,28367,28368],{"class":37,"line":326},[35,28369,1228],{"class":113},[35,28371,28372,28374,28377,28379,28381],{"class":37,"line":332},[35,28373,28213],{"class":113},[35,28375,28376],{"class":234},"\"twirling\"",[35,28378,238],{"class":113},[35,28380,242],{"class":241},[35,28382,1104],{"class":113},[35,28384,28385,28388],{"class":37,"line":338},[35,28386,28387],{"class":234},"    \"num_randomizations\"",[35,28389,28390],{"class":113},": num_randomizations,\n",[35,28392,28393,28396],{"class":37,"line":344},[35,28394,28395],{"class":234},"    \"shots_per_randomization\"",[35,28397,28256],{"class":113},[35,28399,28400,28403,28405,28408],{"class":37,"line":349},[35,28401,28402],{"class":234},"    \"strategy\"",[35,28404,282],{"class":113},[35,28406,28407],{"class":234},"\"all\"",[35,28409,912],{"class":113},[35,28411,28412],{"class":37,"line":367},[35,28413,1228],{"class":113},[35,28415,28416],{"class":37,"line":387},[35,28417,103],{"emptyLinePlaceholder":102},[35,28419,28420,28423,28425,28428,28430,28433,28435,28438,28441,28443],{"class":37,"line":404},[35,28421,28422],{"class":113},"estimator ",[35,28424,242],{"class":241},[35,28426,28427],{"class":271}," Estimator",[35,28429,275],{"class":113},[35,28431,28432],{"class":822},"mode",[35,28434,242],{"class":241},[35,28436,28437],{"class":113},"backend, ",[35,28439,28440],{"class":822},"options",[35,28442,242],{"class":241},[35,28444,28445],{"class":113},"experimental_opts)\n",[35,28447,28448],{"class":37,"line":434},[35,28449,103],{"emptyLinePlaceholder":102},[35,28451,28452],{"class":37,"line":474},[35,28453,103],{"emptyLinePlaceholder":102},[35,28455,28456,28459,28461,28464,28466],{"class":37,"line":546},[35,28457,28458],{"class":113},"job ",[35,28460,242],{"class":241},[35,28462,28463],{"class":113}," estimator.",[35,28465,27007],{"class":271},[35,28467,28468],{"class":113},"([pub])\n",[5019,28470,28472],{"id":28471},"step-4-post-process-and-return-result-in-desired-classical-format","Step 4: Post-process and return result in desired classical format",[25,28474,28476],{"className":27,"code":28475,"language":29,"meta":30,"style":30},"results = job.result()[0]\n",[32,28477,28478],{"__ignoreMap":30},[35,28479,28480,28483,28485,28488,28491,28494,28496],{"class":37,"line":38},[35,28481,28482],{"class":113},"results ",[35,28484,242],{"class":241},[35,28486,28487],{"class":113}," job.",[35,28489,28490],{"class":271},"result",[35,28492,28493],{"class":113},"()[",[35,28495,5312],{"class":245},[35,28497,1074],{"class":113},[7047,28499,28501],{"id":28500},"calculate-effective-hamiltonian-and-overlap-matrices","Calculate Effective Hamiltonian and Overlap matrices",[12,28503,28504,28505,28541],{},"First calculate the phase accumulated by the ",[35,28506,28508,28526],{"className":28507},[7057],[35,28509,28511],{"className":28510},[7061],[7063,28512,28513],{"xmlns":7065},[7067,28514,28515,28523],{},[7070,28516,28517,28519,28521],{},[7076,28518,7283],{"mathvariant":7282},[7227,28520,5312],{},[7223,28522,7290],{"stretchy":7289},[7083,28524,28525],{"encoding":7085},"\\vert 0 \\rangle",[35,28527,28529],{"className":28528,"ariaHidden":7091},[7090],[35,28530,28532,28535,28538],{"className":28531},[7095],[35,28533],{"className":28534,"style":7303},[7099],[35,28536,14253],{"className":28537},[7104],[35,28539,7290],{"className":28540},[7314]," state during the uncontrolled time evolution",[25,28543,28545],{"className":27,"code":28544,"language":29,"meta":30,"style":30},"prefactors = [\n    np.exp(-1j * sum([c for p, c in H_op.to_list() if \"Z\" in p]) * i * dt)\n    for i in range(1, krylov_dim)\n]\n",[32,28546,28547,28556,28611,28628],{"__ignoreMap":30},[35,28548,28549,28552,28554],{"class":37,"line":38},[35,28550,28551],{"class":113},"prefactors ",[35,28553,242],{"class":241},[35,28555,894],{"class":113},[35,28557,28558,28561,28564,28566,28568,28570,28572,28574,28576,28579,28581,28584,28586,28588,28590,28592,28594,28597,28599,28602,28604,28606,28608],{"class":37,"line":45},[35,28559,28560],{"class":113},"    np.",[35,28562,28563],{"class":271},"exp",[35,28565,275],{"class":113},[35,28567,564],{"class":241},[35,28569,500],{"class":245},[35,28571,6920],{"class":109},[35,28573,463],{"class":241},[35,28575,13333],{"class":241},[35,28577,28578],{"class":113},"([c ",[35,28580,1874],{"class":109},[35,28582,28583],{"class":113}," p, c ",[35,28585,1663],{"class":109},[35,28587,6513],{"class":113},[35,28589,6516],{"class":271},[35,28591,2519],{"class":113},[35,28593,1492],{"class":109},[35,28595,28596],{"class":234}," \"Z\"",[35,28598,26493],{"class":109},[35,28600,28601],{"class":113}," p]) ",[35,28603,484],{"class":241},[35,28605,2958],{"class":113},[35,28607,484],{"class":241},[35,28609,28610],{"class":113}," dt)\n",[35,28612,28613,28615,28617,28619,28621,28623,28625],{"class":37,"line":51},[35,28614,1657],{"class":109},[35,28616,2958],{"class":113},[35,28618,1663],{"class":109},[35,28620,12746],{"class":241},[35,28622,275],{"class":113},[35,28624,500],{"class":245},[35,28626,28627],{"class":113},", krylov_dim)\n",[35,28629,28630],{"class":37,"line":57},[35,28631,1074],{"class":113},[12,28633,28634,28635],{},"Once we have the results of the circuit executions we can post-process the data to calculate the matrix elements of ",[35,28636,28638,28651],{"className":28637},[7057],[35,28639,28641],{"className":28640},[7061],[7063,28642,28643],{"xmlns":7065},[7067,28644,28645,28649],{},[7070,28646,28647],{},[7076,28648,10384],{},[7083,28650,10384],{"encoding":7085},[35,28652,28654],{"className":28653,"ariaHidden":7091},[7090],[35,28655,28657,28660],{"className":28656},[7095],[35,28658],{"className":28659,"style":7100},[7099],[35,28661,10384],{"className":28662,"style":10419},[7104,7145],[25,28664,28666],{"className":27,"code":28665,"language":29,"meta":30,"style":30},"# Assemble S, the overlap matrix of dimension D:\nS_first_row = np.zeros(krylov_dim, dtype=complex)\nS_first_row[0] = 1 + 0j\n\n# Add in ancilla-only measurements:\nfor i in range(krylov_dim - 1):\n    # Get expectation values from experiment\n    expval_real = results.data.evs[0][0][\n        i\n    ]  # automatic extrapolated evs if ZNE is used\n    expval_imag = results.data.evs[1][0][\n        i\n    ]  # automatic extrapolated evs if ZNE is used\n\n    # Get expectation values\n    expval = expval_real + 1j * expval_imag\n    S_first_row[i + 1] += prefactors[i] * expval\n\nS_first_row_list = S_first_row.tolist()  # for saving purposes\n\n\nS_circ = np.zeros((krylov_dim, krylov_dim), dtype=complex)\n\n# Distribute entries from first row across matrix:\nfor i, j in it.product(range(krylov_dim), repeat=2):\n    if i >= j:\n        S_circ[j, i] = S_first_row[i - j]\n    else:\n        S_circ[j, i] = np.conj(S_first_row[j - i])\n",[32,28667,28668,28673,28693,28713,28717,28722,28741,28745,28763,28768,28775,28791,28795,28801,28805,28810,28828,28848,28852,28870,28874,28878,28898,28902,28907,28937,28949,28964,28970],{"__ignoreMap":30},[35,28669,28670],{"class":37,"line":38},[35,28671,28672],{"class":41},"# Assemble S, the overlap matrix of dimension D:\n",[35,28674,28675,28678,28680,28682,28684,28687,28689,28691],{"class":37,"line":45},[35,28676,28677],{"class":113},"S_first_row ",[35,28679,242],{"class":241},[35,28681,2412],{"class":113},[35,28683,6717],{"class":271},[35,28685,28686],{"class":113},"(krylov_dim, ",[35,28688,6723],{"class":822},[35,28690,6726],{"class":241},[35,28692,384],{"class":113},[35,28694,28695,28698,28700,28702,28704,28706,28708,28710],{"class":37,"line":51},[35,28696,28697],{"class":113},"S_first_row[",[35,28699,5312],{"class":245},[35,28701,238],{"class":113},[35,28703,242],{"class":241},[35,28705,6079],{"class":245},[35,28707,415],{"class":241},[35,28709,361],{"class":245},[35,28711,28712],{"class":109},"j\n",[35,28714,28715],{"class":37,"line":57},[35,28716,103],{"emptyLinePlaceholder":102},[35,28718,28719],{"class":37,"line":63},[35,28720,28721],{"class":41},"# Add in ancilla-only measurements:\n",[35,28723,28724,28726,28728,28730,28732,28735,28737,28739],{"class":37,"line":69},[35,28725,1874],{"class":109},[35,28727,2958],{"class":113},[35,28729,1663],{"class":109},[35,28731,12746],{"class":241},[35,28733,28734],{"class":113},"(krylov_dim ",[35,28736,564],{"class":241},[35,28738,6079],{"class":245},[35,28740,6479],{"class":113},[35,28742,28743],{"class":37,"line":75},[35,28744,27860],{"class":41},[35,28746,28747,28749,28751,28754,28756,28758,28760],{"class":37,"line":81},[35,28748,27865],{"class":113},[35,28750,242],{"class":241},[35,28752,28753],{"class":113}," results.data.evs[",[35,28755,5312],{"class":245},[35,28757,1997],{"class":113},[35,28759,5312],{"class":245},[35,28761,28762],{"class":113},"][\n",[35,28764,28765],{"class":37,"line":87},[35,28766,28767],{"class":113},"        i\n",[35,28769,28770,28772],{"class":37,"line":93},[35,28771,6788],{"class":113},[35,28773,28774],{"class":41},"# automatic extrapolated evs if ZNE is used\n",[35,28776,28777,28779,28781,28783,28785,28787,28789],{"class":37,"line":99},[35,28778,27908],{"class":113},[35,28780,242],{"class":241},[35,28782,28753],{"class":113},[35,28784,500],{"class":245},[35,28786,1997],{"class":113},[35,28788,5312],{"class":245},[35,28790,28762],{"class":113},[35,28792,28793],{"class":37,"line":106},[35,28794,28767],{"class":113},[35,28796,28797,28799],{"class":37,"line":117},[35,28798,6788],{"class":113},[35,28800,28774],{"class":41},[35,28802,28803],{"class":37,"line":125},[35,28804,103],{"emptyLinePlaceholder":102},[35,28806,28807],{"class":37,"line":139},[35,28808,28809],{"class":41},"    # Get expectation values\n",[35,28811,28812,28814,28816,28818,28820,28822,28824,28826],{"class":37,"line":152},[35,28813,27949],{"class":113},[35,28815,242],{"class":241},[35,28817,27954],{"class":113},[35,28819,445],{"class":241},[35,28821,6079],{"class":245},[35,28823,6920],{"class":109},[35,28825,463],{"class":241},[35,28827,27965],{"class":113},[35,28829,28830,28833,28835,28837,28839,28841,28844,28846],{"class":37,"line":157},[35,28831,28832],{"class":113},"    S_first_row[i ",[35,28834,445],{"class":241},[35,28836,6079],{"class":245},[35,28838,238],{"class":113},[35,28840,6840],{"class":241},[35,28842,28843],{"class":113}," prefactors[i] ",[35,28845,484],{"class":241},[35,28847,27989],{"class":113},[35,28849,28850],{"class":37,"line":171},[35,28851,103],{"emptyLinePlaceholder":102},[35,28853,28854,28857,28859,28862,28864,28867],{"class":37,"line":184},[35,28855,28856],{"class":113},"S_first_row_list ",[35,28858,242],{"class":241},[35,28860,28861],{"class":113}," S_first_row.",[35,28863,2669],{"class":271},[35,28865,28866],{"class":113},"()  ",[35,28868,28869],{"class":41},"# for saving purposes\n",[35,28871,28872],{"class":37,"line":197},[35,28873,103],{"emptyLinePlaceholder":102},[35,28875,28876],{"class":37,"line":210},[35,28877,103],{"emptyLinePlaceholder":102},[35,28879,28880,28883,28885,28887,28889,28892,28894,28896],{"class":37,"line":223},[35,28881,28882],{"class":113},"S_circ ",[35,28884,242],{"class":241},[35,28886,2412],{"class":113},[35,28888,6717],{"class":271},[35,28890,28891],{"class":113},"((krylov_dim, krylov_dim), ",[35,28893,6723],{"class":822},[35,28895,6726],{"class":241},[35,28897,384],{"class":113},[35,28899,28900],{"class":37,"line":228},[35,28901,103],{"emptyLinePlaceholder":102},[35,28903,28904],{"class":37,"line":249},[35,28905,28906],{"class":41},"# Distribute entries from first row across matrix:\n",[35,28908,28909,28911,28914,28916,28918,28921,28923,28925,28928,28931,28933,28935],{"class":37,"line":254},[35,28910,1874],{"class":109},[35,28912,28913],{"class":113}," i, j ",[35,28915,1663],{"class":109},[35,28917,6759],{"class":113},[35,28919,28920],{"class":271},"product",[35,28922,275],{"class":113},[35,28924,6767],{"class":241},[35,28926,28927],{"class":113},"(krylov_dim), ",[35,28929,28930],{"class":822},"repeat",[35,28932,242],{"class":241},[35,28934,460],{"class":245},[35,28936,6479],{"class":113},[35,28938,28939,28941,28943,28946],{"class":37,"line":260},[35,28940,352],{"class":109},[35,28942,2958],{"class":113},[35,28944,28945],{"class":241},">=",[35,28947,28948],{"class":113}," j:\n",[35,28950,28951,28954,28956,28959,28961],{"class":37,"line":265},[35,28952,28953],{"class":113},"        S_circ[j, i] ",[35,28955,242],{"class":241},[35,28957,28958],{"class":113}," S_first_row[i ",[35,28960,564],{"class":241},[35,28962,28963],{"class":113}," j]\n",[35,28965,28966,28968],{"class":37,"line":326},[35,28967,6403],{"class":109},[35,28969,364],{"class":113},[35,28971,28972,28974,28976,28978,28980,28983,28985],{"class":37,"line":332},[35,28973,28953],{"class":113},[35,28975,242],{"class":241},[35,28977,2412],{"class":113},[35,28979,6293],{"class":271},[35,28981,28982],{"class":113},"(S_first_row[j ",[35,28984,564],{"class":241},[35,28986,28987],{"class":113}," i])\n",[25,28989,28991],{"className":27,"code":28990,"language":29,"meta":30,"style":30},"Matrix(S_circ)\n",[32,28992,28993],{"__ignoreMap":30},[35,28994,28995,28998],{"class":37,"line":38},[35,28996,28997],{"class":271},"Matrix",[35,28999,29000],{"class":113},"(S_circ)\n",[12,29002,29003,29004],{},"And the matrix elements of ",[35,29005,29007,29024],{"className":29006},[7057],[35,29008,29010],{"className":29009},[7061],[7063,29011,29012],{"xmlns":7065},[7067,29013,29014,29022],{},[7070,29015,29016],{},[8800,29017,29018,29020],{"accent":7091},[7076,29019,7669],{},[7223,29021,8806],{},[7083,29023,8809],{"encoding":7085},[35,29025,29027],{"className":29026,"ariaHidden":7091},[7090],[35,29028,29030,29033],{"className":29029},[7095],[35,29031],{"className":29032,"style":8819},[7099],[35,29034,29036],{"className":29035},[7104,8823],[35,29037,29039],{"className":29038},[7117],[35,29040,29042],{"className":29041},[7121],[35,29043,29045,29053],{"className":29044,"style":8819},[7125],[35,29046,29047,29050],{"style":8835},[35,29048],{"className":29049,"style":8839},[7133],[35,29051,7669],{"className":29052,"style":7684},[7104,7145],[35,29054,29055,29058],{"style":8845},[35,29056],{"className":29057,"style":8839},[7133],[35,29059,29061],{"className":29060,"style":8853},[8852],[35,29062,8806],{"className":29063},[7104],[25,29065,29067],{"className":27,"code":29066,"language":29,"meta":30,"style":30},"# Assemble S, the overlap matrix of dimension D:\nH_first_row = np.zeros(krylov_dim, dtype=complex)\nH_first_row[0] = H_expval\n\nfor obs_idx, (pauli, coeff) in enumerate(zip(H_op.paulis, H_op.coeffs)):\n    # Add in ancilla-only measurements:\n    for i in range(krylov_dim - 1):\n        # Get expectation values from experiment\n        expval_real = results.data.evs[2 + 2 * obs_idx][0][\n            i\n        ]  # automatic extrapolated evs if ZNE is used\n        expval_imag = results.data.evs[2 + 2 * obs_idx + 1][0][\n            i\n        ]  # automatic extrapolated evs if ZNE is used\n\n        # Get expectation values\n        expval = expval_real + 1j * expval_imag\n        H_first_row[i + 1] += prefactors[i] * coeff * expval\n\nH_first_row_list = H_first_row.tolist()\n\nH_eff_circ = np.zeros((krylov_dim, krylov_dim), dtype=complex)\n\n# Distribute entries from first row across matrix:\nfor i, j in it.product(range(krylov_dim), repeat=2):\n    if i >= j:\n        H_eff_circ[j, i] = H_first_row[i - j]\n    else:\n        H_eff_circ[j, i] = np.conj(H_first_row[j - i])\n",[32,29068,29069,29073,29092,29106,29110,29126,29131,29149,29154,29178,29183,29190,29220,29224,29230,29234,29239,29258,29281,29285,29299,29303,29322,29326,29330,29356,29366,29380,29386],{"__ignoreMap":30},[35,29070,29071],{"class":37,"line":38},[35,29072,28672],{"class":41},[35,29074,29075,29078,29080,29082,29084,29086,29088,29090],{"class":37,"line":45},[35,29076,29077],{"class":113},"H_first_row ",[35,29079,242],{"class":241},[35,29081,2412],{"class":113},[35,29083,6717],{"class":271},[35,29085,28686],{"class":113},[35,29087,6723],{"class":822},[35,29089,6726],{"class":241},[35,29091,384],{"class":113},[35,29093,29094,29097,29099,29101,29103],{"class":37,"line":51},[35,29095,29096],{"class":113},"H_first_row[",[35,29098,5312],{"class":245},[35,29100,238],{"class":113},[35,29102,242],{"class":241},[35,29104,29105],{"class":113}," H_expval\n",[35,29107,29108],{"class":37,"line":57},[35,29109,103],{"emptyLinePlaceholder":102},[35,29111,29112,29114,29116,29118,29120,29122,29124],{"class":37,"line":63},[35,29113,1874],{"class":109},[35,29115,27843],{"class":113},[35,29117,1663],{"class":109},[35,29119,2905],{"class":241},[35,29121,275],{"class":113},[35,29123,27852],{"class":241},[35,29125,27855],{"class":113},[35,29127,29128],{"class":37,"line":69},[35,29129,29130],{"class":41},"    # Add in ancilla-only measurements:\n",[35,29132,29133,29135,29137,29139,29141,29143,29145,29147],{"class":37,"line":75},[35,29134,1657],{"class":109},[35,29136,2958],{"class":113},[35,29138,1663],{"class":109},[35,29140,12746],{"class":241},[35,29142,28734],{"class":113},[35,29144,564],{"class":241},[35,29146,6079],{"class":245},[35,29148,6479],{"class":113},[35,29150,29151],{"class":37,"line":81},[35,29152,29153],{"class":41},"        # Get expectation values from experiment\n",[35,29155,29156,29159,29161,29163,29165,29167,29169,29171,29174,29176],{"class":37,"line":87},[35,29157,29158],{"class":113},"        expval_real ",[35,29160,242],{"class":241},[35,29162,28753],{"class":113},[35,29164,460],{"class":245},[35,29166,415],{"class":241},[35,29168,424],{"class":245},[35,29170,463],{"class":241},[35,29172,29173],{"class":113}," obs_idx][",[35,29175,5312],{"class":245},[35,29177,28762],{"class":113},[35,29179,29180],{"class":37,"line":93},[35,29181,29182],{"class":113},"            i\n",[35,29184,29185,29188],{"class":37,"line":99},[35,29186,29187],{"class":113},"        ]  ",[35,29189,28774],{"class":41},[35,29191,29192,29195,29197,29199,29201,29203,29205,29207,29210,29212,29214,29216,29218],{"class":37,"line":106},[35,29193,29194],{"class":113},"        expval_imag ",[35,29196,242],{"class":241},[35,29198,28753],{"class":113},[35,29200,460],{"class":245},[35,29202,415],{"class":241},[35,29204,424],{"class":245},[35,29206,463],{"class":241},[35,29208,29209],{"class":113}," obs_idx ",[35,29211,445],{"class":241},[35,29213,6079],{"class":245},[35,29215,1997],{"class":113},[35,29217,5312],{"class":245},[35,29219,28762],{"class":113},[35,29221,29222],{"class":37,"line":117},[35,29223,29182],{"class":113},[35,29225,29226,29228],{"class":37,"line":125},[35,29227,29187],{"class":113},[35,29229,28774],{"class":41},[35,29231,29232],{"class":37,"line":139},[35,29233,103],{"emptyLinePlaceholder":102},[35,29235,29236],{"class":37,"line":152},[35,29237,29238],{"class":41},"        # Get expectation values\n",[35,29240,29241,29244,29246,29248,29250,29252,29254,29256],{"class":37,"line":157},[35,29242,29243],{"class":113},"        expval ",[35,29245,242],{"class":241},[35,29247,27954],{"class":113},[35,29249,445],{"class":241},[35,29251,6079],{"class":245},[35,29253,6920],{"class":109},[35,29255,463],{"class":241},[35,29257,27965],{"class":113},[35,29259,29260,29263,29265,29267,29269,29271,29273,29275,29277,29279],{"class":37,"line":171},[35,29261,29262],{"class":113},"        H_first_row[i ",[35,29264,445],{"class":241},[35,29266,6079],{"class":245},[35,29268,238],{"class":113},[35,29270,6840],{"class":241},[35,29272,28843],{"class":113},[35,29274,484],{"class":241},[35,29276,27984],{"class":113},[35,29278,484],{"class":241},[35,29280,27989],{"class":113},[35,29282,29283],{"class":37,"line":184},[35,29284,103],{"emptyLinePlaceholder":102},[35,29286,29287,29290,29292,29295,29297],{"class":37,"line":197},[35,29288,29289],{"class":113},"H_first_row_list ",[35,29291,242],{"class":241},[35,29293,29294],{"class":113}," H_first_row.",[35,29296,2669],{"class":271},[35,29298,2101],{"class":113},[35,29300,29301],{"class":37,"line":210},[35,29302,103],{"emptyLinePlaceholder":102},[35,29304,29305,29308,29310,29312,29314,29316,29318,29320],{"class":37,"line":223},[35,29306,29307],{"class":113},"H_eff_circ ",[35,29309,242],{"class":241},[35,29311,2412],{"class":113},[35,29313,6717],{"class":271},[35,29315,28891],{"class":113},[35,29317,6723],{"class":822},[35,29319,6726],{"class":241},[35,29321,384],{"class":113},[35,29323,29324],{"class":37,"line":228},[35,29325,103],{"emptyLinePlaceholder":102},[35,29327,29328],{"class":37,"line":249},[35,29329,28906],{"class":41},[35,29331,29332,29334,29336,29338,29340,29342,29344,29346,29348,29350,29352,29354],{"class":37,"line":254},[35,29333,1874],{"class":109},[35,29335,28913],{"class":113},[35,29337,1663],{"class":109},[35,29339,6759],{"class":113},[35,29341,28920],{"class":271},[35,29343,275],{"class":113},[35,29345,6767],{"class":241},[35,29347,28927],{"class":113},[35,29349,28930],{"class":822},[35,29351,242],{"class":241},[35,29353,460],{"class":245},[35,29355,6479],{"class":113},[35,29357,29358,29360,29362,29364],{"class":37,"line":260},[35,29359,352],{"class":109},[35,29361,2958],{"class":113},[35,29363,28945],{"class":241},[35,29365,28948],{"class":113},[35,29367,29368,29371,29373,29376,29378],{"class":37,"line":265},[35,29369,29370],{"class":113},"        H_eff_circ[j, i] ",[35,29372,242],{"class":241},[35,29374,29375],{"class":113}," H_first_row[i ",[35,29377,564],{"class":241},[35,29379,28963],{"class":113},[35,29381,29382,29384],{"class":37,"line":326},[35,29383,6403],{"class":109},[35,29385,364],{"class":113},[35,29387,29388,29390,29392,29394,29396,29399,29401],{"class":37,"line":332},[35,29389,29370],{"class":113},[35,29391,242],{"class":241},[35,29393,2412],{"class":113},[35,29395,6293],{"class":271},[35,29397,29398],{"class":113},"(H_first_row[j ",[35,29400,564],{"class":241},[35,29402,28987],{"class":113},[25,29404,29406],{"className":27,"code":29405,"language":29,"meta":30,"style":30},"Matrix(H_eff_circ)\n",[32,29407,29408],{"__ignoreMap":30},[35,29409,29410,29412],{"class":37,"line":38},[35,29411,28997],{"class":271},[35,29413,29414],{"class":113},"(H_eff_circ)\n",[12,29416,29417,29418,3439],{},"Finally, we can solve the generalized eigenvalue problem for ",[35,29419,29421,29438],{"className":29420},[7057],[35,29422,29424],{"className":29423},[7061],[7063,29425,29426],{"xmlns":7065},[7067,29427,29428,29436],{},[7070,29429,29430],{},[8800,29431,29432,29434],{"accent":7091},[7076,29433,7669],{},[7223,29435,8806],{},[7083,29437,8809],{"encoding":7085},[35,29439,29441],{"className":29440,"ariaHidden":7091},[7090],[35,29442,29444,29447],{"className":29443},[7095],[35,29445],{"className":29446,"style":8819},[7099],[35,29448,29450],{"className":29449},[7104,8823],[35,29451,29453],{"className":29452},[7117],[35,29454,29456],{"className":29455},[7121],[35,29457,29459,29467],{"className":29458,"style":8819},[7125],[35,29460,29461,29464],{"style":8835},[35,29462],{"className":29463,"style":8839},[7133],[35,29465,7669],{"className":29466,"style":7684},[7104,7145],[35,29468,29469,29472],{"style":8845},[35,29470],{"className":29471,"style":8839},[7133],[35,29473,29475],{"className":29474,"style":8853},[8852],[35,29476,8806],{"className":29477},[7104],[12,29479,29480],{},[35,29481,29483,29519],{"className":29482},[7057],[35,29484,29486],{"className":29485},[7061],[7063,29487,29488],{"xmlns":7065},[7067,29489,29490,29516],{},[7070,29491,29492,29498,29504,29506,29508,29510],{},[8800,29493,29494,29496],{"accent":7091},[7076,29495,7669],{},[7223,29497,8806],{},[8800,29499,29500,29502],{"accent":7091},[7076,29501,10701],{},[7223,29503,10704],{},[7223,29505,242],{},[7076,29507,10701],{},[7076,29509,10384],{},[8800,29511,29512,29514],{"accent":7091},[7076,29513,10701],{},[7223,29515,10704],{},[7083,29517,29518],{"encoding":7085},"\\tilde{H} \\vec{c} = c S \\vec{c}",[35,29520,29522,29603],{"className":29521,"ariaHidden":7091},[7090],[35,29523,29525,29528,29559,29594,29597,29600],{"className":29524},[7095],[35,29526],{"className":29527,"style":8819},[7099],[35,29529,29531],{"className":29530},[7104,8823],[35,29532,29534],{"className":29533},[7117],[35,29535,29537],{"className":29536},[7121],[35,29538,29540,29548],{"className":29539,"style":8819},[7125],[35,29541,29542,29545],{"style":8835},[35,29543],{"className":29544,"style":8839},[7133],[35,29546,7669],{"className":29547,"style":7684},[7104,7145],[35,29549,29550,29553],{"style":8845},[35,29551],{"className":29552,"style":8839},[7133],[35,29554,29556],{"className":29555,"style":8853},[8852],[35,29557,8806],{"className":29558},[7104],[35,29560,29562],{"className":29561},[7104,8823],[35,29563,29565],{"className":29564},[7117],[35,29566,29568],{"className":29567},[7121],[35,29569,29571,29579],{"className":29570,"style":10784},[7125],[35,29572,29573,29576],{"style":8835},[35,29574],{"className":29575,"style":8839},[7133],[35,29577,10701],{"className":29578},[7104,7145],[35,29580,29581,29584],{"style":8835},[35,29582],{"className":29583,"style":8839},[7133],[35,29585,29587],{"className":29586,"style":10801},[8852],[35,29588,29590],{"className":29589,"style":10806},[10805],[10808,29591,29592],{"xmlns":10810,"width":10811,"height":10812,"style":10813,"viewBox":10814,"preserveAspectRatio":10815},[10817,29593],{"d":10819},[35,29595],{"className":29596,"style":7450},[7248],[35,29598,242],{"className":29599},[7454],[35,29601],{"className":29602,"style":7450},[7248],[35,29604,29606,29609,29612,29615],{"className":29605},[7095],[35,29607],{"className":29608,"style":10784},[7099],[35,29610,10701],{"className":29611},[7104,7145],[35,29613,10384],{"className":29614,"style":10419},[7104,7145],[35,29616,29618],{"className":29617},[7104,8823],[35,29619,29621],{"className":29620},[7117],[35,29622,29624],{"className":29623},[7121],[35,29625,29627,29635],{"className":29626,"style":10784},[7125],[35,29628,29629,29632],{"style":8835},[35,29630],{"className":29631,"style":8839},[7133],[35,29633,10701],{"className":29634},[7104,7145],[35,29636,29637,29640],{"style":8835},[35,29638],{"className":29639,"style":8839},[7133],[35,29641,29643],{"className":29642,"style":10801},[8852],[35,29644,29646],{"className":29645,"style":10806},[10805],[10808,29647,29648],{"xmlns":10810,"width":10811,"height":10812,"style":10813,"viewBox":10814,"preserveAspectRatio":10815},[10817,29649],{"d":10819},[12,29651,29652,29653],{},"and get an estimate of the ground state energy ",[35,29654,29656,29680],{"className":29655},[7057],[35,29657,29659],{"className":29658},[7061],[7063,29660,29661],{"xmlns":7065},[7067,29662,29663,29677],{},[7070,29664,29665],{},[7699,29666,29667,29669],{},[7076,29668,10701],{},[7070,29670,29671,29673,29675],{},[7076,29672,8946],{},[7076,29674,7824],{},[7076,29676,291],{},[7083,29678,29679],{"encoding":7085},"c_{min}",[35,29681,29683],{"className":29682,"ariaHidden":7091},[7090],[35,29684,29686,29690],{"className":29685},[7095],[35,29687],{"className":29688,"style":29689},[7099],"height:0.5806em;vertical-align:-0.15em;",[35,29691,29693,29696],{"className":29692},[7104],[35,29694,10701],{"className":29695},[7104,7145],[35,29697,29699],{"className":29698},[7113],[35,29700,29702,29725],{"className":29701},[7117,7731],[35,29703,29705,29722],{"className":29704},[7121],[35,29706,29708],{"className":29707,"style":11329},[7125],[35,29709,29710,29713],{"style":9926},[35,29711],{"className":29712,"style":7134},[7133],[35,29714,29716],{"className":29715},[7138,7139,7140,7141],[35,29717,29719],{"className":29718},[7104,7141],[35,29720,570],{"className":29721},[7104,7145,7141],[35,29723,7756],{"className":29724},[7755],[35,29726,29728],{"className":29727},[7121],[35,29729,29731],{"className":29730,"style":7763},[7125],[35,29732],{},[25,29734,29736],{"className":27,"code":29735,"language":29,"meta":30,"style":30},"gnd_en_circ_est_list = []\nfor d in range(1, krylov_dim + 1):\n    # Solve generalized eigenvalue problem for different size of the Krylov space\n    gnd_en_circ_est = solve_regularized_gen_eig(\n        H_eff_circ[:d, :d], S_circ[:d, :d], threshold=9e-1\n    )\n    gnd_en_circ_est_list.append(gnd_en_circ_est)\n    print(\"The estimated ground state energy is: \", gnd_en_circ_est)\n",[32,29737,29738,29747,29771,29776,29787,29800,29804,29814],{"__ignoreMap":30},[35,29739,29740,29743,29745],{"class":37,"line":38},[35,29741,29742],{"class":113},"gnd_en_circ_est_list ",[35,29744,242],{"class":241},[35,29746,1850],{"class":113},[35,29748,29749,29751,29754,29756,29758,29760,29762,29765,29767,29769],{"class":37,"line":45},[35,29750,1874],{"class":109},[35,29752,29753],{"class":113}," d ",[35,29755,1663],{"class":109},[35,29757,12746],{"class":241},[35,29759,275],{"class":113},[35,29761,500],{"class":245},[35,29763,29764],{"class":113},", krylov_dim ",[35,29766,445],{"class":241},[35,29768,6079],{"class":245},[35,29770,6479],{"class":113},[35,29772,29773],{"class":37,"line":51},[35,29774,29775],{"class":41},"    # Solve generalized eigenvalue problem for different size of the Krylov space\n",[35,29777,29778,29781,29783,29785],{"class":37,"line":57},[35,29779,29780],{"class":113},"    gnd_en_circ_est ",[35,29782,242],{"class":241},[35,29784,6037],{"class":271},[35,29786,903],{"class":113},[35,29788,29789,29792,29795,29797],{"class":37,"line":63},[35,29790,29791],{"class":113},"        H_eff_circ[:d, :d], S_circ[:d, :d], ",[35,29793,29794],{"class":822},"threshold",[35,29796,242],{"class":241},[35,29798,29799],{"class":245},"9e-1\n",[35,29801,29802],{"class":37,"line":69},[35,29803,6280],{"class":113},[35,29805,29806,29809,29811],{"class":37,"line":75},[35,29807,29808],{"class":113},"    gnd_en_circ_est_list.",[35,29810,1927],{"class":271},[35,29812,29813],{"class":113},"(gnd_en_circ_est)\n",[35,29815,29816,29818,29820,29823],{"class":37,"line":81},[35,29817,6637],{"class":241},[35,29819,275],{"class":113},[35,29821,29822],{"class":234},"\"The estimated ground state energy is: \"",[35,29824,29825],{"class":113},", gnd_en_circ_est)\n",[25,29827,29830],{"className":29828,"code":29829,"language":4991,"meta":30},[4989],"The estimated ground state energy is:  25.0\nThe estimated ground state energy is:  22.572154819954875\nThe estimated ground state energy is:  21.691509219286587\nThe estimated ground state energy is:  21.23882298756386\nThe estimated ground state energy is:  20.965499325470294\n",[32,29831,29829],{"__ignoreMap":30},[12,29833,29834],{},"For a single-particle sector, we can efficiently calculate the ground state of this sector of the Hamiltonian classically",[25,29836,29838],{"className":27,"code":29837,"language":29,"meta":30,"style":30},"gs_en = single_particle_gs(H_op, n_qubits)\n",[32,29839,29840],{"__ignoreMap":30},[35,29841,29842,29845,29847,29849],{"class":37,"line":38},[35,29843,29844],{"class":113},"gs_en ",[35,29846,242],{"class":241},[35,29848,6466],{"class":271},[35,29850,29851],{"class":113},"(H_op, n_qubits)\n",[25,29853,29856],{"className":29854,"code":29855,"language":4991,"meta":30},[4989],"n_sys_qubits 30\nn_exc 1 , subspace dimension 31\nsingle particle ground state energy:  21.021912418526906\n",[32,29857,29855],{"__ignoreMap":30},[25,29859,29861],{"className":27,"code":29860,"language":29,"meta":30,"style":30},"plt.plot(\n    range(1, krylov_dim + 1),\n    gnd_en_circ_est_list,\n    color=\"blue\",\n    linestyle=\"-.\",\n    label=\"KQD estimate\",\n)\nplt.plot(\n    range(1, krylov_dim + 1),\n    [gs_en] * krylov_dim,\n    color=\"red\",\n    linestyle=\"-\",\n    label=\"exact\",\n)\nplt.xticks(range(1, krylov_dim + 1), range(1, krylov_dim + 1))\nplt.legend()\nplt.xlabel(\"Krylov space dimension\")\nplt.ylabel(\"Energy\")\nplt.title(\n    \"Estimating Ground state energy with Krylov Quantum Diagonalization\"\n)\nplt.show()\n",[32,29862,29863,29872,29889,29894,29906,29918,29930,29934,29942,29958,29968,29979,29990,30001,30005,30041,30049,30063,30076,30084,30089,30093],{"__ignoreMap":30},[35,29864,29865,29867,29870],{"class":37,"line":38},[35,29866,2378],{"class":113},[35,29868,29869],{"class":271},"plot",[35,29871,903],{"class":113},[35,29873,29874,29877,29879,29881,29883,29885,29887],{"class":37,"line":45},[35,29875,29876],{"class":241},"    range",[35,29878,275],{"class":113},[35,29880,500],{"class":245},[35,29882,29764],{"class":113},[35,29884,445],{"class":241},[35,29886,6079],{"class":245},[35,29888,18715],{"class":113},[35,29890,29891],{"class":37,"line":51},[35,29892,29893],{"class":113},"    gnd_en_circ_est_list,\n",[35,29895,29896,29899,29901,29904],{"class":37,"line":57},[35,29897,29898],{"class":822},"    color",[35,29900,242],{"class":241},[35,29902,29903],{"class":234},"\"blue\"",[35,29905,912],{"class":113},[35,29907,29908,29911,29913,29916],{"class":37,"line":63},[35,29909,29910],{"class":822},"    linestyle",[35,29912,242],{"class":241},[35,29914,29915],{"class":234},"\"-.\"",[35,29917,912],{"class":113},[35,29919,29920,29923,29925,29928],{"class":37,"line":69},[35,29921,29922],{"class":822},"    label",[35,29924,242],{"class":241},[35,29926,29927],{"class":234},"\"KQD estimate\"",[35,29929,912],{"class":113},[35,29931,29932],{"class":37,"line":75},[35,29933,384],{"class":113},[35,29935,29936,29938,29940],{"class":37,"line":81},[35,29937,2378],{"class":113},[35,29939,29869],{"class":271},[35,29941,903],{"class":113},[35,29943,29944,29946,29948,29950,29952,29954,29956],{"class":37,"line":87},[35,29945,29876],{"class":241},[35,29947,275],{"class":113},[35,29949,500],{"class":245},[35,29951,29764],{"class":113},[35,29953,445],{"class":241},[35,29955,6079],{"class":245},[35,29957,18715],{"class":113},[35,29959,29960,29963,29965],{"class":37,"line":93},[35,29961,29962],{"class":113},"    [gs_en] ",[35,29964,484],{"class":241},[35,29966,29967],{"class":113}," krylov_dim,\n",[35,29969,29970,29972,29974,29977],{"class":37,"line":99},[35,29971,29898],{"class":822},[35,29973,242],{"class":241},[35,29975,29976],{"class":234},"\"red\"",[35,29978,912],{"class":113},[35,29980,29981,29983,29985,29988],{"class":37,"line":106},[35,29982,29910],{"class":822},[35,29984,242],{"class":241},[35,29986,29987],{"class":234},"\"-\"",[35,29989,912],{"class":113},[35,29991,29992,29994,29996,29999],{"class":37,"line":117},[35,29993,29922],{"class":822},[35,29995,242],{"class":241},[35,29997,29998],{"class":234},"\"exact\"",[35,30000,912],{"class":113},[35,30002,30003],{"class":37,"line":125},[35,30004,384],{"class":113},[35,30006,30007,30009,30011,30013,30015,30017,30019,30021,30023,30025,30027,30029,30031,30033,30035,30037,30039],{"class":37,"line":139},[35,30008,2378],{"class":113},[35,30010,2658],{"class":271},[35,30012,275],{"class":113},[35,30014,6767],{"class":241},[35,30016,275],{"class":113},[35,30018,500],{"class":245},[35,30020,29764],{"class":113},[35,30022,445],{"class":241},[35,30024,6079],{"class":245},[35,30026,685],{"class":113},[35,30028,6767],{"class":241},[35,30030,275],{"class":113},[35,30032,500],{"class":245},[35,30034,29764],{"class":113},[35,30036,445],{"class":241},[35,30038,6079],{"class":245},[35,30040,2401],{"class":113},[35,30042,30043,30045,30047],{"class":37,"line":152},[35,30044,2378],{"class":113},[35,30046,3108],{"class":271},[35,30048,2101],{"class":113},[35,30050,30051,30053,30056,30058,30061],{"class":37,"line":157},[35,30052,2378],{"class":113},[35,30054,30055],{"class":271},"xlabel",[35,30057,275],{"class":113},[35,30059,30060],{"class":234},"\"Krylov space dimension\"",[35,30062,384],{"class":113},[35,30064,30065,30067,30069,30071,30074],{"class":37,"line":171},[35,30066,2378],{"class":113},[35,30068,2700],{"class":271},[35,30070,275],{"class":113},[35,30072,30073],{"class":234},"\"Energy\"",[35,30075,384],{"class":113},[35,30077,30078,30080,30082],{"class":37,"line":184},[35,30079,2378],{"class":113},[35,30081,2715],{"class":271},[35,30083,903],{"class":113},[35,30085,30086],{"class":37,"line":197},[35,30087,30088],{"class":234},"    \"Estimating Ground state energy with Krylov Quantum Diagonalization\"\n",[35,30090,30091],{"class":37,"line":210},[35,30092,384],{"class":113},[35,30094,30095,30097,30099],{"class":37,"line":223},[35,30096,2378],{"class":113},[35,30098,2792],{"class":271},[35,30100,2101],{"class":113},[5019,30102,30104],{"id":30103},"appendix-krylov-subspace-from-real-time-evolutions","Appendix: Krylov subspace from real time-evolutions",[12,30106,30107],{},"The unitary Krylov space is defined as",[35,30109,30111],{"className":30110},[7318],[35,30112,30114,30222],{"className":30113},[7057],[35,30115,30117],{"className":30116},[7061],[7063,30118,30119],{"xmlns":7065,"display":7327},[7067,30120,30121,30219],{},[7070,30122,30123,30129,30131,30133,30135,30137,30139,30141,30143,30145,30147],{},[7699,30124,30125,30127],{},[7076,30126,7079],{"mathvariant":7078},[7076,30128,7810],{},[7223,30130,275],{"stretchy":7289},[7076,30132,7669],{},[7223,30134,2877],{"separator":7091},[7076,30136,7283],{"mathvariant":7282},[7076,30138,8322],{},[7223,30140,7290],{"stretchy":7289},[7223,30142,1694],{"stretchy":7289},[7223,30144,242],{},[10694,30146,35],{},[7070,30148,30149,30151,30153,30155,30157,30159,30178,30180,30182,30184,30186,30189,30191,30211,30213,30215,30217],{},[7223,30150,1681],{"fence":7091},[7076,30152,7283],{"mathvariant":7282},[7076,30154,8322],{},[7223,30156,7290],{"stretchy":7289},[7223,30158,2877],{"separator":7091},[7073,30160,30161,30163],{},[7076,30162,7817],{},[7070,30164,30165,30167,30169,30171,30174,30176],{},[7223,30166,7225],{},[7076,30168,7824],{},[7076,30170,7669],{},[10694,30172,30173],{}," ",[7076,30175,9487],{},[7076,30177,7829],{},[7076,30179,7283],{"mathvariant":7282},[7076,30181,8322],{},[7223,30183,7290],{"stretchy":7289},[7223,30185,2877],{"separator":7091},[7223,30187,30188],{},"…",[7223,30190,2877],{"separator":7091},[7073,30192,30193,30195],{},[7076,30194,7817],{},[7070,30196,30197,30199,30201,30203,30205,30207,30209],{},[7223,30198,7225],{},[7076,30200,7824],{},[7076,30202,6778],{},[7076,30204,7669],{},[10694,30206,30173],{},[7076,30208,9487],{},[7076,30210,7829],{},[7076,30212,7283],{"mathvariant":7282},[7076,30214,8322],{},[7223,30216,7290],{"stretchy":7289},[7223,30218,1697],{"fence":7091},[7083,30220,30221],{"encoding":7085},"\\mathcal{K}_U(H, |\\psi\\rangle) = \\text{span}\\left\\{ |\\psi\\rangle,  e^{-iH\\,dt} |\\psi\\rangle, \\dots, e^{-irH\\,dt} |\\psi\\rangle \\right\\}",[35,30223,30225,30302],{"className":30224,"ariaHidden":7091},[7090],[35,30226,30228,30231,30271,30274,30277,30280,30283,30286,30289,30293,30296,30299],{"className":30227},[7095],[35,30229],{"className":30230,"style":7303},[7099],[35,30232,30234,30237],{"className":30233},[7104],[35,30235,7079],{"className":30236,"style":7109},[7104,7108],[35,30238,30240],{"className":30239},[7113],[35,30241,30243,30263],{"className":30242},[7117,7731],[35,30244,30246,30260],{"className":30245},[7121],[35,30247,30249],{"className":30248,"style":7738},[7125],[35,30250,30251,30254],{"style":7741},[35,30252],{"className":30253,"style":7134},[7133],[35,30255,30257],{"className":30256},[7138,7139,7140,7141],[35,30258,7810],{"className":30259,"style":7845},[7104,7145,7141],[35,30261,7756],{"className":30262},[7755],[35,30264,30266],{"className":30265},[7121],[35,30267,30269],{"className":30268,"style":7763},[7125],[35,30270],{},[35,30272,275],{"className":30273},[7472],[35,30275,7669],{"className":30276,"style":7684},[7104,7145],[35,30278,2877],{"className":30279},[7492],[35,30281],{"className":30282,"style":7496},[7248],[35,30284,7283],{"className":30285},[7104],[35,30287,8322],{"className":30288,"style":7310},[7104,7145],[35,30290,30292],{"className":30291},[7314],"⟩)",[35,30294],{"className":30295,"style":7450},[7248],[35,30297,242],{"className":30298},[7454],[35,30300],{"className":30301,"style":7450},[7248],[35,30303,30305,30309,30315,30318],{"className":30304},[7095],[35,30306],{"className":30307,"style":30308},[7099],"height:1.2491em;vertical-align:-0.35em;",[35,30310,30312],{"className":30311},[7104,4991],[35,30313,35],{"className":30314},[7104],[35,30316],{"className":30317,"style":7496},[7248],[35,30319,30321,30327,30330,30333,30336,30339,30342,30390,30393,30396,30399,30402,30405,30408,30411,30414,30417,30467,30470,30473,30476],{"className":30320},[7468],[35,30322,30324],{"className":30323,"style":7474},[7472,7473],[35,30325,1681],{"className":30326},[7478,7479],[35,30328,7283],{"className":30329},[7104],[35,30331,8322],{"className":30332,"style":7310},[7104,7145],[35,30334,7290],{"className":30335},[7314],[35,30337,2877],{"className":30338},[7492],[35,30340],{"className":30341,"style":7496},[7248],[35,30343,30345,30348],{"className":30344},[7104],[35,30346,7817],{"className":30347},[7104,7145],[35,30349,30351],{"className":30350},[7113],[35,30352,30354],{"className":30353},[7117],[35,30355,30357],{"className":30356},[7121],[35,30358,30360],{"className":30359,"style":9568},[7125],[35,30361,30362,30365],{"style":7437},[35,30363],{"className":30364,"style":7134},[7133],[35,30366,30368],{"className":30367},[7138,7139,7140,7141],[35,30369,30371,30374,30377,30380,30384,30387],{"className":30370},[7104,7141],[35,30372,7225],{"className":30373},[7104,7141],[35,30375,7824],{"className":30376},[7104,7145,7141],[35,30378,7669],{"className":30379,"style":7684},[7104,7145,7141],[35,30381],{"className":30382,"style":30383},[7248,7141],"margin-right:0.1952em;",[35,30385,9487],{"className":30386},[7104,7145,7141],[35,30388,7829],{"className":30389},[7104,7145,7141],[35,30391,7283],{"className":30392},[7104],[35,30394,8322],{"className":30395,"style":7310},[7104,7145],[35,30397,7290],{"className":30398},[7314],[35,30400,2877],{"className":30401},[7492],[35,30403],{"className":30404,"style":7496},[7248],[35,30406,30188],{"className":30407},[7468],[35,30409],{"className":30410,"style":7496},[7248],[35,30412,2877],{"className":30413},[7492],[35,30415],{"className":30416,"style":7496},[7248],[35,30418,30420,30423],{"className":30419},[7104],[35,30421,7817],{"className":30422},[7104,7145],[35,30424,30426],{"className":30425},[7113],[35,30427,30429],{"className":30428},[7117],[35,30430,30432],{"className":30431},[7121],[35,30433,30435],{"className":30434,"style":9568},[7125],[35,30436,30437,30440],{"style":7437},[35,30438],{"className":30439,"style":7134},[7133],[35,30441,30443],{"className":30442},[7138,7139,7140,7141],[35,30444,30446,30449,30452,30455,30458,30461,30464],{"className":30445},[7104,7141],[35,30447,7225],{"className":30448},[7104,7141],[35,30450,7824],{"className":30451},[7104,7145,7141],[35,30453,6778],{"className":30454,"style":7146},[7104,7145,7141],[35,30456,7669],{"className":30457,"style":7684},[7104,7145,7141],[35,30459],{"className":30460,"style":30383},[7248,7141],[35,30462,9487],{"className":30463},[7104,7145,7141],[35,30465,7829],{"className":30466},[7104,7145,7141],[35,30468,7283],{"className":30469},[7104],[35,30471,8322],{"className":30472,"style":7310},[7104,7145],[35,30474,7290],{"className":30475},[7314],[35,30477,30479],{"className":30478,"style":7474},[7314,7473],[35,30480,1697],{"className":30481},[7478,7479],[12,30483,30484,30485,30518,30519,30547,30548,30607],{},"for some timestep ",[35,30486,30488,30503],{"className":30487},[7057],[35,30489,30491],{"className":30490},[7061],[7063,30492,30493],{"xmlns":7065},[7067,30494,30495,30501],{},[7070,30496,30497,30499],{},[7076,30498,9487],{},[7076,30500,7829],{},[7083,30502,9991],{"encoding":7085},[35,30504,30506],{"className":30505,"ariaHidden":7091},[7090],[35,30507,30509,30512,30515],{"className":30508},[7095],[35,30510],{"className":30511,"style":9962},[7099],[35,30513,9487],{"className":30514},[7104,7145],[35,30516,7829],{"className":30517},[7104,7145]," that we will determine later. Temporarily assume ",[35,30520,30522,30535],{"className":30521},[7057],[35,30523,30525],{"className":30524},[7061],[7063,30526,30527],{"xmlns":7065},[7067,30528,30529,30533],{},[7070,30530,30531],{},[7076,30532,6778],{},[7083,30534,6778],{"encoding":7085},[35,30536,30538],{"className":30537,"ariaHidden":7091},[7090],[35,30539,30541,30544],{"className":30540},[7095],[35,30542],{"className":30543,"style":7173},[7099],[35,30545,6778],{"className":30546,"style":7146},[7104,7145]," is even: then define ",[35,30549,30551,30573],{"className":30550},[7057],[35,30552,30554],{"className":30553},[7061],[7063,30555,30556],{"xmlns":7065},[7067,30557,30558,30570],{},[7070,30559,30560,30562,30564,30566,30568],{},[7076,30561,9487],{},[7223,30563,242],{},[7076,30565,6778],{},[7076,30567,398],{"mathvariant":7282},[7227,30569,460],{},[7083,30571,30572],{"encoding":7085},"d=r\u002F2",[35,30574,30576,30594],{"className":30575,"ariaHidden":7091},[7090],[35,30577,30579,30582,30585,30588,30591],{"className":30578},[7095],[35,30580],{"className":30581,"style":9962},[7099],[35,30583,9487],{"className":30584},[7104,7145],[35,30586],{"className":30587,"style":7450},[7248],[35,30589,242],{"className":30590},[7454],[35,30592],{"className":30593,"style":7450},[7248],[35,30595,30597,30600,30603],{"className":30596},[7095],[35,30598],{"className":30599,"style":7303},[7099],[35,30601,6778],{"className":30602,"style":7146},[7104,7145],[35,30604,30606],{"className":30605},[7104],"\u002F2",". Notice that when we project the Hamiltonian into the Krylov space above, it is indistinguishable from the Krylov space",[35,30609,30611],{"className":30610},[7318],[35,30612,30614,30792],{"className":30613},[7057],[35,30615,30617],{"className":30616},[7061],[7063,30618,30619],{"xmlns":7065,"display":7327},[7067,30620,30621,30789],{},[7070,30622,30623,30629,30631,30633,30635,30637,30639,30641,30643,30645,30647,30787],{},[7699,30624,30625,30627],{},[7076,30626,7079],{"mathvariant":7078},[7076,30628,7810],{},[7223,30630,275],{"stretchy":7289},[7076,30632,7669],{},[7223,30634,2877],{"separator":7091},[7076,30636,7283],{"mathvariant":7282},[7076,30638,8322],{},[7223,30640,7290],{"stretchy":7289},[7223,30642,1694],{"stretchy":7289},[7223,30644,242],{},[10694,30646,35],{},[7070,30648,30649,30651,30673,30675,30677,30679,30681,30707,30709,30711,30713,30715,30717,30719,30747,30749,30751,30753,30755,30779,30781,30783,30785],{},[7223,30650,1681],{"fence":7091},[7073,30652,30653,30655],{},[7076,30654,7817],{},[7070,30656,30657,30659,30661,30663,30665,30667,30669,30671],{},[7076,30658,7824],{},[10694,30660,30173],{},[7076,30662,9487],{},[10694,30664,30173],{},[7076,30666,7669],{},[10694,30668,30173],{},[7076,30670,9487],{},[7076,30672,7829],{},[7076,30674,7283],{"mathvariant":7282},[7076,30676,8322],{},[7223,30678,7290],{"stretchy":7289},[7223,30680,2877],{"separator":7091},[7073,30682,30683,30685],{},[7076,30684,7817],{},[7070,30686,30687,30689,30691,30693,30695,30697,30699,30701,30703,30705],{},[7076,30688,7824],{},[7223,30690,275],{"stretchy":7289},[7076,30692,9487],{},[7223,30694,7225],{},[7227,30696,500],{},[7223,30698,1694],{"stretchy":7289},[7076,30700,7669],{},[10694,30702,30173],{},[7076,30704,9487],{},[7076,30706,7829],{},[7076,30708,7283],{"mathvariant":7282},[7076,30710,8322],{},[7223,30712,7290],{"stretchy":7289},[7223,30714,2877],{"separator":7091},[7223,30716,30188],{},[7223,30718,2877],{"separator":7091},[7073,30720,30721,30723],{},[7076,30722,7817],{},[7070,30724,30725,30727,30729,30731,30733,30735,30737,30739,30741,30743,30745],{},[7223,30726,7225],{},[7076,30728,7824],{},[7223,30730,275],{"stretchy":7289},[7076,30732,9487],{},[7223,30734,7225],{},[7227,30736,500],{},[7223,30738,1694],{"stretchy":7289},[7076,30740,7669],{},[10694,30742,30173],{},[7076,30744,9487],{},[7076,30746,7829],{},[7076,30748,7283],{"mathvariant":7282},[7076,30750,8322],{},[7223,30752,7290],{"stretchy":7289},[7223,30754,2877],{"separator":7091},[7073,30756,30757,30759],{},[7076,30758,7817],{},[7070,30760,30761,30763,30765,30767,30769,30771,30773,30775,30777],{},[7223,30762,7225],{},[7076,30764,7824],{},[10694,30766,30173],{},[7076,30768,9487],{},[10694,30770,30173],{},[7076,30772,7669],{},[10694,30774,30173],{},[7076,30776,9487],{},[7076,30778,7829],{},[7076,30780,7283],{"mathvariant":7282},[7076,30782,8322],{},[7223,30784,7290],{"stretchy":7289},[7223,30786,1697],{"fence":7091},[7223,30788,2877],{"separator":7091},[7083,30790,30791],{"encoding":7085},"\\mathcal{K}_U(H, |\\psi\\rangle) = \\text{span}\\left\\{ e^{i\\,d\\,H\\,dt}|\\psi\\rangle,  e^{i(d-1)H\\,dt} |\\psi\\rangle, \\dots, e^{-i(d-1)H\\,dt} |\\psi\\rangle, e^{-i\\,d\\,H\\,dt} |\\psi\\rangle \\right\\},",[35,30793,30795,30871],{"className":30794,"ariaHidden":7091},[7090],[35,30796,30798,30801,30841,30844,30847,30850,30853,30856,30859,30862,30865,30868],{"className":30797},[7095],[35,30799],{"className":30800,"style":7303},[7099],[35,30802,30804,30807],{"className":30803},[7104],[35,30805,7079],{"className":30806,"style":7109},[7104,7108],[35,30808,30810],{"className":30809},[7113],[35,30811,30813,30833],{"className":30812},[7117,7731],[35,30814,30816,30830],{"className":30815},[7121],[35,30817,30819],{"className":30818,"style":7738},[7125],[35,30820,30821,30824],{"style":7741},[35,30822],{"className":30823,"style":7134},[7133],[35,30825,30827],{"className":30826},[7138,7139,7140,7141],[35,30828,7810],{"className":30829,"style":7845},[7104,7145,7141],[35,30831,7756],{"className":30832},[7755],[35,30834,30836],{"className":30835},[7121],[35,30837,30839],{"className":30838,"style":7763},[7125],[35,30840],{},[35,30842,275],{"className":30843},[7472],[35,30845,7669],{"className":30846,"style":7684},[7104,7145],[35,30848,2877],{"className":30849},[7492],[35,30851],{"className":30852,"style":7496},[7248],[35,30854,7283],{"className":30855},[7104],[35,30857,8322],{"className":30858,"style":7310},[7104,7145],[35,30860,30292],{"className":30861},[7314],[35,30863],{"className":30864,"style":7450},[7248],[35,30866,242],{"className":30867},[7454],[35,30869],{"className":30870,"style":7450},[7248],[35,30872,30874,30877,30883,30886,31198,31201],{"className":30873},[7095],[35,30875],{"className":30876,"style":18060},[7099],[35,30878,30880],{"className":30879},[7104,4991],[35,30881,35],{"className":30882},[7104],[35,30884],{"className":30885,"style":7496},[7248],[35,30887,30889,30895,30948,30951,30954,30957,30960,30963,31023,31026,31029,31032,31035,31038,31041,31044,31047,31050,31112,31115,31118,31121,31124,31127,31183,31186,31189,31192],{"className":30888},[7468],[35,30890,30892],{"className":30891,"style":7474},[7472,7473],[35,30893,1681],{"className":30894},[7478,14437],[35,30896,30898,30901],{"className":30897},[7104],[35,30899,7817],{"className":30900},[7104,7145],[35,30902,30904],{"className":30903},[7113],[35,30905,30907],{"className":30906},[7117],[35,30908,30910],{"className":30909},[7121],[35,30911,30913],{"className":30912,"style":9568},[7125],[35,30914,30915,30918],{"style":7437},[35,30916],{"className":30917,"style":7134},[7133],[35,30919,30921],{"className":30920},[7138,7139,7140,7141],[35,30922,30924,30927,30930,30933,30936,30939,30942,30945],{"className":30923},[7104,7141],[35,30925,7824],{"className":30926},[7104,7145,7141],[35,30928],{"className":30929,"style":30383},[7248,7141],[35,30931,9487],{"className":30932},[7104,7145,7141],[35,30934],{"className":30935,"style":30383},[7248,7141],[35,30937,7669],{"className":30938,"style":7684},[7104,7145,7141],[35,30940],{"className":30941,"style":30383},[7248,7141],[35,30943,9487],{"className":30944},[7104,7145,7141],[35,30946,7829],{"className":30947},[7104,7145,7141],[35,30949,7283],{"className":30950},[7104],[35,30952,8322],{"className":30953,"style":7310},[7104,7145],[35,30955,7290],{"className":30956},[7314],[35,30958,2877],{"className":30959},[7492],[35,30961],{"className":30962,"style":7496},[7248],[35,30964,30966,30969],{"className":30965},[7104],[35,30967,7817],{"className":30968},[7104,7145],[35,30970,30972],{"className":30971},[7113],[35,30973,30975],{"className":30974},[7117],[35,30976,30978],{"className":30977},[7121],[35,30979,30982],{"className":30980,"style":30981},[7125],"height:0.938em;",[35,30983,30984,30987],{"style":7437},[35,30985],{"className":30986,"style":7134},[7133],[35,30988,30990],{"className":30989},[7138,7139,7140,7141],[35,30991,30993,30996,30999,31002,31005,31008,31011,31014,31017,31020],{"className":30992},[7104,7141],[35,30994,7824],{"className":30995},[7104,7145,7141],[35,30997,275],{"className":30998},[7472,7141],[35,31000,9487],{"className":31001},[7104,7145,7141],[35,31003,7225],{"className":31004},[7253,7141],[35,31006,500],{"className":31007},[7104,7141],[35,31009,1694],{"className":31010},[7314,7141],[35,31012,7669],{"className":31013,"style":7684},[7104,7145,7141],[35,31015],{"className":31016,"style":30383},[7248,7141],[35,31018,9487],{"className":31019},[7104,7145,7141],[35,31021,7829],{"className":31022},[7104,7145,7141],[35,31024,7283],{"className":31025},[7104],[35,31027,8322],{"className":31028,"style":7310},[7104,7145],[35,31030,7290],{"className":31031},[7314],[35,31033,2877],{"className":31034},[7492],[35,31036],{"className":31037,"style":7496},[7248],[35,31039,30188],{"className":31040},[7468],[35,31042],{"className":31043,"style":7496},[7248],[35,31045,2877],{"className":31046},[7492],[35,31048],{"className":31049,"style":7496},[7248],[35,31051,31053,31056],{"className":31052},[7104],[35,31054,7817],{"className":31055},[7104,7145],[35,31057,31059],{"className":31058},[7113],[35,31060,31062],{"className":31061},[7117],[35,31063,31065],{"className":31064},[7121],[35,31066,31068],{"className":31067,"style":30981},[7125],[35,31069,31070,31073],{"style":7437},[35,31071],{"className":31072,"style":7134},[7133],[35,31074,31076],{"className":31075},[7138,7139,7140,7141],[35,31077,31079,31082,31085,31088,31091,31094,31097,31100,31103,31106,31109],{"className":31078},[7104,7141],[35,31080,7225],{"className":31081},[7104,7141],[35,31083,7824],{"className":31084},[7104,7145,7141],[35,31086,275],{"className":31087},[7472,7141],[35,31089,9487],{"className":31090},[7104,7145,7141],[35,31092,7225],{"className":31093},[7253,7141],[35,31095,500],{"className":31096},[7104,7141],[35,31098,1694],{"className":31099},[7314,7141],[35,31101,7669],{"className":31102,"style":7684},[7104,7145,7141],[35,31104],{"className":31105,"style":30383},[7248,7141],[35,31107,9487],{"className":31108},[7104,7145,7141],[35,31110,7829],{"className":31111},[7104,7145,7141],[35,31113,7283],{"className":31114},[7104],[35,31116,8322],{"className":31117,"style":7310},[7104,7145],[35,31119,7290],{"className":31120},[7314],[35,31122,2877],{"className":31123},[7492],[35,31125],{"className":31126,"style":7496},[7248],[35,31128,31130,31133],{"className":31129},[7104],[35,31131,7817],{"className":31132},[7104,7145],[35,31134,31136],{"className":31135},[7113],[35,31137,31139],{"className":31138},[7117],[35,31140,31142],{"className":31141},[7121],[35,31143,31145],{"className":31144,"style":9568},[7125],[35,31146,31147,31150],{"style":7437},[35,31148],{"className":31149,"style":7134},[7133],[35,31151,31153],{"className":31152},[7138,7139,7140,7141],[35,31154,31156,31159,31162,31165,31168,31171,31174,31177,31180],{"className":31155},[7104,7141],[35,31157,7225],{"className":31158},[7104,7141],[35,31160,7824],{"className":31161},[7104,7145,7141],[35,31163],{"className":31164,"style":30383},[7248,7141],[35,31166,9487],{"className":31167},[7104,7145,7141],[35,31169],{"className":31170,"style":30383},[7248,7141],[35,31172,7669],{"className":31173,"style":7684},[7104,7145,7141],[35,31175],{"className":31176,"style":30383},[7248,7141],[35,31178,9487],{"className":31179},[7104,7145,7141],[35,31181,7829],{"className":31182},[7104,7145,7141],[35,31184,7283],{"className":31185},[7104],[35,31187,8322],{"className":31188,"style":7310},[7104,7145],[35,31190,7290],{"className":31191},[7314],[35,31193,31195],{"className":31194,"style":7474},[7314,7473],[35,31196,1697],{"className":31197},[7478,14437],[35,31199],{"className":31200,"style":7496},[7248],[35,31202,2877],{"className":31203},[7492],[12,31205,31206,31207,31235],{},"that is, where all the time-evolutions are shifted backward by ",[35,31208,31210,31223],{"className":31209},[7057],[35,31211,31213],{"className":31212},[7061],[7063,31214,31215],{"xmlns":7065},[7067,31216,31217,31221],{},[7070,31218,31219],{},[7076,31220,9487],{},[7083,31222,9487],{"encoding":7085},[35,31224,31226],{"className":31225,"ariaHidden":7091},[7090],[35,31227,31229,31232],{"className":31228},[7095],[35,31230],{"className":31231,"style":9962},[7099],[35,31233,9487],{"className":31234},[7104,7145]," timesteps.\nThe reason it is indistinguishable is because the matrix elements",[35,31237,31239],{"className":31238},[7318],[35,31240,31242,31374],{"className":31241},[7057],[35,31243,31245],{"className":31244},[7061],[7063,31246,31247],{"xmlns":7065,"display":7327},[7067,31248,31249,31371],{},[7070,31250,31251,31267,31269,31271,31273,31275,31297,31299,31323,31325,31327,31329,31331,31333,31335,31337,31339,31365,31367,31369],{},[7699,31252,31253,31259],{},[8800,31254,31255,31257],{"accent":7091},[7076,31256,7669],{},[7223,31258,8806],{},[7070,31260,31261,31263,31265],{},[7076,31262,6920],{},[7223,31264,2877],{"separator":7091},[7076,31266,279],{},[7223,31268,242],{},[7223,31270,8953],{"stretchy":7289},[7076,31272,8322],{},[7076,31274,7283],{"mathvariant":7282},[7073,31276,31277,31279],{},[7076,31278,7817],{},[7070,31280,31281,31283,31285,31287,31289,31291,31293,31295],{},[7076,31282,7824],{},[10694,31284,30173],{},[7076,31286,6920],{},[10694,31288,30173],{},[7076,31290,7669],{},[10694,31292,30173],{},[7076,31294,9487],{},[7076,31296,7829],{},[7076,31298,7669],{},[7073,31300,31301,31303],{},[7076,31302,7817],{},[7070,31304,31305,31307,31309,31311,31313,31315,31317,31319,31321],{},[7223,31306,7225],{},[7076,31308,7824],{},[10694,31310,30173],{},[7076,31312,279],{},[10694,31314,30173],{},[7076,31316,7669],{},[10694,31318,30173],{},[7076,31320,9487],{},[7076,31322,7829],{},[7076,31324,7283],{"mathvariant":7282},[7076,31326,8322],{},[7223,31328,7290],{"stretchy":7289},[7223,31330,242],{},[7223,31332,8953],{"stretchy":7289},[7076,31334,8322],{},[7076,31336,7283],{"mathvariant":7282},[7076,31338,7669],{},[7073,31340,31341,31343],{},[7076,31342,7817],{},[7070,31344,31345,31347,31349,31351,31353,31355,31357,31359,31361,31363],{},[7076,31346,7824],{},[7223,31348,275],{"stretchy":7289},[7076,31350,6920],{},[7223,31352,7225],{},[7076,31354,279],{},[7223,31356,1694],{"stretchy":7289},[7076,31358,7669],{},[10694,31360,30173],{},[7076,31362,9487],{},[7076,31364,7829],{},[7076,31366,7283],{"mathvariant":7282},[7076,31368,8322],{},[7223,31370,7290],{"stretchy":7289},[7083,31372,31373],{"encoding":7085},"\\tilde{H}_{j,k} = \\langle\\psi|e^{i\\,j\\,H\\,dt}He^{-i\\,k\\,H\\,dt}|\\psi\\rangle=\\langle\\psi|He^{i(j-k)H\\,dt}|\\psi\\rangle",[35,31375,31377,31469,31614],{"className":31376,"ariaHidden":7091},[7090],[35,31378,31380,31383,31460,31463,31466],{"className":31379},[7095],[35,31381],{"className":31382,"style":11282},[7099],[35,31384,31386,31417],{"className":31385},[7104],[35,31387,31389],{"className":31388},[7104,8823],[35,31390,31392],{"className":31391},[7117],[35,31393,31395],{"className":31394},[7121],[35,31396,31398,31406],{"className":31397,"style":8819},[7125],[35,31399,31400,31403],{"style":8835},[35,31401],{"className":31402,"style":8839},[7133],[35,31404,7669],{"className":31405,"style":7684},[7104,7145],[35,31407,31408,31411],{"style":8845},[35,31409],{"className":31410,"style":8839},[7133],[35,31412,31414],{"className":31413,"style":8853},[8852],[35,31415,8806],{"className":31416},[7104],[35,31418,31420],{"className":31419},[7113],[35,31421,31423,31452],{"className":31422},[7117,7731],[35,31424,31426,31449],{"className":31425},[7121],[35,31427,31429],{"className":31428,"style":15760},[7125],[35,31430,31431,31434],{"style":9038},[35,31432],{"className":31433,"style":7134},[7133],[35,31435,31437],{"className":31436},[7138,7139,7140,7141],[35,31438,31440,31443,31446],{"className":31439},[7104,7141],[35,31441,6920],{"className":31442,"style":11350},[7104,7145,7141],[35,31444,2877],{"className":31445},[7492,7141],[35,31447,279],{"className":31448,"style":8221},[7104,7145,7141],[35,31450,7756],{"className":31451},[7755],[35,31453,31455],{"className":31454},[7121],[35,31456,31458],{"className":31457,"style":11360},[7125],[35,31459],{},[35,31461],{"className":31462,"style":7450},[7248],[35,31464,242],{"className":31465},[7454],[35,31467],{"className":31468,"style":7450},[7248],[35,31470,31472,31475,31478,31481,31484,31537,31540,31596,31599,31602,31605,31608,31611],{"className":31471},[7095],[35,31473],{"className":31474,"style":9540},[7099],[35,31476,8953],{"className":31477},[7472],[35,31479,8322],{"className":31480,"style":7310},[7104,7145],[35,31482,7283],{"className":31483},[7104],[35,31485,31487,31490],{"className":31486},[7104],[35,31488,7817],{"className":31489},[7104,7145],[35,31491,31493],{"className":31492},[7113],[35,31494,31496],{"className":31495},[7117],[35,31497,31499],{"className":31498},[7121],[35,31500,31502],{"className":31501,"style":9568},[7125],[35,31503,31504,31507],{"style":7437},[35,31505],{"className":31506,"style":7134},[7133],[35,31508,31510],{"className":31509},[7138,7139,7140,7141],[35,31511,31513,31516,31519,31522,31525,31528,31531,31534],{"className":31512},[7104,7141],[35,31514,7824],{"className":31515},[7104,7145,7141],[35,31517],{"className":31518,"style":30383},[7248,7141],[35,31520,6920],{"className":31521,"style":11350},[7104,7145,7141],[35,31523],{"className":31524,"style":30383},[7248,7141],[35,31526,7669],{"className":31527,"style":7684},[7104,7145,7141],[35,31529],{"className":31530,"style":30383},[7248,7141],[35,31532,9487],{"className":31533},[7104,7145,7141],[35,31535,7829],{"className":31536},[7104,7145,7141],[35,31538,7669],{"className":31539,"style":7684},[7104,7145],[35,31541,31543,31546],{"className":31542},[7104],[35,31544,7817],{"className":31545},[7104,7145],[35,31547,31549],{"className":31548},[7113],[35,31550,31552],{"className":31551},[7117],[35,31553,31555],{"className":31554},[7121],[35,31556,31558],{"className":31557,"style":9568},[7125],[35,31559,31560,31563],{"style":7437},[35,31561],{"className":31562,"style":7134},[7133],[35,31564,31566],{"className":31565},[7138,7139,7140,7141],[35,31567,31569,31572,31575,31578,31581,31584,31587,31590,31593],{"className":31568},[7104,7141],[35,31570,7225],{"className":31571},[7104,7141],[35,31573,7824],{"className":31574},[7104,7145,7141],[35,31576],{"className":31577,"style":30383},[7248,7141],[35,31579,279],{"className":31580,"style":8221},[7104,7145,7141],[35,31582],{"className":31583,"style":30383},[7248,7141],[35,31585,7669],{"className":31586,"style":7684},[7104,7145,7141],[35,31588],{"className":31589,"style":30383},[7248,7141],[35,31591,9487],{"className":31592},[7104,7145,7141],[35,31594,7829],{"className":31595},[7104,7145,7141],[35,31597,7283],{"className":31598},[7104],[35,31600,8322],{"className":31601,"style":7310},[7104,7145],[35,31603,7290],{"className":31604},[7314],[35,31606],{"className":31607,"style":7450},[7248],[35,31609,242],{"className":31610},[7454],[35,31612],{"className":31613,"style":7450},[7248],[35,31615,31617,31621,31624,31627,31630,31633,31692,31695,31698],{"className":31616},[7095],[35,31618],{"className":31619,"style":31620},[7099],"height:1.188em;vertical-align:-0.25em;",[35,31622,8953],{"className":31623},[7472],[35,31625,8322],{"className":31626,"style":7310},[7104,7145],[35,31628,7283],{"className":31629},[7104],[35,31631,7669],{"className":31632,"style":7684},[7104,7145],[35,31634,31636,31639],{"className":31635},[7104],[35,31637,7817],{"className":31638},[7104,7145],[35,31640,31642],{"className":31641},[7113],[35,31643,31645],{"className":31644},[7117],[35,31646,31648],{"className":31647},[7121],[35,31649,31651],{"className":31650,"style":30981},[7125],[35,31652,31653,31656],{"style":7437},[35,31654],{"className":31655,"style":7134},[7133],[35,31657,31659],{"className":31658},[7138,7139,7140,7141],[35,31660,31662,31665,31668,31671,31674,31677,31680,31683,31686,31689],{"className":31661},[7104,7141],[35,31663,7824],{"className":31664},[7104,7145,7141],[35,31666,275],{"className":31667},[7472,7141],[35,31669,6920],{"className":31670,"style":11350},[7104,7145,7141],[35,31672,7225],{"className":31673},[7253,7141],[35,31675,279],{"className":31676,"style":8221},[7104,7145,7141],[35,31678,1694],{"className":31679},[7314,7141],[35,31681,7669],{"className":31682,"style":7684},[7104,7145,7141],[35,31684],{"className":31685,"style":30383},[7248,7141],[35,31687,9487],{"className":31688},[7104,7145,7141],[35,31690,7829],{"className":31691},[7104,7145,7141],[35,31693,7283],{"className":31694},[7104],[35,31696,8322],{"className":31697,"style":7310},[7104,7145],[35,31699,7290],{"className":31700},[7314],[12,31702,31703,31704,31732,31733,5122],{},"are invariant under overall shifts of the evolution time, since the time-evolutions commute with the Hamiltonian. For odd ",[35,31705,31707,31720],{"className":31706},[7057],[35,31708,31710],{"className":31709},[7061],[7063,31711,31712],{"xmlns":7065},[7067,31713,31714,31718],{},[7070,31715,31716],{},[7076,31717,6778],{},[7083,31719,6778],{"encoding":7085},[35,31721,31723],{"className":31722,"ariaHidden":7091},[7090],[35,31724,31726,31729],{"className":31725},[7095],[35,31727],{"className":31728,"style":7173},[7099],[35,31730,6778],{"className":31731,"style":7146},[7104,7145],", we can use the analysis for ",[35,31734,31736,31753],{"className":31735},[7057],[35,31737,31739],{"className":31738},[7061],[7063,31740,31741],{"xmlns":7065},[7067,31742,31743,31751],{},[7070,31744,31745,31747,31749],{},[7076,31746,6778],{},[7223,31748,7225],{},[7227,31750,500],{},[7083,31752,7231],{"encoding":7085},[35,31754,31756,31774],{"className":31755,"ariaHidden":7091},[7090],[35,31757,31759,31762,31765,31768,31771],{"className":31758},[7095],[35,31760],{"className":31761,"style":7241},[7099],[35,31763,6778],{"className":31764,"style":7146},[7104,7145],[35,31766],{"className":31767,"style":7249},[7248],[35,31769,7225],{"className":31770},[7253],[35,31772],{"className":31773,"style":7249},[7248],[35,31775,31777,31780],{"className":31776},[7095],[35,31778],{"className":31779,"style":7263},[7099],[35,31781,500],{"className":31782},[7104],[12,31784,31785,31786,3439],{},"We want to show that somewhere in this Krylov space, there is guaranteed to be a low-energy state. We do so by way of the following result, which is derived from Theorem 3.1 in ",[19,31787,31788],{"href":5756},"[3]",[12,31790,31791,31795,31796,31825,31826,31854],{},[31792,31793,31794],"strong",{},"Claim 1:"," there exists a function ",[35,31797,31799,31812],{"className":31798},[7057],[35,31800,31802],{"className":31801},[7061],[7063,31803,31804],{"xmlns":7065},[7067,31805,31806,31810],{},[7070,31807,31808],{},[7076,31809,1675],{},[7083,31811,1675],{"encoding":7085},[35,31813,31815],{"className":31814,"ariaHidden":7091},[7090],[35,31816,31818,31821],{"className":31817},[7095],[35,31819],{"className":31820,"style":11562},[7099],[35,31822,1675],{"className":31823,"style":31824},[7104,7145],"margin-right:0.1076em;"," such that for energies ",[35,31827,31829,31842],{"className":31828},[7057],[35,31830,31832],{"className":31831},[7061],[7063,31833,31834],{"xmlns":7065},[7067,31835,31836,31840],{},[7070,31837,31838],{},[7076,31839,10709],{},[7083,31841,10709],{"encoding":7085},[35,31843,31845],{"className":31844,"ariaHidden":7091},[7090],[35,31846,31848,31851],{"className":31847},[7095],[35,31849],{"className":31850,"style":7100},[7099],[35,31852,10709],{"className":31853,"style":10419},[7104,7145]," in the spectral range of the Hamiltonian (that is, between the ground state energy and the maximum energy)...",[5024,31856,31857,31966,32274],{},[5027,31858,31859],{},[35,31860,31862,31890],{"className":31861},[7057],[35,31863,31865],{"className":31864},[7061],[7063,31866,31867],{"xmlns":7065},[7067,31868,31869,31887],{},[7070,31870,31871,31873,31875,31881,31883,31885],{},[7076,31872,1675],{},[7223,31874,275],{"stretchy":7289},[7699,31876,31877,31879],{},[7076,31878,10709],{},[7227,31880,5312],{},[7223,31882,1694],{"stretchy":7289},[7223,31884,242],{},[7227,31886,500],{},[7083,31888,31889],{"encoding":7085},"f(E_0)=1",[35,31891,31893,31957],{"className":31892,"ariaHidden":7091},[7090],[35,31894,31896,31899,31902,31905,31945,31948,31951,31954],{"className":31895},[7095],[35,31897],{"className":31898,"style":7303},[7099],[35,31900,1675],{"className":31901,"style":31824},[7104,7145],[35,31903,275],{"className":31904},[7472],[35,31906,31908,31911],{"className":31907},[7104],[35,31909,10709],{"className":31910,"style":10419},[7104,7145],[35,31912,31914],{"className":31913},[7113],[35,31915,31917,31937],{"className":31916},[7117,7731],[35,31918,31920,31934],{"className":31919},[7121],[35,31921,31923],{"className":31922,"style":18794},[7125],[35,31924,31925,31928],{"style":10541},[35,31926],{"className":31927,"style":7134},[7133],[35,31929,31931],{"className":31930},[7138,7139,7140,7141],[35,31932,5312],{"className":31933},[7104,7141],[35,31935,7756],{"className":31936},[7755],[35,31938,31940],{"className":31939},[7121],[35,31941,31943],{"className":31942,"style":7763},[7125],[35,31944],{},[35,31946,1694],{"className":31947},[7314],[35,31949],{"className":31950,"style":7450},[7248],[35,31952,242],{"className":31953},[7454],[35,31955],{"className":31956,"style":7450},[7248],[35,31958,31960,31963],{"className":31959},[7095],[35,31961],{"className":31962,"style":7263},[7099],[35,31964,500],{"className":31965},[7104],[5027,31967,31968,32128,32129,32157,32158,32203,32204,32273],{},[35,31969,31971,32021],{"className":31970},[7057],[35,31972,31974],{"className":31973},[7061],[7063,31975,31976],{"xmlns":7065},[7067,31977,31978,32018],{},[7070,31979,31980,31982,31984,31986,31988,31990,31992,31995,31997],{},[7076,31981,7283],{"mathvariant":7282},[7076,31983,1675],{},[7223,31985,275],{"stretchy":7289},[7076,31987,10709],{},[7223,31989,1694],{"stretchy":7289},[7076,31991,7283],{"mathvariant":7282},[7223,31993,31994],{},"≤",[7227,31996,460],{},[7073,31998,31999,32012],{},[7070,32000,32001,32003,32005,32007,32010],{},[7223,32002,275],{"fence":7091},[7227,32004,500],{},[7223,32006,445],{},[7076,32008,32009],{},"δ",[7223,32011,1694],{"fence":7091},[7070,32013,32014,32016],{},[7223,32015,7225],{},[7076,32017,9487],{},[7083,32019,32020],{"encoding":7085},"|f(E)|\\le2\\left(1 + \\delta\\right)^{-d}",[35,32022,32024,32057],{"className":32023,"ariaHidden":7091},[7090],[35,32025,32027,32030,32033,32036,32039,32042,32045,32048,32051,32054],{"className":32026},[7095],[35,32028],{"className":32029,"style":7303},[7099],[35,32031,7283],{"className":32032},[7104],[35,32034,1675],{"className":32035,"style":31824},[7104,7145],[35,32037,275],{"className":32038},[7472],[35,32040,10709],{"className":32041,"style":10419},[7104,7145],[35,32043,1694],{"className":32044},[7314],[35,32046,7283],{"className":32047},[7104],[35,32049],{"className":32050,"style":7450},[7248],[35,32052,31994],{"className":32053},[7454],[35,32055],{"className":32056,"style":7450},[7248],[35,32058,32060,32064,32067,32070],{"className":32059},[7095],[35,32061],{"className":32062,"style":32063},[7099],"height:1.239em;vertical-align:-0.25em;",[35,32065,460],{"className":32066},[7104],[35,32068],{"className":32069,"style":7496},[7248],[35,32071,32073,32098],{"className":32072},[7468],[35,32074,32076,32079,32082,32085,32088,32091,32095],{"className":32075},[7468],[35,32077,275],{"className":32078,"style":7474},[7472,7473],[35,32080,500],{"className":32081},[7104],[35,32083],{"className":32084,"style":7249},[7248],[35,32086,445],{"className":32087},[7253],[35,32089],{"className":32090,"style":7249},[7248],[35,32092,32009],{"className":32093,"style":32094},[7104,7145],"margin-right:0.0379em;",[35,32096,1694],{"className":32097,"style":7474},[7314,7473],[35,32099,32101],{"className":32100},[7113],[35,32102,32104],{"className":32103},[7117],[35,32105,32107],{"className":32106},[7121],[35,32108,32111],{"className":32109,"style":32110},[7125],"height:0.989em;",[35,32112,32113,32116],{"style":12371},[35,32114],{"className":32115,"style":7134},[7133],[35,32117,32119],{"className":32118},[7138,7139,7140,7141],[35,32120,32122,32125],{"className":32121},[7104,7141],[35,32123,7225],{"className":32124},[7104,7141],[35,32126,9487],{"className":32127},[7104,7145,7141]," for all values of ",[35,32130,32132,32145],{"className":32131},[7057],[35,32133,32135],{"className":32134},[7061],[7063,32136,32137],{"xmlns":7065},[7067,32138,32139,32143],{},[7070,32140,32141],{},[7076,32142,10709],{},[7083,32144,10709],{"encoding":7085},[35,32146,32148],{"className":32147,"ariaHidden":7091},[7090],[35,32149,32151,32154],{"className":32150},[7095],[35,32152],{"className":32153,"style":7100},[7099],[35,32155,10709],{"className":32156,"style":10419},[7104,7145]," that lie ",[35,32159,32161,32178],{"className":32160},[7057],[35,32162,32164],{"className":32163},[7061],[7063,32165,32166],{"xmlns":7065},[7067,32167,32168,32175],{},[7070,32169,32170,32173],{},[7223,32171,32172],{},"≥",[7076,32174,32009],{},[7083,32176,32177],{"encoding":7085},"\\ge\\delta",[35,32179,32181,32194],{"className":32180,"ariaHidden":7091},[7090],[35,32182,32184,32188,32191],{"className":32183},[7095],[35,32185],{"className":32186,"style":32187},[7099],"height:0.7719em;vertical-align:-0.136em;",[35,32189,32172],{"className":32190},[7454],[35,32192],{"className":32193,"style":7450},[7248],[35,32195,32197,32200],{"className":32196},[7095],[35,32198],{"className":32199,"style":9962},[7099],[35,32201,32009],{"className":32202,"style":32094},[7104,7145]," away from ",[35,32205,32207,32224],{"className":32206},[7057],[35,32208,32210],{"className":32209},[7061],[7063,32211,32212],{"xmlns":7065},[7067,32213,32214,32222],{},[7070,32215,32216],{},[7699,32217,32218,32220],{},[7076,32219,10709],{},[7227,32221,5312],{},[7083,32223,18766],{"encoding":7085},[35,32225,32227],{"className":32226,"ariaHidden":7091},[7090],[35,32228,32230,32233],{"className":32229},[7095],[35,32231],{"className":32232,"style":7718},[7099],[35,32234,32236,32239],{"className":32235},[7104],[35,32237,10709],{"className":32238,"style":10419},[7104,7145],[35,32240,32242],{"className":32241},[7113],[35,32243,32245,32265],{"className":32244},[7117,7731],[35,32246,32248,32262],{"className":32247},[7121],[35,32249,32251],{"className":32250,"style":18794},[7125],[35,32252,32253,32256],{"style":10541},[35,32254],{"className":32255,"style":7134},[7133],[35,32257,32259],{"className":32258},[7138,7139,7140,7141],[35,32260,5312],{"className":32261},[7104,7141],[35,32263,7756],{"className":32264},[7755],[35,32266,32268],{"className":32267},[7121],[35,32269,32271],{"className":32270,"style":7763},[7125],[35,32272],{},", that is, it is exponentially suppressed",[5027,32275,32276,32320,32321,32408,32409],{},[35,32277,32279,32299],{"className":32278},[7057],[35,32280,32282],{"className":32281},[7061],[7063,32283,32284],{"xmlns":7065},[7067,32285,32286,32296],{},[7070,32287,32288,32290,32292,32294],{},[7076,32289,1675],{},[7223,32291,275],{"stretchy":7289},[7076,32293,10709],{},[7223,32295,1694],{"stretchy":7289},[7083,32297,32298],{"encoding":7085},"f(E)",[35,32300,32302],{"className":32301,"ariaHidden":7091},[7090],[35,32303,32305,32308,32311,32314,32317],{"className":32304},[7095],[35,32306],{"className":32307,"style":7303},[7099],[35,32309,1675],{"className":32310,"style":31824},[7104,7145],[35,32312,275],{"className":32313},[7472],[35,32315,10709],{"className":32316,"style":10419},[7104,7145],[35,32318,1694],{"className":32319},[7314]," is a linear combination of ",[35,32322,32324,32354],{"className":32323},[7057],[35,32325,32327],{"className":32326},[7061],[7063,32328,32329],{"xmlns":7065},[7067,32330,32331,32351],{},[7070,32332,32333],{},[7073,32334,32335,32337],{},[7076,32336,7817],{},[7070,32338,32339,32341,32343,32345,32347,32349],{},[7076,32340,7824],{},[7076,32342,6920],{},[7076,32344,10709],{},[10694,32346,30173],{},[7076,32348,9487],{},[7076,32350,7829],{},[7083,32352,32353],{"encoding":7085},"e^{ijE\\,dt}",[35,32355,32357],{"className":32356,"ariaHidden":7091},[7090],[35,32358,32360,32363],{"className":32359},[7095],[35,32361],{"className":32362,"style":8191},[7099],[35,32364,32366,32369],{"className":32365},[7104],[35,32367,7817],{"className":32368},[7104,7145],[35,32370,32372],{"className":32371},[7113],[35,32373,32375],{"className":32374},[7117],[35,32376,32378],{"className":32377},[7121],[35,32379,32381],{"className":32380,"style":8191},[7125],[35,32382,32383,32386],{"style":7129},[35,32384],{"className":32385,"style":7134},[7133],[35,32387,32389],{"className":32388},[7138,7139,7140,7141],[35,32390,32392,32396,32399,32402,32405],{"className":32391},[7104,7141],[35,32393,32395],{"className":32394,"style":11350},[7104,7145,7141],"ij",[35,32397,10709],{"className":32398,"style":10419},[7104,7145,7141],[35,32400],{"className":32401,"style":30383},[7248,7141],[35,32403,9487],{"className":32404},[7104,7145,7141],[35,32406,7829],{"className":32407},[7104,7145,7141]," for ",[35,32410,32412,32462],{"className":32411},[7057],[35,32413,32415],{"className":32414},[7061],[7063,32416,32417],{"xmlns":7065},[7067,32418,32419,32459],{},[7070,32420,32421,32423,32425,32427,32429,32431,32433,32435,32437,32439,32441,32443,32445,32447,32449,32451,32453,32455,32457],{},[7076,32422,6920],{},[7223,32424,242],{},[7223,32426,7225],{},[7076,32428,9487],{},[7223,32430,2877],{"separator":7091},[7223,32432,7225],{},[7076,32434,9487],{},[7223,32436,445],{},[7227,32438,500],{},[7223,32440,2877],{"separator":7091},[7076,32442,5122],{"mathvariant":7282},[7076,32444,5122],{"mathvariant":7282},[7076,32446,5122],{"mathvariant":7282},[7223,32448,2877],{"separator":7091},[7076,32450,9487],{},[7223,32452,7225],{},[7227,32454,500],{},[7223,32456,2877],{"separator":7091},[7076,32458,9487],{},[7083,32460,32461],{"encoding":7085},"j=-d,-d+1,...,d-1,d",[35,32463,32465,32484,32517,32553],{"className":32464,"ariaHidden":7091},[7090],[35,32466,32468,32472,32475,32478,32481],{"className":32467},[7095],[35,32469],{"className":32470,"style":32471},[7099],"height:0.854em;vertical-align:-0.1944em;",[35,32473,6920],{"className":32474,"style":11350},[7104,7145],[35,32476],{"className":32477,"style":7450},[7248],[35,32479,242],{"className":32480},[7454],[35,32482],{"className":32483,"style":7450},[7248],[35,32485,32487,32490,32493,32496,32499,32502,32505,32508,32511,32514],{"className":32486},[7095],[35,32488],{"className":32489,"style":11562},[7099],[35,32491,7225],{"className":32492},[7104],[35,32494,9487],{"className":32495},[7104,7145],[35,32497,2877],{"className":32498},[7492],[35,32500],{"className":32501,"style":7496},[7248],[35,32503,7225],{"className":32504},[7104],[35,32506,9487],{"className":32507},[7104,7145],[35,32509],{"className":32510,"style":7249},[7248],[35,32512,445],{"className":32513},[7253],[35,32515],{"className":32516,"style":7249},[7248],[35,32518,32520,32523,32526,32529,32532,32535,32538,32541,32544,32547,32550],{"className":32519},[7095],[35,32521],{"className":32522,"style":11562},[7099],[35,32524,500],{"className":32525},[7104],[35,32527,2877],{"className":32528},[7492],[35,32530],{"className":32531,"style":7496},[7248],[35,32533,7563],{"className":32534},[7104],[35,32536,2877],{"className":32537},[7492],[35,32539],{"className":32540,"style":7496},[7248],[35,32542,9487],{"className":32543},[7104,7145],[35,32545],{"className":32546,"style":7249},[7248],[35,32548,7225],{"className":32549},[7253],[35,32551],{"className":32552,"style":7249},[7248],[35,32554,32556,32559,32562,32565,32568],{"className":32555},[7095],[35,32557],{"className":32558,"style":11562},[7099],[35,32560,500],{"className":32561},[7104],[35,32563,2877],{"className":32564},[7492],[35,32566],{"className":32567,"style":7496},[7248],[35,32569,9487],{"className":32570},[7104,7145],[12,32572,32573,32574,32633,32634,32673],{},"We give a proof below, but that can be safely skipped unless one wants to understand the full, rigorous argument. For now we focus on the implications of the above claim. By property 3 above, we can see that the shifted Krylov space above contains the state ",[35,32575,32577,32603],{"className":32576},[7057],[35,32578,32580],{"className":32579},[7061],[7063,32581,32582],{"xmlns":7065},[7067,32583,32584,32600],{},[7070,32585,32586,32588,32590,32592,32594,32596,32598],{},[7076,32587,1675],{},[7223,32589,275],{"stretchy":7289},[7076,32591,7669],{},[7223,32593,1694],{"stretchy":7289},[7076,32595,7283],{"mathvariant":7282},[7076,32597,8322],{},[7223,32599,7290],{"stretchy":7289},[7083,32601,32602],{"encoding":7085},"f(H)|\\psi\\rangle",[35,32604,32606],{"className":32605,"ariaHidden":7091},[7090],[35,32607,32609,32612,32615,32618,32621,32624,32627,32630],{"className":32608},[7095],[35,32610],{"className":32611,"style":7303},[7099],[35,32613,1675],{"className":32614,"style":31824},[7104,7145],[35,32616,275],{"className":32617},[7472],[35,32619,7669],{"className":32620,"style":7684},[7104,7145],[35,32622,1694],{"className":32623},[7314],[35,32625,7283],{"className":32626},[7104],[35,32628,8322],{"className":32629,"style":7310},[7104,7145],[35,32631,7290],{"className":32632},[7314],". This is our low-energy state. To see why, write ",[35,32635,32637,32655],{"className":32636},[7057],[35,32638,32640],{"className":32639},[7061],[7063,32641,32642],{"xmlns":7065},[7067,32643,32644,32652],{},[7070,32645,32646,32648,32650],{},[7076,32647,7283],{"mathvariant":7282},[7076,32649,8322],{},[7223,32651,7290],{"stretchy":7289},[7083,32653,32654],{"encoding":7085},"|\\psi\\rangle",[35,32656,32658],{"className":32657,"ariaHidden":7091},[7090],[35,32659,32661,32664,32667,32670],{"className":32660},[7095],[35,32662],{"className":32663,"style":7303},[7099],[35,32665,7283],{"className":32666},[7104],[35,32668,8322],{"className":32669,"style":7310},[7104,7145],[35,32671,7290],{"className":32672},[7314]," in the energy eigenbasis:",[35,32675,32677],{"className":32676},[7318],[35,32678,32680,32734],{"className":32679},[7057],[35,32681,32683],{"className":32682},[7061],[7063,32684,32685],{"xmlns":7065,"display":7327},[7067,32686,32687,32731],{},[7070,32688,32689,32691,32693,32695,32697,32712,32719,32721,32727,32729],{},[7076,32690,7283],{"mathvariant":7282},[7076,32692,8322],{},[7223,32694,7290],{"stretchy":7289},[7223,32696,242],{},[32698,32699,32700,32702,32710],"munderover",{},[7223,32701,12053],{},[7070,32703,32704,32706,32708],{},[7076,32705,279],{},[7223,32707,242],{},[7227,32709,5312],{},[7076,32711,12207],{},[7699,32713,32714,32717],{},[7076,32715,32716],{},"γ",[7076,32718,279],{},[7076,32720,7283],{"mathvariant":7282},[7699,32722,32723,32725],{},[7076,32724,10709],{},[7076,32726,279],{},[7223,32728,7290],{"stretchy":7289},[7223,32730,2877],{"separator":7091},[7083,32732,32733],{"encoding":7085},"|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_k|E_k\\rangle,",[35,32735,32737,32761],{"className":32736,"ariaHidden":7091},[7090],[35,32738,32740,32743,32746,32749,32752,32755,32758],{"className":32739},[7095],[35,32741],{"className":32742,"style":7303},[7099],[35,32744,7283],{"className":32745},[7104],[35,32747,8322],{"className":32748,"style":7310},[7104,7145],[35,32750,7290],{"className":32751},[7314],[35,32753],{"className":32754,"style":7450},[7248],[35,32756,242],{"className":32757},[7454],[35,32759],{"className":32760,"style":7450},[7248],[35,32762,32764,32768,32843,32846,32888,32891,32931,32934],{"className":32763},[7095],[35,32765],{"className":32766,"style":32767},[7099],"height:3.1304em;vertical-align:-1.3021em;",[35,32769,32772],{"className":32770},[12096,32771],"op-limits",[35,32773,32775,32834],{"className":32774},[7117,7731],[35,32776,32778,32831],{"className":32777},[7121],[35,32779,32782,32804,32816],{"className":32780,"style":32781},[7125],"height:1.8283em;",[35,32783,32785,32789],{"style":32784},"top:-1.8479em;margin-left:0em;",[35,32786],{"className":32787,"style":32788},[7133],"height:3.05em;",[35,32790,32792],{"className":32791},[7138,7139,7140,7141],[35,32793,32795,32798,32801],{"className":32794},[7104,7141],[35,32796,279],{"className":32797,"style":8221},[7104,7145,7141],[35,32799,242],{"className":32800},[7454,7141],[35,32802,5312],{"className":32803},[7104,7141],[35,32805,32807,32810],{"style":32806},"top:-3.05em;",[35,32808],{"className":32809,"style":32788},[7133],[35,32811,32812],{},[35,32813,12053],{"className":32814},[12096,12100,32815],"large-op",[35,32817,32819,32822],{"style":32818},"top:-4.3em;margin-left:0em;",[35,32820],{"className":32821,"style":32788},[7133],[35,32823,32825],{"className":32824},[7138,7139,7140,7141],[35,32826,32828],{"className":32827},[7104,7141],[35,32829,12207],{"className":32830,"style":7845},[7104,7145,7141],[35,32832,7756],{"className":32833},[7755],[35,32835,32837],{"className":32836},[7121],[35,32838,32841],{"className":32839,"style":32840},[7125],"height:1.3021em;",[35,32842],{},[35,32844],{"className":32845,"style":7496},[7248],[35,32847,32849,32853],{"className":32848},[7104],[35,32850,32716],{"className":32851,"style":32852},[7104,7145],"margin-right:0.0556em;",[35,32854,32856],{"className":32855},[7113],[35,32857,32859,32880],{"className":32858},[7117,7731],[35,32860,32862,32877],{"className":32861},[7121],[35,32863,32865],{"className":32864,"style":15760},[7125],[35,32866,32868,32871],{"style":32867},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[35,32869],{"className":32870,"style":7134},[7133],[35,32872,32874],{"className":32873},[7138,7139,7140,7141],[35,32875,279],{"className":32876,"style":8221},[7104,7145,7141],[35,32878,7756],{"className":32879},[7755],[35,32881,32883],{"className":32882},[7121],[35,32884,32886],{"className":32885,"style":7763},[7125],[35,32887],{},[35,32889,7283],{"className":32890},[7104],[35,32892,32894,32897],{"className":32893},[7104],[35,32895,10709],{"className":32896,"style":10419},[7104,7145],[35,32898,32900],{"className":32899},[7113],[35,32901,32903,32923],{"className":32902},[7117,7731],[35,32904,32906,32920],{"className":32905},[7121],[35,32907,32909],{"className":32908,"style":15760},[7125],[35,32910,32911,32914],{"style":10541},[35,32912],{"className":32913,"style":7134},[7133],[35,32915,32917],{"className":32916},[7138,7139,7140,7141],[35,32918,279],{"className":32919,"style":8221},[7104,7145,7141],[35,32921,7756],{"className":32922},[7755],[35,32924,32926],{"className":32925},[7121],[35,32927,32929],{"className":32928,"style":7763},[7125],[35,32930],{},[35,32932,7290],{"className":32933},[7314],[35,32935,2877],{"className":32936},[7492],[12,32938,32939,32940,33020,33021,33092,33093,33131,33132,33190],{},"where ",[35,32941,32943,32965],{"className":32942},[7057],[35,32944,32946],{"className":32945},[7061],[7063,32947,32948],{"xmlns":7065},[7067,32949,32950,32962],{},[7070,32951,32952,32954,32960],{},[7076,32953,7283],{"mathvariant":7282},[7699,32955,32956,32958],{},[7076,32957,10709],{},[7076,32959,279],{},[7223,32961,7290],{"stretchy":7289},[7083,32963,32964],{"encoding":7085},"|E_k\\rangle",[35,32966,32968],{"className":32967,"ariaHidden":7091},[7090],[35,32969,32971,32974,32977,33017],{"className":32970},[7095],[35,32972],{"className":32973,"style":7303},[7099],[35,32975,7283],{"className":32976},[7104],[35,32978,32980,32983],{"className":32979},[7104],[35,32981,10709],{"className":32982,"style":10419},[7104,7145],[35,32984,32986],{"className":32985},[7113],[35,32987,32989,33009],{"className":32988},[7117,7731],[35,32990,32992,33006],{"className":32991},[7121],[35,32993,32995],{"className":32994,"style":15760},[7125],[35,32996,32997,33000],{"style":10541},[35,32998],{"className":32999,"style":7134},[7133],[35,33001,33003],{"className":33002},[7138,7139,7140,7141],[35,33004,279],{"className":33005,"style":8221},[7104,7145,7141],[35,33007,7756],{"className":33008},[7755],[35,33010,33012],{"className":33011},[7121],[35,33013,33015],{"className":33014,"style":7763},[7125],[35,33016],{},[35,33018,7290],{"className":33019},[7314]," is the kth energy eigenstate and ",[35,33022,33024,33042],{"className":33023},[7057],[35,33025,33027],{"className":33026},[7061],[7063,33028,33029],{"xmlns":7065},[7067,33030,33031,33039],{},[7070,33032,33033],{},[7699,33034,33035,33037],{},[7076,33036,32716],{},[7076,33038,279],{},[7083,33040,33041],{"encoding":7085},"\\gamma_k",[35,33043,33045],{"className":33044,"ariaHidden":7091},[7090],[35,33046,33048,33052],{"className":33047},[7095],[35,33049],{"className":33050,"style":33051},[7099],"height:0.625em;vertical-align:-0.1944em;",[35,33053,33055,33058],{"className":33054},[7104],[35,33056,32716],{"className":33057,"style":32852},[7104,7145],[35,33059,33061],{"className":33060},[7113],[35,33062,33064,33084],{"className":33063},[7117,7731],[35,33065,33067,33081],{"className":33066},[7121],[35,33068,33070],{"className":33069,"style":15760},[7125],[35,33071,33072,33075],{"style":32867},[35,33073],{"className":33074,"style":7134},[7133],[35,33076,33078],{"className":33077},[7138,7139,7140,7141],[35,33079,279],{"className":33080,"style":8221},[7104,7145,7141],[35,33082,7756],{"className":33083},[7755],[35,33085,33087],{"className":33086},[7121],[35,33088,33090],{"className":33089,"style":7763},[7125],[35,33091],{}," is its amplitude in the initial state ",[35,33094,33096,33113],{"className":33095},[7057],[35,33097,33099],{"className":33098},[7061],[7063,33100,33101],{"xmlns":7065},[7067,33102,33103,33111],{},[7070,33104,33105,33107,33109],{},[7076,33106,7283],{"mathvariant":7282},[7076,33108,8322],{},[7223,33110,7290],{"stretchy":7289},[7083,33112,32654],{"encoding":7085},[35,33114,33116],{"className":33115,"ariaHidden":7091},[7090],[35,33117,33119,33122,33125,33128],{"className":33118},[7095],[35,33120],{"className":33121,"style":7303},[7099],[35,33123,7283],{"className":33124},[7104],[35,33126,8322],{"className":33127,"style":7310},[7104,7145],[35,33129,7290],{"className":33130},[7314],". Expressed in terms of this, ",[35,33133,33135,33160],{"className":33134},[7057],[35,33136,33138],{"className":33137},[7061],[7063,33139,33140],{"xmlns":7065},[7067,33141,33142,33158],{},[7070,33143,33144,33146,33148,33150,33152,33154,33156],{},[7076,33145,1675],{},[7223,33147,275],{"stretchy":7289},[7076,33149,7669],{},[7223,33151,1694],{"stretchy":7289},[7076,33153,7283],{"mathvariant":7282},[7076,33155,8322],{},[7223,33157,7290],{"stretchy":7289},[7083,33159,32602],{"encoding":7085},[35,33161,33163],{"className":33162,"ariaHidden":7091},[7090],[35,33164,33166,33169,33172,33175,33178,33181,33184,33187],{"className":33165},[7095],[35,33167],{"className":33168,"style":7303},[7099],[35,33170,1675],{"className":33171,"style":31824},[7104,7145],[35,33173,275],{"className":33174},[7472],[35,33176,7669],{"className":33177,"style":7684},[7104,7145],[35,33179,1694],{"className":33180},[7314],[35,33182,7283],{"className":33183},[7104],[35,33185,8322],{"className":33186,"style":7310},[7104,7145],[35,33188,7290],{"className":33189},[7314]," is given by",[35,33192,33194],{"className":33193},[7318],[35,33195,33197,33269],{"className":33196},[7057],[35,33198,33200],{"className":33199},[7061],[7063,33201,33202],{"xmlns":7065,"display":7327},[7067,33203,33204,33266],{},[7070,33205,33206,33208,33210,33212,33214,33216,33218,33220,33222,33236,33242,33244,33246,33252,33254,33256,33262,33264],{},[7076,33207,1675],{},[7223,33209,275],{"stretchy":7289},[7076,33211,7669],{},[7223,33213,1694],{"stretchy":7289},[7076,33215,7283],{"mathvariant":7282},[7076,33217,8322],{},[7223,33219,7290],{"stretchy":7289},[7223,33221,242],{},[32698,33223,33224,33226,33234],{},[7223,33225,12053],{},[7070,33227,33228,33230,33232],{},[7076,33229,279],{},[7223,33231,242],{},[7227,33233,5312],{},[7076,33235,12207],{},[7699,33237,33238,33240],{},[7076,33239,32716],{},[7076,33241,279],{},[7076,33243,1675],{},[7223,33245,275],{"stretchy":7289},[7699,33247,33248,33250],{},[7076,33249,10709],{},[7076,33251,279],{},[7223,33253,1694],{"stretchy":7289},[7076,33255,7283],{"mathvariant":7282},[7699,33257,33258,33260],{},[7076,33259,10709],{},[7076,33261,279],{},[7223,33263,7290],{"stretchy":7289},[7223,33265,2877],{"separator":7091},[7083,33267,33268],{"encoding":7085},"f(H)|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_kf(E_k)|E_k\\rangle,",[35,33270,33272,33308],{"className":33271,"ariaHidden":7091},[7090],[35,33273,33275,33278,33281,33284,33287,33290,33293,33296,33299,33302,33305],{"className":33274},[7095],[35,33276],{"className":33277,"style":7303},[7099],[35,33279,1675],{"className":33280,"style":31824},[7104,7145],[35,33282,275],{"className":33283},[7472],[35,33285,7669],{"className":33286,"style":7684},[7104,7145],[35,33288,1694],{"className":33289},[7314],[35,33291,7283],{"className":33292},[7104],[35,33294,8322],{"className":33295,"style":7310},[7104,7145],[35,33297,7290],{"className":33298},[7314],[35,33300],{"className":33301,"style":7450},[7248],[35,33303,242],{"className":33304},[7454],[35,33306],{"className":33307,"style":7450},[7248],[35,33309,33311,33314,33381,33384,33424,33427,33430,33470,33473,33476,33516,33519],{"className":33310},[7095],[35,33312],{"className":33313,"style":32767},[7099],[35,33315,33317],{"className":33316},[12096,32771],[35,33318,33320,33373],{"className":33319},[7117,7731],[35,33321,33323,33370],{"className":33322},[7121],[35,33324,33326,33346,33356],{"className":33325,"style":32781},[7125],[35,33327,33328,33331],{"style":32784},[35,33329],{"className":33330,"style":32788},[7133],[35,33332,33334],{"className":33333},[7138,7139,7140,7141],[35,33335,33337,33340,33343],{"className":33336},[7104,7141],[35,33338,279],{"className":33339,"style":8221},[7104,7145,7141],[35,33341,242],{"className":33342},[7454,7141],[35,33344,5312],{"className":33345},[7104,7141],[35,33347,33348,33351],{"style":32806},[35,33349],{"className":33350,"style":32788},[7133],[35,33352,33353],{},[35,33354,12053],{"className":33355},[12096,12100,32815],[35,33357,33358,33361],{"style":32818},[35,33359],{"className":33360,"style":32788},[7133],[35,33362,33364],{"className":33363},[7138,7139,7140,7141],[35,33365,33367],{"className":33366},[7104,7141],[35,33368,12207],{"className":33369,"style":7845},[7104,7145,7141],[35,33371,7756],{"className":33372},[7755],[35,33374,33376],{"className":33375},[7121],[35,33377,33379],{"className":33378,"style":32840},[7125],[35,33380],{},[35,33382],{"className":33383,"style":7496},[7248],[35,33385,33387,33390],{"className":33386},[7104],[35,33388,32716],{"className":33389,"style":32852},[7104,7145],[35,33391,33393],{"className":33392},[7113],[35,33394,33396,33416],{"className":33395},[7117,7731],[35,33397,33399,33413],{"className":33398},[7121],[35,33400,33402],{"className":33401,"style":15760},[7125],[35,33403,33404,33407],{"style":32867},[35,33405],{"className":33406,"style":7134},[7133],[35,33408,33410],{"className":33409},[7138,7139,7140,7141],[35,33411,279],{"className":33412,"style":8221},[7104,7145,7141],[35,33414,7756],{"className":33415},[7755],[35,33417,33419],{"className":33418},[7121],[35,33420,33422],{"className":33421,"style":7763},[7125],[35,33423],{},[35,33425,1675],{"className":33426,"style":31824},[7104,7145],[35,33428,275],{"className":33429},[7472],[35,33431,33433,33436],{"className":33432},[7104],[35,33434,10709],{"className":33435,"style":10419},[7104,7145],[35,33437,33439],{"className":33438},[7113],[35,33440,33442,33462],{"className":33441},[7117,7731],[35,33443,33445,33459],{"className":33444},[7121],[35,33446,33448],{"className":33447,"style":15760},[7125],[35,33449,33450,33453],{"style":10541},[35,33451],{"className":33452,"style":7134},[7133],[35,33454,33456],{"className":33455},[7138,7139,7140,7141],[35,33457,279],{"className":33458,"style":8221},[7104,7145,7141],[35,33460,7756],{"className":33461},[7755],[35,33463,33465],{"className":33464},[7121],[35,33466,33468],{"className":33467,"style":7763},[7125],[35,33469],{},[35,33471,1694],{"className":33472},[7314],[35,33474,7283],{"className":33475},[7104],[35,33477,33479,33482],{"className":33478},[7104],[35,33480,10709],{"className":33481,"style":10419},[7104,7145],[35,33483,33485],{"className":33484},[7113],[35,33486,33488,33508],{"className":33487},[7117,7731],[35,33489,33491,33505],{"className":33490},[7121],[35,33492,33494],{"className":33493,"style":15760},[7125],[35,33495,33496,33499],{"style":10541},[35,33497],{"className":33498,"style":7134},[7133],[35,33500,33502],{"className":33501},[7138,7139,7140,7141],[35,33503,279],{"className":33504,"style":8221},[7104,7145,7141],[35,33506,7756],{"className":33507},[7755],[35,33509,33511],{"className":33510},[7121],[35,33512,33514],{"className":33513,"style":7763},[7125],[35,33515],{},[35,33517,7290],{"className":33518},[7314],[35,33520,2877],{"className":33521},[7492],[12,33523,33524,33525,33553,33554,33624,33625,33704],{},"using the fact that we can replace ",[35,33526,33528,33541],{"className":33527},[7057],[35,33529,33531],{"className":33530},[7061],[7063,33532,33533],{"xmlns":7065},[7067,33534,33535,33539],{},[7070,33536,33537],{},[7076,33538,7669],{},[7083,33540,7669],{"encoding":7085},[35,33542,33544],{"className":33543,"ariaHidden":7091},[7090],[35,33545,33547,33550],{"className":33546},[7095],[35,33548],{"className":33549,"style":7100},[7099],[35,33551,7669],{"className":33552,"style":7684},[7104,7145]," by ",[35,33555,33557,33575],{"className":33556},[7057],[35,33558,33560],{"className":33559},[7061],[7063,33561,33562],{"xmlns":7065},[7067,33563,33564,33572],{},[7070,33565,33566],{},[7699,33567,33568,33570],{},[7076,33569,10709],{},[7076,33571,279],{},[7083,33573,33574],{"encoding":7085},"E_k",[35,33576,33578],{"className":33577,"ariaHidden":7091},[7090],[35,33579,33581,33584],{"className":33580},[7095],[35,33582],{"className":33583,"style":7718},[7099],[35,33585,33587,33590],{"className":33586},[7104],[35,33588,10709],{"className":33589,"style":10419},[7104,7145],[35,33591,33593],{"className":33592},[7113],[35,33594,33596,33616],{"className":33595},[7117,7731],[35,33597,33599,33613],{"className":33598},[7121],[35,33600,33602],{"className":33601,"style":15760},[7125],[35,33603,33604,33607],{"style":10541},[35,33605],{"className":33606,"style":7134},[7133],[35,33608,33610],{"className":33609},[7138,7139,7140,7141],[35,33611,279],{"className":33612,"style":8221},[7104,7145,7141],[35,33614,7756],{"className":33615},[7755],[35,33617,33619],{"className":33618},[7121],[35,33620,33622],{"className":33621,"style":7763},[7125],[35,33623],{}," when it acts on the eigenstate ",[35,33626,33628,33649],{"className":33627},[7057],[35,33629,33631],{"className":33630},[7061],[7063,33632,33633],{"xmlns":7065},[7067,33634,33635,33647],{},[7070,33636,33637,33639,33645],{},[7076,33638,7283],{"mathvariant":7282},[7699,33640,33641,33643],{},[7076,33642,10709],{},[7076,33644,279],{},[7223,33646,7290],{"stretchy":7289},[7083,33648,32964],{"encoding":7085},[35,33650,33652],{"className":33651,"ariaHidden":7091},[7090],[35,33653,33655,33658,33661,33701],{"className":33654},[7095],[35,33656],{"className":33657,"style":7303},[7099],[35,33659,7283],{"className":33660},[7104],[35,33662,33664,33667],{"className":33663},[7104],[35,33665,10709],{"className":33666,"style":10419},[7104,7145],[35,33668,33670],{"className":33669},[7113],[35,33671,33673,33693],{"className":33672},[7117,7731],[35,33674,33676,33690],{"className":33675},[7121],[35,33677,33679],{"className":33678,"style":15760},[7125],[35,33680,33681,33684],{"style":10541},[35,33682],{"className":33683,"style":7134},[7133],[35,33685,33687],{"className":33686},[7138,7139,7140,7141],[35,33688,279],{"className":33689,"style":8221},[7104,7145,7141],[35,33691,7756],{"className":33692},[7755],[35,33694,33696],{"className":33695},[7121],[35,33697,33699],{"className":33698,"style":7763},[7125],[35,33700],{},[35,33702,7290],{"className":33703},[7314],". The energy error of this state is therefore",[35,33706,33708],{"className":33707},[7318],[35,33709,33711,33800],{"className":33710},[7057],[35,33712,33714],{"className":33713},[7061],[7063,33715,33716],{"xmlns":7065,"display":7327},[7067,33717,33718,33797],{},[7070,33719,33720,33723,33725],{},[10694,33721,33722],{},"energy error",[7223,33724,242],{},[14009,33726,33727,33771],{},[7070,33728,33729,33731,33733,33735,33737,33739,33741,33743,33745,33747,33749,33755,33757,33759,33761,33763,33765,33767,33769],{},[7223,33730,8953],{"stretchy":7289},[7076,33732,8322],{},[7076,33734,7283],{"mathvariant":7282},[7076,33736,1675],{},[7223,33738,275],{"stretchy":7289},[7076,33740,7669],{},[7223,33742,1694],{"stretchy":7289},[7223,33744,275],{"stretchy":7289},[7076,33746,7669],{},[7223,33748,7225],{},[7699,33750,33751,33753],{},[7076,33752,10709],{},[7227,33754,5312],{},[7223,33756,1694],{"stretchy":7289},[7076,33758,1675],{},[7223,33760,275],{"stretchy":7289},[7076,33762,7669],{},[7223,33764,1694],{"stretchy":7289},[7076,33766,7283],{"mathvariant":7282},[7076,33768,8322],{},[7223,33770,7290],{"stretchy":7289},[7070,33772,33773,33775,33777,33779,33781,33783,33785,33791,33793,33795],{},[7223,33774,8953],{"stretchy":7289},[7076,33776,8322],{},[7076,33778,7283],{"mathvariant":7282},[7076,33780,1675],{},[7223,33782,275],{"stretchy":7289},[7076,33784,7669],{},[7073,33786,33787,33789],{},[7223,33788,1694],{"stretchy":7289},[7227,33790,460],{},[7076,33792,7283],{"mathvariant":7282},[7076,33794,8322],{},[7223,33796,7290],{"stretchy":7289},[7083,33798,33799],{"encoding":7085},"\\text{energy error} = 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turn this into an upper bound that is easier to understand, we first separate the sum in the numerator into terms with ",[35,34791,34793,34823],{"className":34792},[7057],[35,34794,34796],{"className":34795},[7061],[7063,34797,34798],{"xmlns":7065},[7067,34799,34800,34820],{},[7070,34801,34802,34808,34810,34816,34818],{},[7699,34803,34804,34806],{},[7076,34805,10709],{},[7076,34807,279],{},[7223,34809,7225],{},[7699,34811,34812,34814],{},[7076,34813,10709],{},[7227,34815,5312],{},[7223,34817,31994],{},[7076,34819,32009],{},[7083,34821,34822],{"encoding":7085},"E_k-E_0\\le\\delta",[35,34824,34826,34881,34936],{"className":34825,"ariaHidden":7091},[7090],[35,34827,34829,34832,34872,34875,34878],{"className":34828},[7095],[35,34830],{"className":34831,"style":7718},[7099],[35,34833,34835,34838],{"className":34834},[7104],[35,34836,10709],{"className":34837,"style":10419},[7104,7145],[35,34839,34841],{"className":34840},[7113],[35,34842,34844,34864],{"className":34843},[7117,7731],[35,34845,34847,34861],{"className":34846},[7121],[35,34848,34850],{"className":34849,"style":15760},[7125],[35,34851,34852,34855],{"style":10541},[35,34853],{"className":34854,"style":7134},[7133],[35,34856,34858],{"className":34857},[7138,7139,7140,7141],[35,34859,279],{"className":34860,"style":8221},[7104,7145,7141],[35,34862,7756],{"className":34863},[7755],[35,34865,34867],{"className":34866},[7121],[35,34868,34870],{"className":34869,"style":7763},[7125],[35,34871],{},[35,34873],{"className":34874,"style":7249},[7248],[35,34876,7225],{"className":34877},[7253],[35,34879],{"className":34880,"style":7249},[7248],[35,34882,34884,34887,34927,34930,34933],{"className":34883},[7095],[35,34885],{"className":34886,"style":7718},[7099],[35,34888,34890,34893],{"className":34889},[7104],[35,34891,10709],{"className":34892,"style":10419},[7104,7145],[35,34894,34896],{"className":34895},[7113],[35,34897,34899,34919],{"className":34898},[7117,7731],[35,34900,34902,34916],{"className":34901},[7121],[35,34903,34905],{"className":34904,"style":18794},[7125],[35,34906,34907,34910],{"style":10541},[35,34908],{"className":34909,"style":7134},[7133],[35,34911,34913],{"className":34912},[7138,7139,7140,7141],[35,34914,5312],{"className":34915},[7104,7141],[35,34917,7756],{"className":34918},[7755],[35,34920,34922],{"className":34921},[7121],[35,34923,34925],{"className":34924,"style":7763},[7125],[35,34926],{},[35,34928],{"className":34929,"style":7450},[7248],[35,34931,31994],{"className":34932},[7454],[35,34934],{"className":34935,"style":7450},[7248],[35,34937,34939,34942],{"className":34938},[7095],[35,34940],{"className":34941,"style":9962},[7099],[35,34943,32009],{"className":34944,"style":32094},[7104,7145]," and terms with ",[35,34947,34949,34979],{"className":34948},[7057],[35,34950,34952],{"className":34951},[7061],[7063,34953,34954],{"xmlns":7065},[7067,34955,34956,34976],{},[7070,34957,34958,34964,34966,34972,34974],{},[7699,34959,34960,34962],{},[7076,34961,10709],{},[7076,34963,279],{},[7223,34965,7225],{},[7699,34967,34968,34970],{},[7076,34969,10709],{},[7227,34971,5312],{},[7223,34973,1497],{},[7076,34975,32009],{},[7083,34977,34978],{"encoding":7085},"E_k-E_0>\\delta",[35,34980,34982,35037,35092],{"className":34981,"ariaHidden":7091},[7090],[35,34983,34985,34988,35028,35031,35034],{"className":34984},[7095],[35,34986],{"className":34987,"style":7718},[7099],[35,34989,34991,34994],{"className":34990},[7104],[35,34992,10709],{"className":34993,"style":10419},[7104,7145],[35,34995,34997],{"className":34996},[7113],[35,34998,35000,35020],{"className":34999},[7117,7731],[35,35001,35003,35017],{"className":35002},[7121],[35,35004,35006],{"className":35005,"style":15760},[7125],[35,35007,35008,35011],{"style":10541},[35,35009],{"className":35010,"style":7134},[7133],[35,35012,35014],{"className":35013},[7138,7139,7140,7141],[35,35015,279],{"className":35016,"style":8221},[7104,7145,7141],[35,35018,7756],{"className":35019},[7755],[35,35021,35023],{"className":35022},[7121],[35,35024,35026],{"className":35025,"style":7763},[7125],[35,35027],{},[35,35029],{"className":35030,"style":7249},[7248],[35,35032,7225],{"className":35033},[7253],[35,35035],{"className":35036,"style":7249},[7248],[35,35038,35040,35043,35083,35086,35089],{"className":35039},[7095],[35,35041],{"className":35042,"style":7718},[7099],[35,35044,35046,35049],{"className":35045},[7104],[35,35047,10709],{"className":35048,"style":10419},[7104,7145],[35,35050,35052],{"className":35051},[7113],[35,35053,35055,35075],{"className":35054},[7117,7731],[35,35056,35058,35072],{"className":35057},[7121],[35,35059,35061],{"className":35060,"style":18794},[7125],[35,35062,35063,35066],{"style":10541},[35,35064],{"className":35065,"style":7134},[7133],[35,35067,35069],{"className":35068},[7138,7139,7140,7141],[35,35070,5312],{"className":35071},[7104,7141],[35,35073,7756],{"className":35074},[7755],[35,35076,35078],{"className":35077},[7121],[35,35079,35081],{"className":35080,"style":7763},[7125],[35,35082],{},[35,35084],{"className":35085,"style":7450},[7248],[35,35087,1497],{"className":35088},[7454],[35,35090],{"className":35091,"style":7450},[7248],[35,35093,35095,35098],{"className":35094},[7095],[35,35096],{"className":35097,"style":9962},[7099],[35,35099,32009],{"className":35100,"style":32094},[7104,7145],[35,35102,35104],{"className":35103},[7318],[35,35105,35107,35372],{"className":35106},[7057],[35,35108,35110],{"className":35109},[7061],[7063,35111,35112],{"xmlns":7065,"display":7327},[7067,35113,35114,35369],{},[7070,35115,35116,35118,35120,35243,35245,35367],{},[10694,35117,33722],{},[7223,35119,242],{},[14009,35121,35122,35197],{},[7070,35123,35124,35149,35151,35157,35163,35165,35167,35173,35179,35181,35187,35189,35195],{},[35125,35126,35127,35129],"munder",{},[7223,35128,12053],{},[7070,35130,35131,35137,35139,35145,35147],{},[7699,35132,35133,35135],{},[7076,35134,10709],{},[7076,35136,279],{},[7223,35138,31994],{},[7699,35140,35141,35143],{},[7076,35142,10709],{},[7227,35144,5312],{},[7223,35146,445],{},[7076,35148,32009],{},[7076,35150,7283],{"mathvariant":7282},[7699,35152,35153,35155],{},[7076,35154,32716],{},[7076,35156,279],{},[7073,35158,35159,35161],{},[7076,35160,7283],{"mathvariant":7282},[7227,35162,460],{},[7076,35164,1675],{},[7223,35166,275],{"stretchy":7289},[7699,35168,35169,35171],{},[7076,35170,10709],{},[7076,35172,279],{},[7073,35174,35175,35177],{},[7223,35176,1694],{"stretchy":7289},[7227,35178,460],{},[7223,35180,275],{"stretchy":7289},[7699,35182,35183,35185],{},[7076,35184,10709],{},[7076,35186,279],{},[7223,35188,7225],{},[7699,35190,35191,35193],{},[7076,35192,10709],{},[7227,35194,5312],{},[7223,35196,1694],{"stretchy":7289},[7070,35198,35199,35213,35215,35221,35227,35229,35231,35237],{},[32698,35200,35201,35203,35211],{},[7223,35202,12053],{},[7070,35204,35205,35207,35209],{},[7076,35206,279],{},[7223,35208,242],{},[7227,35210,5312],{},[7076,35212,12207],{},[7076,35214,7283],{"mathvariant":7282},[7699,35216,35217,35219],{},[7076,35218,32716],{},[7076,35220,279],{},[7073,35222,35223,35225],{},[7076,35224,7283],{"mathvariant":7282},[7227,35226,460],{},[7076,35228,1675],{},[7223,35230,275],{"stretchy":7289},[7699,35232,35233,35235],{},[7076,35234,10709],{},[7076,35236,279],{},[7073,35238,35239,35241],{},[7223,35240,1694],{"stretchy":7289},[7227,35242,460],{},[7223,35244,445],{},[14009,35246,35247,35321],{},[7070,35248,35249,35273,35275,35281,35287,35289,35291,35297,35303,35305,35311,35313,35319],{},[35125,35250,35251,35253],{},[7223,35252,12053],{},[7070,35254,35255,35261,35263,35269,35271],{},[7699,35256,35257,35259],{},[7076,35258,10709],{},[7076,35260,279],{},[7223,35262,1497],{},[7699,35264,35265,35267],{},[7076,35266,10709],{},[7227,35268,5312],{},[7223,35270,445],{},[7076,35272,32009],{},[7076,35274,7283],{"mathvariant":7282},[7699,35276,35277,35279],{},[7076,35278,32716],{},[7076,35280,279],{},[7073,35282,35283,35285],{},[7076,35284,7283],{"mathvariant":7282},[7227,35286,460],{},[7076,35288,1675],{},[7223,35290,275],{"stretchy":7289},[7699,35292,35293,35295],{},[7076,35294,10709],{},[7076,35296,279],{},[7073,35298,35299,35301],{},[7223,35300,1694],{"stretchy":7289},[7227,35302,460],{},[7223,35304,275],{"stretchy":7289},[7699,35306,35307,35309],{},[7076,35308,10709],{},[7076,35310,279],{},[7223,35312,7225],{},[7699,35314,35315,35317],{},[7076,35316,10709],{},[7227,35318,5312],{},[7223,35320,1694],{"stretchy":7289},[7070,35322,35323,35337,35339,35345,35351,35353,35355,35361],{},[32698,35324,35325,35327,35335],{},[7223,35326,12053],{},[7070,35328,35329,35331,35333],{},[7076,35330,279],{},[7223,35332,242],{},[7227,35334,5312],{},[7076,35336,12207],{},[7076,35338,7283],{"mathvariant":7282},[7699,35340,35341,35343],{},[7076,35342,32716],{},[7076,35344,279],{},[7073,35346,35347,35349],{},[7076,35348,7283],{"mathvariant":7282},[7227,35350,460],{},[7076,35352,1675],{},[7223,35354,275],{"stretchy":7289},[7699,35356,35357,35359],{},[7076,35358,10709],{},[7076,35360,279],{},[7073,35362,35363,35365],{},[7223,35364,1694],{"stretchy":7289},[7227,35366,460],{},[7076,35368,5122],{"mathvariant":7282},[7083,35370,35371],{"encoding":7085},"\\text{energy error} = \\frac{\\sum_{E_k\\le E_0+\\delta}|\\gamma_k|^2f(E_k)^2(E_k-E_0)}{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2} + \\frac{\\sum_{E_k> 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for every 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in the sum, and the second step follows because the sum in the numerator is a subset of the sum in the denominator. For the second term, first we lower bound the denominator by ",[35,38473,38475,38501],{"className":38474},[7057],[35,38476,38478],{"className":38477},[7061],[7063,38479,38480],{"xmlns":7065},[7067,38481,38482,38498],{},[7070,38483,38484,38486,38492],{},[7076,38485,7283],{"mathvariant":7282},[7699,38487,38488,38490],{},[7076,38489,32716],{},[7227,38491,5312],{},[7073,38493,38494,38496],{},[7076,38495,7283],{"mathvariant":7282},[7227,38497,460],{},[7083,38499,38500],{"encoding":7085},"|\\gamma_0|^2",[35,38502,38504],{"className":38503,"ariaHidden":7091},[7090],[35,38505,38507,38510,38513,38553],{"className":38506},[7095],[35,38508],{"className":38509,"style":17717},[7099],[35,38511,7283],{"className":38512},[7104],[35,38514,38516,38519],{"className":38515},[7104],[35,38517,32716],{"className":38518,"style":32852},[7104,7145],[35,38520,38522],{"className":38521},[7113],[35,38523,38525,38545],{"className":38524},[7117,7731],[35,38526,38528,38542],{"className":38527},[7121],[35,38529,38531],{"className":38530,"style":18794},[7125],[35,38532,38533,38536],{"style":32867},[35,38534],{"className":38535,"style":7134},[7133],[35,38537,38539],{"className":38538},[7138,7139,7140,7141],[35,38540,5312],{"className":38541},[7104,7141],[35,38543,7756],{"className":38544},[7755],[35,38546,38548],{"className":38547},[7121],[35,38549,38551],{"className":38550,"style":7763},[7125],[35,38552],{},[35,38554,38556,38559],{"className":38555},[7104],[35,38557,7283],{"className":38558},[7104],[35,38560,38562],{"className":38561},[7113],[35,38563,38565],{"className":38564},[7117],[35,38566,38568],{"className":38567},[7121],[35,38569,38571],{"className":38570,"style":17739},[7125],[35,38572,38573,38576],{"style":7129},[35,38574],{"className":38575,"style":7134},[7133],[35,38577,38579],{"className":38578},[7138,7139,7140,7141],[35,38580,460],{"className":38581},[7104,7141],", since ",[35,38584,38586,38618],{"className":38585},[7057],[35,38587,38589],{"className":38588},[7061],[7063,38590,38591],{"xmlns":7065},[7067,38592,38593,38615],{},[7070,38594,38595,38597,38599,38605,38611,38613],{},[7076,38596,1675],{},[7223,38598,275],{"stretchy":7289},[7699,38600,38601,38603],{},[7076,38602,10709],{},[7227,38604,5312],{},[7073,38606,38607,38609],{},[7223,38608,1694],{"stretchy":7289},[7227,38610,460],{},[7223,38612,242],{},[7227,38614,500],{},[7083,38616,38617],{"encoding":7085},"f(E_0)^2=1",[35,38619,38621,38711],{"className":38620,"ariaHidden":7091},[7090],[35,38622,38624,38627,38630,38633,38673,38702,38705,38708],{"className":38623},[7095],[35,38625],{"className":38626,"style":17717},[7099],[35,38628,1675],{"className":38629,"style":31824},[7104,7145],[35,38631,275],{"className":38632},[7472],[35,38634,38636,38639],{"className":38635},[7104],[35,38637,10709],{"className":38638,"style":10419},[7104,7145],[35,38640,38642],{"className":38641},[7113],[35,38643,38645,38665],{"className":38644},[7117,7731],[35,38646,38648,38662],{"className":38647},[7121],[35,38649,38651],{"className":38650,"style":18794},[7125],[35,38652,38653,38656],{"style":10541},[35,38654],{"className":38655,"style":7134},[7133],[35,38657,38659],{"className":38658},[7138,7139,7140,7141],[35,38660,5312],{"className":38661},[7104,7141],[35,38663,7756],{"className":38664},[7755],[35,38666,38668],{"className":38667},[7121],[35,38669,38671],{"className":38670,"style":7763},[7125],[35,38672],{},[35,38674,38676,38679],{"className":38675},[7314],[35,38677,1694],{"className":38678},[7314],[35,38680,38682],{"className":38681},[7113],[35,38683,38685],{"className":38684},[7117],[35,38686,38688],{"className":38687},[7121],[35,38689,38691],{"className":38690,"style":17739},[7125],[35,38692,38693,38696],{"style":7129},[35,38694],{"className":38695,"style":7134},[7133],[35,38697,38699],{"className":38698},[7138,7139,7140,7141],[35,38700,460],{"className":38701},[7104,7141],[35,38703],{"className":38704,"style":7450},[7248],[35,38706,242],{"className":38707},[7454],[35,38709],{"className":38710,"style":7450},[7248],[35,38712,38714,38717],{"className":38713},[7095],[35,38715],{"className":38716,"style":7263},[7099],[35,38718,500],{"className":38719},[7104],": adding everything back together, this gives",[35,38722,38724],{"className":38723},[7318],[35,38725,38727,38841],{"className":38726},[7057],[35,38728,38730],{"className":38729},[7061],[7063,38731,38732],{"xmlns":7065,"display":7327},[7067,38733,38734,38838],{},[7070,38735,38736,38738,38740,38742,38744,38764,38788,38790,38796,38802,38804,38806,38812,38818,38820,38826,38828,38834,38836],{},[10694,38737,33722],{},[7223,38739,31994],{},[7076,38741,32009],{},[7223,38743,445],{},[14009,38745,38746,38748],{},[7227,38747,500],{},[7070,38749,38750,38752,38758],{},[7076,38751,7283],{"mathvariant":7282},[7699,38753,38754,38756],{},[7076,38755,32716],{},[7227,38757,5312],{},[7073,38759,38760,38762],{},[7076,38761,7283],{"mathvariant":7282},[7227,38763,460],{},[35125,38765,38766,38768],{},[7223,38767,12053],{},[7070,38769,38770,38776,38778,38784,38786],{},[7699,38771,38772,38774],{},[7076,38773,10709],{},[7076,38775,279],{},[7223,38777,1497],{},[7699,38779,38780,38782],{},[7076,38781,10709],{},[7227,38783,5312],{},[7223,38785,445],{},[7076,38787,32009],{},[7076,38789,7283],{"mathvariant":7282},[7699,38791,38792,38794],{},[7076,38793,32716],{},[7076,38795,279],{},[7073,38797,38798,38800],{},[7076,38799,7283],{"mathvariant":7282},[7227,38801,460],{},[7076,38803,1675],{},[7223,38805,275],{"stretchy":7289},[7699,38807,38808,38810],{},[7076,38809,10709],{},[7076,38811,279],{},[7073,38813,38814,38816],{},[7223,38815,1694],{"stretchy":7289},[7227,38817,460],{},[7223,38819,275],{"stretchy":7289},[7699,38821,38822,38824],{},[7076,38823,10709],{},[7076,38825,279],{},[7223,38827,7225],{},[7699,38829,38830,38832],{},[7076,38831,10709],{},[7227,38833,5312],{},[7223,38835,1694],{"stretchy":7289},[7076,38837,5122],{"mathvariant":7282},[7083,38839,38840],{"encoding":7085},"\\text{energy error} \\le \\delta + \\frac{1}{|\\gamma_0|^2}\\sum_{E_k>E_0+\\delta}|\\gamma_k|^2f(E_k)^2(E_k-E_0).",[35,38842,38844,38866,38885,39363],{"className":38843,"ariaHidden":7091},[7090],[35,38845,38847,38851,38857,38860,38863],{"className":38846},[7095],[35,38848],{"className":38849,"style":38850},[7099],"height:0.8304em;vertical-align:-0.1944em;",[35,38852,38854],{"className":38853},[7104,4991],[35,38855,33722],{"className":38856},[7104],[35,38858],{"className":38859,"style":7450},[7248],[35,38861,31994],{"className":38862},[7454],[35,38864],{"className":38865,"style":7450},[7248],[35,38867,38869,38873,38876,38879,38882],{"className":38868},[7095],[35,38870],{"className":38871,"style":38872},[7099],"height:0.7778em;vertical-align:-0.0833em;",[35,38874,32009],{"className":38875,"style":32094},[7104,7145],[35,38877],{"className":38878,"style":7249},[7248],[35,38880,445],{"className":38881},[7253],[35,38883],{"className":38884,"style":7249},[7248],[35,38886,38888,38892,39023,39026,39161,39164,39167,39207,39236,39239,39242,39282,39311,39314,39354,39357,39360],{"className":38887},[7095],[35,38889],{"className":38890,"style":38891},[7099],"height:2.7294em;vertical-align:-1.408em;",[35,38893,38895,38898,39020],{"className":38894},[7104],[35,38896],{"className":38897},[7472,14311],[35,38899,38901],{"className":38900},[14009],[35,38902,38904,39012],{"className":38903},[7117,7731],[35,38905,38907,39009],{"className":38906},[7121],[35,38908,38910,38990,38998],{"className":38909,"style":14324},[7125],[35,38911,38912,38915],{"style":16234},[35,38913],{"className":38914,"style":8839},[7133],[35,38916,38918,38921,38961],{"className":38917},[7104],[35,38919,7283],{"className":38920},[7104],[35,38922,38924,38927],{"className":38923},[7104],[35,38925,32716],{"className":38926,"style":32852},[7104,7145],[35,38928,38930],{"className":38929},[7113],[35,38931,38933,38953],{"className":38932},[7117,7731],[35,38934,38936,38950],{"className":38935},[7121],[35,38937,38939],{"className":38938,"style":18794},[7125],[35,38940,38941,38944],{"style":32867},[35,38942],{"className":38943,"style":7134},[7133],[35,38945,38947],{"className":38946},[7138,7139,7140,7141],[35,38948,5312],{"className":38949},[7104,7141],[35,38951,7756],{"className":38952},[7755],[35,38954,38956],{"className":38955},[7121],[35,38957,38959],{"className":38958,"style":7763},[7125],[35,38960],{},[35,38962,38964,38967],{"className":38963},[7104],[35,38965,7283],{"className":38966},[7104],[35,38968,38970],{"className":38969},[7113],[35,38971,38973],{"className":38972},[7117],[35,38974,38976],{"className":38975},[7121],[35,38977,38979],{"className":38978,"style":33894},[7125],[35,38980,38981,38984],{"style":33897},[35,38982],{"className":38983,"style":7134},[7133],[35,38985,38987],{"className":38986},[7138,7139,7140,7141],[35,38988,460],{"className":38989},[7104,7141],[35,38991,38992,38995],{"style":14395},[35,38993],{"className":38994,"style":8839},[7133],[35,38996],{"className":38997,"style":14403},[14402],[35,38999,39000,39003],{"style":14406},[35,39001],{"className":39002,"style":8839},[7133],[35,39004,39006],{"className":39005},[7104],[35,39007,500],{"className":39008},[7104],[35,39010,7756],{"className":39011},[7755],[35,39013,39015],{"className":39014},[7121],[35,39016,39018],{"className":39017,"style":34041},[7125],[35,39019],{},[35,39021],{"className":39022},[7314,14311],[35,39024],{"className":39025,"style":7496},[7248],[35,39027,39029],{"className":39028},[12096,32771],[35,39030,39032,39152],{"className":39031},[7117,7731],[35,39033,39035,39149],{"className":39034},[7121],[35,39036,39039,39139],{"className":39037,"style":39038},[7125],"height:1.05em;",[35,39040,39041,39044],{"style":32784},[35,39042],{"className":39043,"style":32788},[7133],[35,39045,39047],{"className":39046},[7138,7139,7140,7141],[35,39048,39050,39090,39093,39133,39136],{"className":39049},[7104,7141],[35,39051,39053,39056],{"className":39052},[7104,7141],[35,39054,10709],{"className":39055,"style":10419},[7104,7145,7141],[35,39057,39059],{"className":39058},[7113],[35,39060,39062,39082],{"className":39061},[7117,7731],[35,39063,39065,39079],{"className":39064},[7121],[35,39066,39068],{"className":39067,"style":35708},[7125],[35,39069,39070,39073],{"style":35711},[35,39071],{"className":39072,"style":9340},[7133],[35,39074,39076],{"className":39075},[7138,9344,7479,7141],[35,39077,279],{"className":39078,"style":8221},[7104,7145,7141],[35,39080,7756],{"className":39081},[7755],[35,39083,39085],{"className":39084},[7121],[35,39086,39088],{"className":39087,"style":35730},[7125],[35,39089],{},[35,39091,1497],{"className":39092},[7454,7141],[35,39094,39096,39099],{"className":39095},[7104,7141],[35,39097,10709],{"className":39098,"style":10419},[7104,7145,7141],[35,39100,39102],{"className":39101},[7113],[35,39103,39105,39125],{"className":39104},[7117,7731],[35,39106,39108,39122],{"className":39107},[7121],[35,39109,39111],{"className":39110,"style":35754},[7125],[35,39112,39113,39116],{"style":35757},[35,39114],{"className":39115,"style":9340},[7133],[35,39117,39119],{"className":39118},[7138,9344,7479,7141],[35,39120,5312],{"className":39121},[7104,7141],[35,39123,7756],{"className":39124},[7755],[35,39126,39128],{"className":39127},[7121],[35,39129,39131],{"className":39130,"style":9357},[7125],[35,39132],{},[35,39134,445],{"className":39135},[7253,7141],[35,39137,32009],{"className":39138,"style":32094},[7104,7145,7141],[35,39140,39141,39144],{"style":32806},[35,39142],{"className":39143,"style":32788},[7133],[35,39145,39146],{},[35,39147,12053],{"className":39148},[12096,12100,32815],[35,39150,7756],{"className":39151},[7755],[35,39153,39155],{"className":39154},[7121],[35,39156,39159],{"className":39157,"style":39158},[7125],"height:1.408em;",[35,39160],{},[35,39162],{"className":39163,"style":7496},[7248],[35,39165,7283],{"className":39166},[7104],[35,39168,39170,39173],{"className":39169},[7104],[35,39171,32716],{"className":39172,"style":32852},[7104,7145],[35,39174,39176],{"className":39175},[7113],[35,39177,39179,39199],{"className":39178},[7117,7731],[35,39180,39182,39196],{"className":39181},[7121],[35,39183,39185],{"className":39184,"style":15760},[7125],[35,39186,39187,39190],{"style":32867},[35,39188],{"className":39189,"style":7134},[7133],[35,39191,39193],{"className":39192},[7138,7139,7140,7141],[35,39194,279],{"className":39195,"style":8221},[7104,7145,7141],[35,39197,7756],{"className":39198},[7755],[35,39200,39202],{"className":39201},[7121],[35,39203,39205],{"className":39204,"style":7763},[7125],[35,39206],{},[35,39208,39210,39213],{"className":39209},[7104],[35,39211,7283],{"className":39212},[7104],[35,39214,39216],{"className":39215},[7113],[35,39217,39219],{"className":39218},[7117],[35,39220,39222],{"className":39221},[7121],[35,39223,39225],{"className":39224,"style":7533},[7125],[35,39226,39227,39230],{"style":7437},[35,39228],{"className":39229,"style":7134},[7133],[35,39231,39233],{"className":39232},[7138,7139,7140,7141],[35,39234,460],{"className":39235},[7104,7141],[35,39237,1675],{"className":39238,"style":31824},[7104,7145],[35,39240,275],{"className":39241},[7472],[35,39243,39245,39248],{"className":39244},[7104],[35,39246,10709],{"className":39247,"style":10419},[7104,7145],[35,39249,39251],{"className":39250},[7113],[35,39252,39254,39274],{"className":39253},[7117,7731],[35,39255,39257,39271],{"className":39256},[7121],[35,39258,39260],{"className":39259,"style":15760},[7125],[35,39261,39262,39265],{"style":10541},[35,39263],{"className":39264,"style":7134},[7133],[35,39266,39268],{"className":39267},[7138,7139,7140,7141],[35,39269,279],{"className":39270,"style":8221},[7104,7145,7141],[35,39272,7756],{"className":39273},[7755],[35,39275,39277],{"className":39276},[7121],[35,39278,39280],{"className":39279,"style":7763},[7125],[35,39281],{},[35,39283,39285,39288],{"className":39284},[7314],[35,39286,1694],{"className":39287},[7314],[35,39289,39291],{"className":39290},[7113],[35,39292,39294],{"className":39293},[7117],[35,39295,39297],{"className":39296},[7121],[35,39298,39300],{"className":39299,"style":7533},[7125],[35,39301,39302,39305],{"style":7437},[35,39303],{"className":39304,"style":7134},[7133],[35,39306,39308],{"className":39307},[7138,7139,7140,7141],[35,39309,460],{"className":39310},[7104,7141],[35,39312,275],{"className":39313},[7472],[35,39315,39317,39320],{"className":39316},[7104],[35,39318,10709],{"className":39319,"style":10419},[7104,7145],[35,39321,39323],{"className":39322},[7113],[35,39324,39326,39346],{"className":39325},[7117,7731],[35,39327,39329,39343],{"className":39328},[7121],[35,39330,39332],{"className":39331,"style":15760},[7125],[35,39333,39334,39337],{"style":10541},[35,39335],{"className":39336,"style":7134},[7133],[35,39338,39340],{"className":39339},[7138,7139,7140,7141],[35,39341,279],{"className":39342,"style":8221},[7104,7145,7141],[35,39344,7756],{"className":39345},[7755],[35,39347,39349],{"className":39348},[7121],[35,39350,39352],{"className":39351,"style":7763},[7125],[35,39353],{},[35,39355],{"className":39356,"style":7249},[7248],[35,39358,7225],{"className":39359},[7253],[35,39361],{"className":39362,"style":7249},[7248],[35,39364,39366,39369,39409,39412],{"className":39365},[7095],[35,39367],{"className":39368,"style":7303},[7099],[35,39370,39372,39375],{"className":39371},[7104],[35,39373,10709],{"className":39374,"style":10419},[7104,7145],[35,39376,39378],{"className":39377},[7113],[35,39379,39381,39401],{"className":39380},[7117,7731],[35,39382,39384,39398],{"className":39383},[7121],[35,39385,39387],{"className":39386,"style":18794},[7125],[35,39388,39389,39392],{"style":10541},[35,39390],{"className":39391,"style":7134},[7133],[35,39393,39395],{"className":39394},[7138,7139,7140,7141],[35,39396,5312],{"className":39397},[7104,7141],[35,39399,7756],{"className":39400},[7755],[35,39402,39404],{"className":39403},[7121],[35,39405,39407],{"className":39406,"style":7763},[7125],[35,39408],{},[35,39410,1694],{"className":39411},[7314],[35,39413,5122],{"className":39414},[7104],[12,39416,39417,39418,39487,39488,39516,39517,39738,39739,39907,39908,40197],{},"To simplify what is left, notice that for all these ",[35,39419,39421,39438],{"className":39420},[7057],[35,39422,39424],{"className":39423},[7061],[7063,39425,39426],{"xmlns":7065},[7067,39427,39428,39436],{},[7070,39429,39430],{},[7699,39431,39432,39434],{},[7076,39433,10709],{},[7076,39435,279],{},[7083,39437,33574],{"encoding":7085},[35,39439,39441],{"className":39440,"ariaHidden":7091},[7090],[35,39442,39444,39447],{"className":39443},[7095],[35,39445],{"className":39446,"style":7718},[7099],[35,39448,39450,39453],{"className":39449},[7104],[35,39451,10709],{"className":39452,"style":10419},[7104,7145],[35,39454,39456],{"className":39455},[7113],[35,39457,39459,39479],{"className":39458},[7117,7731],[35,39460,39462,39476],{"className":39461},[7121],[35,39463,39465],{"className":39464,"style":15760},[7125],[35,39466,39467,39470],{"style":10541},[35,39468],{"className":39469,"style":7134},[7133],[35,39471,39473],{"className":39472},[7138,7139,7140,7141],[35,39474,279],{"className":39475,"style":8221},[7104,7145,7141],[35,39477,7756],{"className":39478},[7755],[35,39480,39482],{"className":39481},[7121],[35,39483,39485],{"className":39484,"style":7763},[7125],[35,39486],{},", by the definition of ",[35,39489,39491,39504],{"className":39490},[7057],[35,39492,39494],{"className":39493},[7061],[7063,39495,39496],{"xmlns":7065},[7067,39497,39498,39502],{},[7070,39499,39500],{},[7076,39501,1675],{},[7083,39503,1675],{"encoding":7085},[35,39505,39507],{"className":39506,"ariaHidden":7091},[7090],[35,39508,39510,39513],{"className":39509},[7095],[35,39511],{"className":39512,"style":11562},[7099],[35,39514,1675],{"className":39515,"style":31824},[7104,7145]," we know that ",[35,39518,39520,39574],{"className":39519},[7057],[35,39521,39523],{"className":39522},[7061],[7063,39524,39525],{"xmlns":7065},[7067,39526,39527,39571],{},[7070,39528,39529,39531,39533,39539,39545,39547,39549],{},[7076,39530,1675],{},[7223,39532,275],{"stretchy":7289},[7699,39534,39535,39537],{},[7076,39536,10709],{},[7076,39538,279],{},[7073,39540,39541,39543],{},[7223,39542,1694],{"stretchy":7289},[7227,39544,460],{},[7223,39546,31994],{},[7227,39548,528],{},[7073,39550,39551,39563],{},[7070,39552,39553,39555,39557,39559,39561],{},[7223,39554,275],{"fence":7091},[7227,39556,500],{},[7223,39558,445],{},[7076,39560,32009],{},[7223,39562,1694],{"fence":7091},[7070,39564,39565,39567,39569],{},[7223,39566,7225],{},[7227,39568,460],{},[7076,39570,9487],{},[7083,39572,39573],{"encoding":7085},"f(E_k)^2 \\le 4\\left(1 + \\delta\\right)^{-2d}",[35,39575,39577,39667],{"className":39576,"ariaHidden":7091},[7090],[35,39578,39580,39583,39586,39589,39629,39658,39661,39664],{"className":39579},[7095],[35,39581],{"className":39582,"style":17717},[7099],[35,39584,1675],{"className":39585,"style":31824},[7104,7145],[35,39587,275],{"className":39588},[7472],[35,39590,39592,39595],{"className":39591},[7104],[35,39593,10709],{"className":39594,"style":10419},[7104,7145],[35,39596,39598],{"className":39597},[7113],[35,39599,39601,39621],{"className":39600},[7117,7731],[35,39602,39604,39618],{"className":39603},[7121],[35,39605,39607],{"className":39606,"style":15760},[7125],[35,39608,39609,39612],{"style":10541},[35,39610],{"className":39611,"style":7134},[7133],[35,39613,39615],{"className":39614},[7138,7139,7140,7141],[35,39616,279],{"className":39617,"style":8221},[7104,7145,7141],[35,39619,7756],{"className":39620},[7755],[35,39622,39624],{"className":39623},[7121],[35,39625,39627],{"className":39626,"style":7763},[7125],[35,39628],{},[35,39630,39632,39635],{"className":39631},[7314],[35,39633,1694],{"className":39634},[7314],[35,39636,39638],{"className":39637},[7113],[35,39639,39641],{"className":39640},[7117],[35,39642,39644],{"className":39643},[7121],[35,39645,39647],{"className":39646,"style":17739},[7125],[35,39648,39649,39652],{"style":7129},[35,39650],{"className":39651,"style":7134},[7133],[35,39653,39655],{"className":39654},[7138,7139,7140,7141],[35,39656,460],{"className":39657},[7104,7141],[35,39659],{"className":39660,"style":7450},[7248],[35,39662,31994],{"className":39663},[7454],[35,39665],{"className":39666,"style":7450},[7248],[35,39668,39670,39673,39676,39679],{"className":39669},[7095],[35,39671],{"className":39672,"style":32063},[7099],[35,39674,528],{"className":39675},[7104],[35,39677],{"className":39678,"style":7496},[7248],[35,39680,39682,39706],{"className":39681},[7468],[35,39683,39685,39688,39691,39694,39697,39700,39703],{"className":39684},[7468],[35,39686,275],{"className":39687,"style":7474},[7472,7473],[35,39689,500],{"className":39690},[7104],[35,39692],{"className":39693,"style":7249},[7248],[35,39695,445],{"className":39696},[7253],[35,39698],{"className":39699,"style":7249},[7248],[35,39701,32009],{"className":39702,"style":32094},[7104,7145],[35,39704,1694],{"className":39705,"style":7474},[7314,7473],[35,39707,39709],{"className":39708},[7113],[35,39710,39712],{"className":39711},[7117],[35,39713,39715],{"className":39714},[7121],[35,39716,39718],{"className":39717,"style":32110},[7125],[35,39719,39720,39723],{"style":12371},[35,39721],{"className":39722,"style":7134},[7133],[35,39724,39726],{"className":39725},[7138,7139,7140,7141],[35,39727,39729,39732,39735],{"className":39728},[7104,7141],[35,39730,7225],{"className":39731},[7104,7141],[35,39733,460],{"className":39734},[7104,7141],[35,39736,9487],{"className":39737},[7104,7145,7141],". Additionally upper bounding ",[35,39740,39742,39778],{"className":39741},[7057],[35,39743,39745],{"className":39744},[7061],[7063,39746,39747],{"xmlns":7065},[7067,39748,39749,39775],{},[7070,39750,39751,39757,39759,39765,39767,39769,39771,39773],{},[7699,39752,39753,39755],{},[7076,39754,10709],{},[7076,39756,279],{},[7223,39758,7225],{},[7699,39760,39761,39763],{},[7076,39762,10709],{},[7227,39764,5312],{},[7223,39766,10205],{},[7227,39768,460],{},[7076,39770,16674],{"mathvariant":7282},[7076,39772,7669],{},[7076,39774,16674],{"mathvariant":7282},[7083,39776,39777],{"encoding":7085},"E_k-E_0\u003C2\\|H\\|",[35,39779,39781,39836,39891],{"className":39780,"ariaHidden":7091},[7090],[35,39782,39784,39787,39827,39830,39833],{"className":39783},[7095],[35,39785],{"className":39786,"style":7718},[7099],[35,39788,39790,39793],{"className":39789},[7104],[35,39791,10709],{"className":39792,"style":10419},[7104,7145],[35,39794,39796],{"className":39795},[7113],[35,39797,39799,39819],{"className":39798},[7117,7731],[35,39800,39802,39816],{"className":39801},[7121],[35,39803,39805],{"className":39804,"style":15760},[7125],[35,39806,39807,39810],{"style":10541},[35,39808],{"className":39809,"style":7134},[7133],[35,39811,39813],{"className":39812},[7138,7139,7140,7141],[35,39814,279],{"className":39815,"style":8221},[7104,7145,7141],[35,39817,7756],{"className":39818},[7755],[35,39820,39822],{"className":39821},[7121],[35,39823,39825],{"className":39824,"style":7763},[7125],[35,39826],{},[35,39828],{"className":39829,"style":7249},[7248],[35,39831,7225],{"className":39832},[7253],[35,39834],{"className":39835,"style":7249},[7248],[35,39837,39839,39842,39882,39885,39888],{"className":39838},[7095],[35,39840],{"className":39841,"style":7718},[7099],[35,39843,39845,39848],{"className":39844},[7104],[35,39846,10709],{"className":39847,"style":10419},[7104,7145],[35,39849,39851],{"className":39850},[7113],[35,39852,39854,39874],{"className":39853},[7117,7731],[35,39855,39857,39871],{"className":39856},[7121],[35,39858,39860],{"className":39859,"style":18794},[7125],[35,39861,39862,39865],{"style":10541},[35,39863],{"className":39864,"style":7134},[7133],[35,39866,39868],{"className":39867},[7138,7139,7140,7141],[35,39869,5312],{"className":39870},[7104,7141],[35,39872,7756],{"className":39873},[7755],[35,39875,39877],{"className":39876},[7121],[35,39878,39880],{"className":39879,"style":7763},[7125],[35,39881],{},[35,39883],{"className":39884,"style":7450},[7248],[35,39886,10205],{"className":39887},[7454],[35,39889],{"className":39890,"style":7450},[7248],[35,39892,39894,39897,39901,39904],{"className":39893},[7095],[35,39895],{"className":39896,"style":7303},[7099],[35,39898,39900],{"className":39899},[7104],"2∥",[35,39902,7669],{"className":39903,"style":7684},[7104,7145],[35,39905,16674],{"className":39906},[7104]," and upper bounding 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gives",[35,40199,40201],{"className":40200},[7318],[35,40202,40204,40275],{"className":40203},[7057],[35,40205,40207],{"className":40206},[7061],[7063,40208,40209],{"xmlns":7065,"display":7327},[7067,40210,40211,40272],{},[7070,40212,40213,40215,40217,40219,40221,40242,40244,40246,40248,40270],{},[10694,40214,33722],{},[7223,40216,31994],{},[7076,40218,32009],{},[7223,40220,445],{},[14009,40222,40223,40226],{},[7227,40224,40225],{},"8",[7070,40227,40228,40230,40236],{},[7076,40229,7283],{"mathvariant":7282},[7699,40231,40232,40234],{},[7076,40233,32716],{},[7227,40235,5312],{},[7073,40237,40238,40240],{},[7076,40239,7283],{"mathvariant":7282},[7227,40241,460],{},[7076,40243,16674],{"mathvariant":7282},[7076,40245,7669],{},[7076,40247,16674],{"mathvariant":7282},[7073,40249,40250,40262],{},[7070,40251,40252,40254,40256,40258,40260],{},[7223,40253,275],{"fence":7091},[7227,40255,500],{},[7223,40257,445],{},[7076,40259,32009],{},[7223,40261,1694],{"fence":7091},[7070,40263,40264,40266,40268],{},[7223,40265,7225],{},[7227,40267,460],{},[7076,40269,9487],{},[7076,40271,5122],{"mathvariant":7282},[7083,40273,40274],{"encoding":7085},"\\text{energy error} \\le \\delta + \\frac{8}{|\\gamma_0|^2}\\|H\\|\\left(1 + \\delta\\right)^{-2d}.",[35,40276,40278,40299,40317],{"className":40277,"ariaHidden":7091},[7090],[35,40279,40281,40284,40290,40293,40296],{"className":40280},[7095],[35,40282],{"className":40283,"style":38850},[7099],[35,40285,40287],{"className":40286},[7104,4991],[35,40288,33722],{"className":40289},[7104],[35,40291],{"className":40292,"style":7450},[7248],[35,40294,31994],{"className":40295},[7454],[35,40297],{"className":40298,"style":7450},[7248],[35,40300,40302,40305,40308,40311,40314],{"className":40301},[7095],[35,40303],{"className":40304,"style":38872},[7099],[35,40306,32009],{"className":40307,"style":32094},[7104,7145],[35,40309],{"className":40310,"style":7249},[7248],[35,40312,445],{"className":40313},[7253],[35,40315],{"className":40316,"style":7249},[7248],[35,40318,40320,40324,40455,40458,40461,40464,40467,40526,40529],{"className":40319},[7095],[35,40321],{"className":40322,"style":40323},[7099],"height:2.2574em;vertical-align:-0.936em;",[35,40325,40327,40330,40452],{"className":40326},[7104],[35,40328],{"className":40329},[7472,14311],[35,40331,40333],{"className":40332},[14009],[35,40334,40336,40444],{"className":40335},[7117,7731],[35,40337,40339,40441],{"className":40338},[7121],[35,40340,40342,40422,40430],{"className":40341,"style":14324},[7125],[35,40343,40344,40347],{"style":16234},[35,40345],{"className":40346,"style":8839},[7133],[35,40348,40350,40353,40393],{"className":40349},[7104],[35,40351,7283],{"className":40352},[7104],[35,40354,40356,40359],{"className":40355},[7104],[35,40357,32716],{"className":40358,"style":32852},[7104,7145],[35,40360,40362],{"className":40361},[7113],[35,40363,40365,40385],{"className":40364},[7117,7731],[35,40366,40368,40382],{"className":40367},[7121],[35,40369,40371],{"className":40370,"style":18794},[7125],[35,40372,40373,40376],{"style":32867},[35,40374],{"className":40375,"style":7134},[7133],[35,40377,40379],{"className":40378},[7138,7139,7140,7141],[35,40380,5312],{"className":40381},[7104,7141],[35,40383,7756],{"className":40384},[7755],[35,40386,40388],{"className":40387},[7121],[35,40389,40391],{"className":40390,"style":7763},[7125],[35,40392],{},[35,40394,40396,40399],{"className":40395},[7104],[35,40397,7283],{"className":40398},[7104],[35,40400,40402],{"className":40401},[7113],[35,40403,40405],{"className":40404},[7117],[35,40406,40408],{"className":40407},[7121],[35,40409,40411],{"className":40410,"style":33894},[7125],[35,40412,40413,40416],{"style":33897},[35,40414],{"className":40415,"style":7134},[7133],[35,40417,40419],{"className":40418},[7138,7139,7140,7141],[35,40420,460],{"className":40421},[7104,7141],[35,40423,40424,40427],{"style":14395},[35,40425],{"className":40426,"style":8839},[7133],[35,40428],{"className":40429,"style":14403},[14402],[35,40431,40432,40435],{"style":14406},[35,40433],{"className":40434,"style":8839},[7133],[35,40436,40438],{"className":40437},[7104],[35,40439,40225],{"className":40440},[7104],[35,40442,7756],{"className":40443},[7755],[35,40445,40447],{"className":40446},[7121],[35,40448,40450],{"className":40449,"style":34041},[7125],[35,40451],{},[35,40453],{"className":40454},[7314,14311],[35,40456,16674],{"className":40457},[7104],[35,40459,7669],{"className":40460,"style":7684},[7104,7145],[35,40462,16674],{"className":40463},[7104],[35,40465],{"className":40466,"style":7496},[7248],[35,40468,40470,40494],{"className":40469},[7468],[35,40471,40473,40476,40479,40482,40485,40488,40491],{"className":40472},[7468],[35,40474,275],{"className":40475,"style":7474},[7472,7473],[35,40477,500],{"className":40478},[7104],[35,40480],{"className":40481,"style":7249},[7248],[35,40483,445],{"className":40484},[7253],[35,40486],{"className":40487,"style":7249},[7248],[35,40489,32009],{"className":40490,"style":32094},[7104,7145],[35,40492,1694],{"className":40493,"style":7474},[7314,7473],[35,40495,40497],{"className":40496},[7113],[35,40498,40500],{"className":40499},[7117],[35,40501,40503],{"className":40502},[7121],[35,40504,40506],{"className":40505,"style":32110},[7125],[35,40507,40508,40511],{"style":12371},[35,40509],{"className":40510,"style":7134},[7133],[35,40512,40514],{"className":40513},[7138,7139,7140,7141],[35,40515,40517,40520,40523],{"className":40516},[7104,7141],[35,40518,7225],{"className":40519},[7104,7141],[35,40521,460],{"className":40522},[7104,7141],[35,40524,9487],{"className":40525},[7104,7145,7141],[35,40527],{"className":40528,"style":7496},[7248],[35,40530,5122],{"className":40531},[7104],[12,40533,40534,40535,40587,40588,40616,40617,40673,40674,40829,40830,40858],{},"This holds for any ",[35,40536,40538,40556],{"className":40537},[7057],[35,40539,40541],{"className":40540},[7061],[7063,40542,40543],{"xmlns":7065},[7067,40544,40545,40553],{},[7070,40546,40547,40549,40551],{},[7076,40548,32009],{},[7223,40550,1497],{},[7227,40552,5312],{},[7083,40554,40555],{"encoding":7085},"\\delta>0",[35,40557,40559,40578],{"className":40558,"ariaHidden":7091},[7090],[35,40560,40562,40566,40569,40572,40575],{"className":40561},[7095],[35,40563],{"className":40564,"style":40565},[7099],"height:0.7335em;vertical-align:-0.0391em;",[35,40567,32009],{"className":40568,"style":32094},[7104,7145],[35,40570],{"className":40571,"style":7450},[7248],[35,40573,1497],{"className":40574},[7454],[35,40576],{"className":40577,"style":7450},[7248],[35,40579,40581,40584],{"className":40580},[7095],[35,40582],{"className":40583,"style":7263},[7099],[35,40585,5312],{"className":40586},[7104],", so if we set ",[35,40589,40591,40604],{"className":40590},[7057],[35,40592,40594],{"className":40593},[7061],[7063,40595,40596],{"xmlns":7065},[7067,40597,40598,40602],{},[7070,40599,40600],{},[7076,40601,32009],{},[7083,40603,36735],{"encoding":7085},[35,40605,40607],{"className":40606,"ariaHidden":7091},[7090],[35,40608,40610,40613],{"className":40609},[7095],[35,40611],{"className":40612,"style":9962},[7099],[35,40614,32009],{"className":40615,"style":32094},[7104,7145]," equal to our goal error, then the error bound above converges towards that exponentially with the Krylov dimension ",[35,40618,40620,40640],{"className":40619},[7057],[35,40621,40623],{"className":40622},[7061],[7063,40624,40625],{"xmlns":7065},[7067,40626,40627,40637],{},[7070,40628,40629,40631,40633,40635],{},[7227,40630,460],{},[7076,40632,9487],{},[7223,40634,242],{},[7076,40636,6778],{},[7083,40638,40639],{"encoding":7085},"2d=r",[35,40641,40643,40664],{"className":40642,"ariaHidden":7091},[7090],[35,40644,40646,40649,40652,40655,40658,40661],{"className":40645},[7095],[35,40647],{"className":40648,"style":9962},[7099],[35,40650,460],{"className":40651},[7104],[35,40653,9487],{"className":40654},[7104,7145],[35,40656],{"className":40657,"style":7450},[7248],[35,40659,242],{"className":40660},[7454],[35,40662],{"className":40663,"style":7450},[7248],[35,40665,40667,40670],{"className":40666},[7095],[35,40668],{"className":40669,"style":7173},[7099],[35,40671,6778],{"className":40672,"style":7146},[7104,7145],". Also note that if ",[35,40675,40677,40707],{"className":40676},[7057],[35,40678,40680],{"className":40679},[7061],[7063,40681,40682],{"xmlns":7065},[7067,40683,40684,40704],{},[7070,40685,40686,40688,40690,40696,40698],{},[7076,40687,32009],{},[7223,40689,10205],{},[7699,40691,40692,40694],{},[7076,40693,10709],{},[7227,40695,500],{},[7223,40697,7225],{},[7699,40699,40700,40702],{},[7076,40701,10709],{},[7227,40703,5312],{},[7083,40705,40706],{"encoding":7085},"\\delta\u003CE_1-E_0",[35,40708,40710,40728,40783],{"className":40709,"ariaHidden":7091},[7090],[35,40711,40713,40716,40719,40722,40725],{"className":40712},[7095],[35,40714],{"className":40715,"style":40565},[7099],[35,40717,32009],{"className":40718,"style":32094},[7104,7145],[35,40720],{"className":40721,"style":7450},[7248],[35,40723,10205],{"className":40724},[7454],[35,40726],{"className":40727,"style":7450},[7248],[35,40729,40731,40734,40774,40777,40780],{"className":40730},[7095],[35,40732],{"className":40733,"style":7718},[7099],[35,40735,40737,40740],{"className":40736},[7104],[35,40738,10709],{"className":40739,"style":10419},[7104,7145],[35,40741,40743],{"className":40742},[7113],[35,40744,40746,40766],{"className":40745},[7117,7731],[35,40747,40749,40763],{"className":40748},[7121],[35,40750,40752],{"className":40751,"style":18794},[7125],[35,40753,40754,40757],{"style":10541},[35,40755],{"className":40756,"style":7134},[7133],[35,40758,40760],{"className":40759},[7138,7139,7140,7141],[35,40761,500],{"className":40762},[7104,7141],[35,40764,7756],{"className":40765},[7755],[35,40767,40769],{"className":40768},[7121],[35,40770,40772],{"className":40771,"style":7763},[7125],[35,40773],{},[35,40775],{"className":40776,"style":7249},[7248],[35,40778,7225],{"className":40779},[7253],[35,40781],{"className":40782,"style":7249},[7248],[35,40784,40786,40789],{"className":40785},[7095],[35,40787],{"className":40788,"style":7718},[7099],[35,40790,40792,40795],{"className":40791},[7104],[35,40793,10709],{"className":40794,"style":10419},[7104,7145],[35,40796,40798],{"className":40797},[7113],[35,40799,40801,40821],{"className":40800},[7117,7731],[35,40802,40804,40818],{"className":40803},[7121],[35,40805,40807],{"className":40806,"style":18794},[7125],[35,40808,40809,40812],{"style":10541},[35,40810],{"className":40811,"style":7134},[7133],[35,40813,40815],{"className":40814},[7138,7139,7140,7141],[35,40816,5312],{"className":40817},[7104,7141],[35,40819,7756],{"className":40820},[7755],[35,40822,40824],{"className":40823},[7121],[35,40825,40827],{"className":40826,"style":7763},[7125],[35,40828],{}," then the ",[35,40831,40833,40846],{"className":40832},[7057],[35,40834,40836],{"className":40835},[7061],[7063,40837,40838],{"xmlns":7065},[7067,40839,40840,40844],{},[7070,40841,40842],{},[7076,40843,32009],{},[7083,40845,36735],{"encoding":7085},[35,40847,40849],{"className":40848,"ariaHidden":7091},[7090],[35,40850,40852,40855],{"className":40851},[7095],[35,40853],{"className":40854,"style":9962},[7099],[35,40856,32009],{"className":40857,"style":32094},[7104,7145]," term actually goes away entirely in the above bound.",[12,40860,40861,40862,40920],{},"To complete the argument, we first note that the above is just the energy error of the particular state ",[35,40863,40865,40890],{"className":40864},[7057],[35,40866,40868],{"className":40867},[7061],[7063,40869,40870],{"xmlns":7065},[7067,40871,40872,40888],{},[7070,40873,40874,40876,40878,40880,40882,40884,40886],{},[7076,40875,1675],{},[7223,40877,275],{"stretchy":7289},[7076,40879,7669],{},[7223,40881,1694],{"stretchy":7289},[7076,40883,7283],{"mathvariant":7282},[7076,40885,8322],{},[7223,40887,7290],{"stretchy":7289},[7083,40889,32602],{"encoding":7085},[35,40891,40893],{"className":40892,"ariaHidden":7091},[7090],[35,40894,40896,40899,40902,40905,40908,40911,40914,40917],{"className":40895},[7095],[35,40897],{"className":40898,"style":7303},[7099],[35,40900,1675],{"className":40901,"style":31824},[7104,7145],[35,40903,275],{"className":40904},[7472],[35,40906,7669],{"className":40907,"style":7684},[7104,7145],[35,40909,1694],{"className":40910},[7314],[35,40912,7283],{"className":40913},[7104],[35,40915,8322],{"className":40916,"style":7310},[7104,7145],[35,40918,7290],{"className":40919},[7314],", rather than the energy error of the lowest energy state in the Krylov space. However, by the (Rayleigh-Ritz) variational principle, the energy error of the lowest energy state in the Krylov space is upper bounded by the energy error of any state in the Krylov space, so the above is also an upper bound on the energy error of the lowest energy state, that is, the output of the Krylov quantum diagonalization algorithm.",[12,40922,40923,40924,15316,40926,40929],{},"A similar analysis as the above can be carried out that additionally accounts for noise and the thresholding procedure discussed in the notebook. See ",[19,40925,13065],{"href":5756},[19,40927,40928],{"href":5756},"[4]"," for this analysis.",[5019,40931,40933],{"id":40932},"appendix-proof-of-claim-1","Appendix: proof of Claim 1",[12,40935,40936,40937,40939,40940,41014,41015,41103,41104,41132],{},"The following is mostly derived from ",[19,40938,31788],{"href":5756},", Theorem 3.1: let ",[35,40941,40943,40965],{"className":40942},[7057],[35,40944,40946],{"className":40945},[7061],[7063,40947,40948],{"xmlns":7065},[7067,40949,40950,40962],{},[7070,40951,40952,40954,40956,40958,40960],{},[7227,40953,5312],{},[7223,40955,10205],{},[7076,40957,19],{},[7223,40959,10205],{},[7076,40961,17617],{},[7083,40963,40964],{"encoding":7085},"0 \u003C a \u003C b",[35,40966,40968,40987,41005],{"className":40967,"ariaHidden":7091},[7090],[35,40969,40971,40975,40978,40981,40984],{"className":40970},[7095],[35,40972],{"className":40973,"style":40974},[7099],"height:0.6835em;vertical-align:-0.0391em;",[35,40976,5312],{"className":40977},[7104],[35,40979],{"className":40980,"style":7450},[7248],[35,40982,10205],{"className":40983},[7454],[35,40985],{"className":40986,"style":7450},[7248],[35,40988,40990,40993,40996,40999,41002],{"className":40989},[7095],[35,40991],{"className":40992,"style":10222},[7099],[35,40994,19],{"className":40995},[7104,7145],[35,40997],{"className":40998,"style":7450},[7248],[35,41000,10205],{"className":41001},[7454],[35,41003],{"className":41004,"style":7450},[7248],[35,41006,41008,41011],{"className":41007},[7095],[35,41009],{"className":41010,"style":9962},[7099],[35,41012,17617],{"className":41013},[7104,7145]," and let ",[35,41016,41018,41039],{"className":41017},[7057],[35,41019,41021],{"className":41020},[7061],[7063,41022,41023],{"xmlns":7065},[7067,41024,41025,41036],{},[7070,41026,41027],{},[8304,41028,41029,41032,41034],{},[7076,41030,41031],{"mathvariant":7282},"Π",[7076,41033,9487],{},[7223,41035,25282],{},[7083,41037,41038],{"encoding":7085},"\\Pi^*_d",[35,41040,41042],{"className":41041,"ariaHidden":7091},[7090],[35,41043,41045,41049],{"className":41044},[7095],[35,41046],{"className":41047,"style":41048},[7099],"height:0.9718em;vertical-align:-0.2831em;",[35,41050,41052,41055],{"className":41051},[7104],[35,41053,41031],{"className":41054},[7104],[35,41056,41058],{"className":41057},[7113],[35,41059,41061,41094],{"className":41060},[7117,7731],[35,41062,41064,41091],{"className":41063},[7121],[35,41065,41068,41080],{"className":41066,"style":41067},[7125],"height:0.6887em;",[35,41069,41071,41074],{"style":41070},"top:-2.4169em;margin-left:0em;margin-right:0.05em;",[35,41072],{"className":41073,"style":7134},[7133],[35,41075,41077],{"className":41076},[7138,7139,7140,7141],[35,41078,9487],{"className":41079},[7104,7145,7141],[35,41081,41082,41085],{"style":7129},[35,41083],{"className":41084,"style":7134},[7133],[35,41086,41088],{"className":41087},[7138,7139,7140,7141],[35,41089,25282],{"className":41090},[7253,7141],[35,41092,7756],{"className":41093},[7755],[35,41095,41097],{"className":41096},[7121],[35,41098,41101],{"className":41099,"style":41100},[7125],"height:0.2831em;",[35,41102],{}," be the space of residual polynomials (polynomials whose value at 0 is 1) of degree at most ",[35,41105,41107,41120],{"className":41106},[7057],[35,41108,41110],{"className":41109},[7061],[7063,41111,41112],{"xmlns":7065},[7067,41113,41114,41118],{},[7070,41115,41116],{},[7076,41117,9487],{},[7083,41119,9487],{"encoding":7085},[35,41121,41123],{"className":41122,"ariaHidden":7091},[7090],[35,41124,41126,41129],{"className":41125},[7095],[35,41127],{"className":41128,"style":9962},[7099],[35,41130,9487],{"className":41131},[7104,7145],". The solution to",[35,41134,41136],{"className":41135},[7318],[35,41137,41139,41232],{"className":41138},[7057],[35,41140,41142],{"className":41141},[7061],[7063,41143,41144],{"xmlns":7065,"display":7327},[7067,41145,41146,41229],{},[7070,41147,41148,41151,41153,41155,41157,41159,41161,41163,41165,41167,41191,41215,41217,41219,41221,41223,41225,41227],{},[7076,41149,41150],{},"β",[7223,41152,275],{"stretchy":7289},[7076,41154,19],{},[7223,41156,2877],{"separator":7091},[7076,41158,17617],{},[7223,41160,2877],{"separator":7091},[7076,41162,9487],{},[7223,41164,1694],{"stretchy":7289},[7223,41166,242],{},[35125,41168,41169,41176],{},[7070,41170,41171,41173],{},[7076,41172,570],{},[7223,41174,41175],{},"⁡",[7070,41177,41178,41180,41183],{},[7076,41179,12],{},[7223,41181,41182],{},"∈",[8304,41184,41185,41187,41189],{},[7076,41186,41031],{"mathvariant":7282},[7076,41188,9487],{},[7223,41190,25282],{},[35125,41192,41193,41199],{},[7070,41194,41195,41197],{},[7076,41196,554],{},[7223,41198,41175],{},[7070,41200,41201,41203,41205,41207,41209,41211,41213],{},[7076,41202,13657],{},[7223,41204,41182],{},[7223,41206,1254],{"stretchy":7289},[7076,41208,19],{},[7223,41210,2877],{"separator":7091},[7076,41212,17617],{},[7223,41214,16788],{"stretchy":7289},[7076,41216,7283],{"mathvariant":7282},[7076,41218,12],{},[7223,41220,275],{"stretchy":7289},[7076,41222,13657],{},[7223,41224,1694],{"stretchy":7289},[7076,41226,7283],{"mathvariant":7282},[7248,41228],{"width":13973},[7083,41230,41231],{"encoding":7085},"\\beta(a, b, d) = \\min_{p \\in \\Pi^*_d} \\max_{x \\in [a, b]} |p(x)| \\quad",[35,41233,41235,41281],{"className":41234,"ariaHidden":7091},[7090],[35,41236,41238,41241,41245,41248,41251,41254,41257,41260,41263,41266,41269,41272,41275,41278],{"className":41237},[7095],[35,41239],{"className":41240,"style":7303},[7099],[35,41242,41150],{"className":41243,"style":41244},[7104,7145],"margin-right:0.0528em;",[35,41246,275],{"className":41247},[7472],[35,41249,19],{"className":41250},[7104,7145],[35,41252,2877],{"className":41253},[7492],[35,41255],{"className":41256,"style":7496},[7248],[35,41258,17617],{"className":41259},[7104,7145],[35,41261,2877],{"className":41262},[7492],[35,41264],{"className":41265,"style":7496},[7248],[35,41267,9487],{"className":41268},[7104,7145],[35,41270,1694],{"className":41271},[7314],[35,41273],{"className":41274,"style":7450},[7248],[35,41276,242],{"className":41277},[7454],[35,41279],{"className":41280,"style":7450},[7248],[35,41282,41284,41288,41396,41399,41466,41469,41472,41475,41478,41481,41484,41487],{"className":41283},[7095],[35,41285],{"className":41286,"style":41287},[7099],"height:1.7374em;vertical-align:-0.9874em;",[35,41289,41291],{"className":41290},[12096,32771],[35,41292,41294,41387],{"className":41293},[7117,7731],[35,41295,41297,41384],{"className":41296},[7121],[35,41298,41301,41374],{"className":41299,"style":41300},[7125],"height:0.6679em;",[35,41302,41304,41307],{"style":41303},"top:-2.3557em;margin-left:0em;",[35,41305],{"className":41306,"style":8839},[7133],[35,41308,41310],{"className":41309},[7138,7139,7140,7141],[35,41311,41313,41316,41319],{"className":41312},[7104,7141],[35,41314,12],{"className":41315},[7104,7145,7141],[35,41317,41182],{"className":41318},[7454,7141],[35,41320,41322,41325],{"className":41321},[7104,7141],[35,41323,41031],{"className":41324},[7104,7141],[35,41326,41328],{"className":41327},[7113],[35,41329,41331,41365],{"className":41330},[7117,7731],[35,41332,41334,41362],{"className":41333},[7121],[35,41335,41338,41350],{"className":41336,"style":41337},[7125],"height:0.6771em;",[35,41339,41341,41344],{"style":41340},"top:-2.1528em;margin-left:0em;margin-right:0.0714em;",[35,41342],{"className":41343,"style":9340},[7133],[35,41345,41347],{"className":41346},[7138,9344,7479,7141],[35,41348,9487],{"className":41349},[7104,7145,7141],[35,41351,41353,41356],{"style":41352},"top:-2.8448em;margin-right:0.0714em;",[35,41354],{"className":41355,"style":9340},[7133],[35,41357,41359],{"className":41358},[7138,9344,7479,7141],[35,41360,25282],{"className":41361},[7253,7141],[35,41363,7756],{"className":41364},[7755],[35,41366,41368],{"className":41367},[7121],[35,41369,41372],{"className":41370,"style":41371},[7125],"height:0.3472em;",[35,41373],{},[35,41375,41376,41379],{"style":8835},[35,41377],{"className":41378,"style":8839},[7133],[35,41380,41381],{},[35,41382,570],{"className":41383},[12096],[35,41385,7756],{"className":41386},[7755],[35,41388,41390],{"className":41389},[7121],[35,41391,41394],{"className":41392,"style":41393},[7125],"height:0.9874em;",[35,41395],{},[35,41397],{"className":41398,"style":7496},[7248],[35,41400,41402],{"className":41401},[12096,32771],[35,41403,41405,41457],{"className":41404},[7117,7731],[35,41406,41408,41454],{"className":41407},[7121],[35,41409,41411,41444],{"className":41410,"style":7173},[7125],[35,41412,41414,41417],{"style":41413},"top:-2.309em;margin-left:0em;",[35,41415],{"className":41416,"style":8839},[7133],[35,41418,41420],{"className":41419},[7138,7139,7140,7141],[35,41421,41423,41426,41429,41432,41435,41438,41441],{"className":41422},[7104,7141],[35,41424,13657],{"className":41425},[7104,7145,7141],[35,41427,41182],{"className":41428},[7454,7141],[35,41430,1254],{"className":41431},[7472,7141],[35,41433,19],{"className":41434},[7104,7145,7141],[35,41436,2877],{"className":41437},[7492,7141],[35,41439,17617],{"className":41440},[7104,7145,7141],[35,41442,16788],{"className":41443},[7314,7141],[35,41445,41446,41449],{"style":8835},[35,41447],{"className":41448,"style":8839},[7133],[35,41450,41451],{},[35,41452,554],{"className":41453},[12096],[35,41455,7756],{"className":41456},[7755],[35,41458,41460],{"className":41459},[7121],[35,41461,41464],{"className":41462,"style":41463},[7125],"height:0.966em;",[35,41465],{},[35,41467],{"className":41468,"style":7496},[7248],[35,41470,7283],{"className":41471},[7104],[35,41473,12],{"className":41474},[7104,7145],[35,41476,275],{"className":41477},[7472],[35,41479,13657],{"className":41480},[7104,7145],[35,41482,1694],{"className":41483},[7314],[35,41485,7283],{"className":41486},[7104],[35,41488],{"className":41489,"style":14292},[7248],[12,41491,41492],{},"is",[35,41494,41496],{"className":41495},[7318],[35,41497,41499,41602],{"className":41498},[7057],[35,41500,41502],{"className":41501},[7061],[7063,41503,41504],{"xmlns":7065,"display":7327},[7067,41505,41506,41599],{},[7070,41507,41508,41514,41516,41518,41520,41522,41595,41597],{},[7073,41509,41510,41512],{},[7076,41511,12],{},[7223,41513,25282],{},[7223,41515,275],{"stretchy":7289},[7076,41517,13657],{},[7223,41519,1694],{"stretchy":7289},[7223,41521,242],{},[14009,41523,41524,41563],{},[7070,41525,41526,41533],{},[7699,41527,41528,41531],{},[7076,41529,41530],{},"T",[7076,41532,9487],{},[7070,41534,41535,41537,41561],{},[7223,41536,275],{"fence":7091},[14009,41538,41539,41553],{},[7070,41540,41541,41543,41545,41547,41549,41551],{},[7076,41542,17617],{},[7223,41544,445],{},[7076,41546,19],{},[7223,41548,7225],{},[7227,41550,460],{},[7076,41552,13657],{},[7070,41554,41555,41557,41559],{},[7076,41556,17617],{},[7223,41558,7225],{},[7076,41560,19],{},[7223,41562,1694],{"fence":7091},[7070,41564,41565,41571],{},[7699,41566,41567,41569],{},[7076,41568,41530],{},[7076,41570,9487],{},[7070,41572,41573,41575,41593],{},[7223,41574,275],{"fence":7091},[14009,41576,41577,41585],{},[7070,41578,41579,41581,41583],{},[7076,41580,17617],{},[7223,41582,445],{},[7076,41584,19],{},[7070,41586,41587,41589,41591],{},[7076,41588,17617],{},[7223,41590,7225],{},[7076,41592,19],{},[7223,41594,1694],{"fence":7091},[7223,41596,2877],{"separator":7091},[7248,41598],{"width":13973},[7083,41600,41601],{"encoding":7085},"p^*(x) = \\frac{T_d\\left(\\frac{b + a - 2x}{b - a}\\right)}{T_d\\left(\\frac{b + a}{b - a}\\right)}, \\quad",[35,41603,41605,41659],{"className":41604,"ariaHidden":7091},[7090],[35,41606,41608,41611,41641,41644,41647,41650,41653,41656],{"className":41607},[7095],[35,41609],{"className":41610,"style":7303},[7099],[35,41612,41614,41617],{"className":41613},[7104],[35,41615,12],{"className":41616},[7104,7145],[35,41618,41620],{"className":41619},[7113],[35,41621,41623],{"className":41622},[7117],[35,41624,41626],{"className":41625},[7121],[35,41627,41630],{"className":41628,"style":41629},[7125],"height:0.7387em;",[35,41631,41632,41635],{"style":7437},[35,41633],{"className":41634,"style":7134},[7133],[35,41636,41638],{"className":41637},[7138,7139,7140,7141],[35,41639,25282],{"className":41640},[7253,7141],[35,41642,275],{"className":41643},[7472],[35,41645,13657],{"className":41646},[7104,7145],[35,41648,1694],{"className":41649},[7314],[35,41651],{"className":41652,"style":7450},[7248],[35,41654,242],{"className":41655},[7454],[35,41657],{"className":41658,"style":7450},[7248],[35,41660,41662,41666,42015,42018],{"className":41661},[7095],[35,41663],{"className":41664,"style":41665},[7099],"height:2.8469em;vertical-align:-1.1734em;",[35,41667,41669,41672,42012],{"className":41668},[7104],[35,41670],{"className":41671},[7472,14311],[35,41673,41675],{"className":41674},[14009],[35,41676,41678,42003],{"className":41677},[7117,7731],[35,41679,41681,42000],{"className":41680},[7121],[35,41682,41685,41836,41844],{"className":41683,"style":41684},[7125],"height:1.6734em;",[35,41686,41688,41691],{"style":41687},"top:-2.2299em;",[35,41689],{"className":41690,"style":8839},[7133],[35,41692,41694,41734,41737],{"className":41693},[7104],[35,41695,41697,41700],{"className":41696},[7104],[35,41698,41530],{"className":41699,"style":7751},[7104,7145],[35,41701,41703],{"className":41702},[7113],[35,41704,41706,41726],{"className":41705},[7117,7731],[35,41707,41709,41723],{"className":41708},[7121],[35,41710,41712],{"className":41711,"style":15760},[7125],[35,41713,41714,41717],{"style":12163},[35,41715],{"className":41716,"style":7134},[7133],[35,41718,41720],{"className":41719},[7138,7139,7140,7141],[35,41721,9487],{"className":41722},[7104,7145,7141],[35,41724,7756],{"className":41725},[7755],[35,41727,41729],{"className":41728},[7121],[35,41730,41732],{"className":41731,"style":7763},[7125],[35,41733],{},[35,41735],{"className":41736,"style":7496},[7248],[35,41738,41740,41746,41830],{"className":41739},[7468],[35,41741,41743],{"className":41742,"style":7474},[7472,7473],[35,41744,275],{"className":41745},[7478,7479],[35,41747,41749,41752,41827],{"className":41748},[7104],[35,41750],{"className":41751},[7472,14311],[35,41753,41755],{"className":41754},[14009],[35,41756,41758,41818],{"className":41757},[7117,7731],[35,41759,41761,41815],{"className":41760},[7121],[35,41762,41765,41786,41794],{"className":41763,"style":41764},[7125],"height:0.8801em;",[35,41766,41768,41771],{"style":41767},"top:-2.655em;",[35,41769],{"className":41770,"style":8839},[7133],[35,41772,41774],{"className":41773},[7138,7139,7140,7141],[35,41775,41777,41780,41783],{"className":41776},[7104,7141],[35,41778,17617],{"className":41779},[7104,7145,7141],[35,41781,7225],{"className":41782},[7253,7141],[35,41784,19],{"className":41785},[7104,7145,7141],[35,41787,41788,41791],{"style":14395},[35,41789],{"className":41790,"style":8839},[7133],[35,41792],{"className":41793,"style":14403},[14402],[35,41795,41797,41800],{"style":41796},"top:-3.394em;",[35,41798],{"className":41799,"style":8839},[7133],[35,41801,41803],{"className":41802},[7138,7139,7140,7141],[35,41804,41806,41809,41812],{"className":41805},[7104,7141],[35,41807,17617],{"className":41808},[7104,7145,7141],[35,41810,445],{"className":41811},[7253,7141],[35,41813,19],{"className":41814},[7104,7145,7141],[35,41816,7756],{"className":41817},[7755],[35,41819,41821],{"className":41820},[7121],[35,41822,41825],{"className":41823,"style":41824},[7125],"height:0.4033em;",[35,41826],{},[35,41828],{"className":41829},[7314,14311],[35,41831,41833],{"className":41832,"style":7474},[7314,7473],[35,41834,1694],{"className":41835},[7478,7479],[35,41837,41838,41841],{"style":14395},[35,41839],{"className":41840,"style":8839},[7133],[35,41842],{"className":41843,"style":14403},[14402],[35,41845,41847,41850],{"style":41846},"top:-3.7933em;",[35,41848],{"className":41849,"style":8839},[7133],[35,41851,41853,41893,41896],{"className":41852},[7104],[35,41854,41856,41859],{"className":41855},[7104],[35,41857,41530],{"className":41858,"style":7751},[7104,7145],[35,41860,41862],{"className":41861},[7113],[35,41863,41865,41885],{"className":41864},[7117,7731],[35,41866,41868,41882],{"className":41867},[7121],[35,41869,41871],{"className":41870,"style":15760},[7125],[35,41872,41873,41876],{"style":12163},[35,41874],{"className":41875,"style":7134},[7133],[35,41877,41879],{"className":41878},[7138,7139,7140,7141],[35,41880,9487],{"className":41881},[7104,7145,7141],[35,41883,7756],{"className":41884},[7755],[35,41886,41888],{"className":41887},[7121],[35,41889,41891],{"className":41890,"style":7763},[7125],[35,41892],{},[35,41894],{"className":41895,"style":7496},[7248],[35,41897,41899,41905,41994],{"className":41898},[7468],[35,41900,41902],{"className":41901,"style":7474},[7472,7473],[35,41903,275],{"className":41904},[7478,7479],[35,41906,41908,41911,41991],{"className":41907},[7104],[35,41909],{"className":41910},[7472,14311],[35,41912,41914],{"className":41913},[14009],[35,41915,41917,41983],{"className":41916},[7117,7731],[35,41918,41920,41980],{"className":41919},[7121],[35,41921,41923,41943,41951],{"className":41922,"style":41764},[7125],[35,41924,41925,41928],{"style":41767},[35,41926],{"className":41927,"style":8839},[7133],[35,41929,41931],{"className":41930},[7138,7139,7140,7141],[35,41932,41934,41937,41940],{"className":41933},[7104,7141],[35,41935,17617],{"className":41936},[7104,7145,7141],[35,41938,7225],{"className":41939},[7253,7141],[35,41941,19],{"className":41942},[7104,7145,7141],[35,41944,41945,41948],{"style":14395},[35,41946],{"className":41947,"style":8839},[7133],[35,41949],{"className":41950,"style":14403},[14402],[35,41952,41953,41956],{"style":41796},[35,41954],{"className":41955,"style":8839},[7133],[35,41957,41959],{"className":41958},[7138,7139,7140,7141],[35,41960,41962,41965,41968,41971,41974,41977],{"className":41961},[7104,7141],[35,41963,17617],{"className":41964},[7104,7145,7141],[35,41966,445],{"className":41967},[7253,7141],[35,41969,19],{"className":41970},[7104,7145,7141],[35,41972,7225],{"className":41973},[7253,7141],[35,41975,460],{"className":41976},[7104,7141],[35,41978,13657],{"className":41979},[7104,7145,7141],[35,41981,7756],{"className":41982},[7755],[35,41984,41986],{"className":41985},[7121],[35,41987,41989],{"className":41988,"style":41824},[7125],[35,41990],{},[35,41992],{"className":41993},[7314,14311],[35,41995,41997],{"className":41996,"style":7474},[7314,7473],[35,41998,1694],{"className":41999},[7478,7479],[35,42001,7756],{"className":42002},[7755],[35,42004,42006],{"className":42005},[7121],[35,42007,42010],{"className":42008,"style":42009},[7125],"height:1.1734em;",[35,42011],{},[35,42013],{"className":42014},[7314,14311],[35,42016,2877],{"className":42017},[7492],[35,42019],{"className":42020,"style":14292},[7248],[12,42022,42023],{},"and the corresponding minimal value is",[35,42025,42027],{"className":42026},[7318],[35,42028,42030,42098],{"className":42029},[7057],[35,42031,42033],{"className":42032},[7061],[7063,42034,42035],{"xmlns":7065,"display":7327},[7067,42036,42037,42095],{},[7070,42038,42039,42041,42043,42045,42047,42049,42051,42053,42055,42057,42069,42093],{},[7076,42040,41150],{},[7223,42042,275],{"stretchy":7289},[7076,42044,19],{},[7223,42046,2877],{"separator":7091},[7076,42048,17617],{},[7223,42050,2877],{"separator":7091},[7076,42052,9487],{},[7223,42054,1694],{"stretchy":7289},[7223,42056,242],{},[8304,42058,42059,42061,42063],{},[7076,42060,41530],{},[7076,42062,9487],{},[7070,42064,42065,42067],{},[7223,42066,7225],{},[7227,42068,500],{},[7070,42070,42071,42073,42091],{},[7223,42072,275],{"fence":7091},[14009,42074,42075,42083],{},[7070,42076,42077,42079,42081],{},[7076,42078,17617],{},[7223,42080,445],{},[7076,42082,19],{},[7070,42084,42085,42087,42089],{},[7076,42086,17617],{},[7223,42088,7225],{},[7076,42090,19],{},[7223,42092,1694],{"fence":7091},[7076,42094,5122],{"mathvariant":7282},[7083,42096,42097],{"encoding":7085},"\\beta(a, b, d) = T_d^{-1}\\left(\\frac{b + a}{b - a}\\right).",[35,42099,42101,42146],{"className":42100,"ariaHidden":7091},[7090],[35,42102,42104,42107,42110,42113,42116,42119,42122,42125,42128,42131,42134,42137,42140,42143],{"className":42103},[7095],[35,42105],{"className":42106,"style":7303},[7099],[35,42108,41150],{"className":42109,"style":41244},[7104,7145],[35,42111,275],{"className":42112},[7472],[35,42114,19],{"className":42115},[7104,7145],[35,42117,2877],{"className":42118},[7492],[35,42120],{"className":42121,"style":7496},[7248],[35,42123,17617],{"className":42124},[7104,7145],[35,42126,2877],{"className":42127},[7492],[35,42129],{"className":42130,"style":7496},[7248],[35,42132,9487],{"className":42133},[7104,7145],[35,42135,1694],{"className":42136},[7314],[35,42138],{"className":42139,"style":7450},[7248],[35,42141,242],{"className":42142},[7454],[35,42144],{"className":42145,"style":7450},[7248],[35,42147,42149,42153,42212,42215,42318,42321],{"className":42148},[7095],[35,42150],{"className":42151,"style":42152},[7099],"height:2.4em;vertical-align:-0.95em;",[35,42154,42156,42159],{"className":42155},[7104],[35,42157,41530],{"className":42158,"style":7751},[7104,7145],[35,42160,42162],{"className":42161},[7113],[35,42163,42165,42203],{"className":42164},[7117,7731],[35,42166,42168,42200],{"className":42167},[7121],[35,42169,42171,42183],{"className":42170,"style":7533},[7125],[35,42172,42174,42177],{"style":42173},"top:-2.4086em;margin-left:-0.1389em;margin-right:0.05em;",[35,42175],{"className":42176,"style":7134},[7133],[35,42178,42180],{"className":42179},[7138,7139,7140,7141],[35,42181,9487],{"className":42182},[7104,7145,7141],[35,42184,42185,42188],{"style":7437},[35,42186],{"className":42187,"style":7134},[7133],[35,42189,42191],{"className":42190},[7138,7139,7140,7141],[35,42192,42194,42197],{"className":42193},[7104,7141],[35,42195,7225],{"className":42196},[7104,7141],[35,42198,500],{"className":42199},[7104,7141],[35,42201,7756],{"className":42202},[7755],[35,42204,42206],{"className":42205},[7121],[35,42207,42210],{"className":42208,"style":42209},[7125],"height:0.2914em;",[35,42211],{},[35,42213],{"className":42214,"style":7496},[7248],[35,42216,42218,42224,42312],{"className":42217},[7468],[35,42219,42221],{"className":42220,"style":7474},[7472,7473],[35,42222,275],{"className":42223},[7478,7140],[35,42225,42227,42230,42309],{"className":42226},[7104],[35,42228],{"className":42229},[7472,14311],[35,42231,42233],{"className":42232},[14009],[35,42234,42236,42300],{"className":42235},[7117,7731],[35,42237,42239,42297],{"className":42238},[7121],[35,42240,42243,42266,42274],{"className":42241,"style":42242},[7125],"height:1.3714em;",[35,42244,42245,42248],{"style":16234},[35,42246],{"className":42247,"style":8839},[7133],[35,42249,42251,42254,42257,42260,42263],{"className":42250},[7104],[35,42252,17617],{"className":42253},[7104,7145],[35,42255],{"className":42256,"style":7249},[7248],[35,42258,7225],{"className":42259},[7253],[35,42261],{"className":42262,"style":7249},[7248],[35,42264,19],{"className":42265},[7104,7145],[35,42267,42268,42271],{"style":14395},[35,42269],{"className":42270,"style":8839},[7133],[35,42272],{"className":42273,"style":14403},[14402],[35,42275,42276,42279],{"style":14406},[35,42277],{"className":42278,"style":8839},[7133],[35,42280,42282,42285,42288,42291,42294],{"className":42281},[7104],[35,42283,17617],{"className":42284},[7104,7145],[35,42286],{"className":42287,"style":7249},[7248],[35,42289,445],{"className":42290},[7253],[35,42292],{"className":42293,"style":7249},[7248],[35,42295,19],{"className":42296},[7104,7145],[35,42298,7756],{"className":42299},[7755],[35,42301,42303],{"className":42302},[7121],[35,42304,42307],{"className":42305,"style":42306},[7125],"height:0.7693em;",[35,42308],{},[35,42310],{"className":42311},[7314,14311],[35,42313,42315],{"className":42314,"style":7474},[7314,7473],[35,42316,1694],{"className":42317},[7478,7140],[35,42319],{"className":42320,"style":7496},[7248],[35,42322,5122],{"className":42323},[7104],[12,42325,42326,42327,42379],{},"We want to convert this into a function that can be expressed naturally in terms of complex exponentials, because those are the real time-evolutions that generate the quantum Krylov space.\nTo do so, it is convenient to introduce the following transformation of energies within the spectral range of the Hamiltonian to numbers in the range ",[35,42328,42330,42352],{"className":42329},[7057],[35,42331,42333],{"className":42332},[7061],[7063,42334,42335],{"xmlns":7065},[7067,42336,42337,42349],{},[7070,42338,42339,42341,42343,42345,42347],{},[7223,42340,1254],{"stretchy":7289},[7227,42342,5312],{},[7223,42344,2877],{"separator":7091},[7227,42346,500],{},[7223,42348,16788],{"stretchy":7289},[7083,42350,42351],{"encoding":7085},"[0,1]",[35,42353,42355],{"className":42354,"ariaHidden":7091},[7090],[35,42356,42358,42361,42364,42367,42370,42373,42376],{"className":42357},[7095],[35,42359],{"className":42360,"style":7303},[7099],[35,42362,1254],{"className":42363},[7472],[35,42365,5312],{"className":42366},[7104],[35,42368,2877],{"className":42369},[7492],[35,42371],{"className":42372,"style":7496},[7248],[35,42374,500],{"className":42375},[7104],[35,42377,16788],{"className":42378},[7314],": define",[35,42381,42383],{"className":42382},[7318],[35,42384,42386,42449],{"className":42385},[7057],[35,42387,42389],{"className":42388},[7061],[7063,42390,42391],{"xmlns":7065,"display":7327},[7067,42392,42393,42446],{},[7070,42394,42395,42398,42400,42402,42404,42406,42444],{},[7076,42396,42397],{},"g",[7223,42399,275],{"stretchy":7289},[7076,42401,10709],{},[7223,42403,1694],{"stretchy":7289},[7223,42405,242],{},[14009,42407,42408,42442],{},[7070,42409,42410,42412,42414,42417,42419,42422,42424,42426,42428,42434,42436,42438,42440],{},[7227,42411,500],{},[7223,42413,7225],{},[7076,42415,42416],{},"cos",[7223,42418,41175],{},[7223,42420,275],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},"1.2em",[7223,42423,275],{"stretchy":7289},[7076,42425,10709],{},[7223,42427,7225],{},[7699,42429,42430,42432],{},[7076,42431,10709],{},[7227,42433,5312],{},[7223,42435,1694],{"stretchy":7289},[7076,42437,9487],{},[7076,42439,7829],{},[7223,42441,1694],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7227,42443,460],{},[7223,42445,2877],{"separator":7091},[7083,42447,42448],{"encoding":7085},"g(E) = \\frac{1-\\cos\\big((E-E_0)dt\\big)}{2},",[35,42450,42452,42479],{"className":42451,"ariaHidden":7091},[7090],[35,42453,42455,42458,42461,42464,42467,42470,42473,42476],{"className":42454},[7095],[35,42456],{"className":42457,"style":7303},[7099],[35,42459,42397],{"className":42460,"style":7310},[7104,7145],[35,42462,275],{"className":42463},[7472],[35,42465,10709],{"className":42466,"style":10419},[7104,7145],[35,42468,1694],{"className":42469},[7314],[35,42471],{"className":42472,"style":7450},[7248],[35,42474,242],{"className":42475},[7454],[35,42477],{"className":42478,"style":7450},[7248],[35,42480,42482,42486,42641],{"className":42481},[7095],[35,42483],{"className":42484,"style":42485},[7099],"height:2.276em;vertical-align:-0.686em;",[35,42487,42489,42492,42638],{"className":42488},[7104],[35,42490],{"className":42491},[7472,14311],[35,42493,42495],{"className":42494},[14009],[35,42496,42498,42630],{"className":42497},[7117,7731],[35,42499,42501,42627],{"className":42500},[7121],[35,42502,42505,42516,42524],{"className":42503,"style":42504},[7125],"height:1.59em;",[35,42506,42507,42510],{"style":16234},[35,42508],{"className":42509,"style":8839},[7133],[35,42511,42513],{"className":42512},[7104],[35,42514,460],{"className":42515},[7104],[35,42517,42518,42521],{"style":14395},[35,42519],{"className":42520,"style":8839},[7133],[35,42522],{"className":42523,"style":14403},[14402],[35,42525,42527,42530],{"style":42526},"top:-3.74em;",[35,42528],{"className":42529,"style":8839},[7133],[35,42531,42533,42536,42539,42542,42545,42548,42551,42557,42560,42563,42566,42569,42572,42612,42615,42618,42621],{"className":42532},[7104],[35,42534,500],{"className":42535},[7104],[35,42537],{"className":42538,"style":7249},[7248],[35,42540,7225],{"className":42541},[7253],[35,42543],{"className":42544,"style":7249},[7248],[35,42546,42416],{"className":42547},[12096],[35,42549],{"className":42550,"style":7496},[7248],[35,42552,42554],{"className":42553},[7104],[35,42555,275],{"className":42556},[7478,7479],[35,42558,275],{"className":42559},[7472],[35,42561,10709],{"className":42562,"style":10419},[7104,7145],[35,42564],{"className":42565,"style":7249},[7248],[35,42567,7225],{"className":42568},[7253],[35,42570],{"className":42571,"style":7249},[7248],[35,42573,42575,42578],{"className":42574},[7104],[35,42576,10709],{"className":42577,"style":10419},[7104,7145],[35,42579,42581],{"className":42580},[7113],[35,42582,42584,42604],{"className":42583},[7117,7731],[35,42585,42587,42601],{"className":42586},[7121],[35,42588,42590],{"className":42589,"style":18794},[7125],[35,42591,42592,42595],{"style":10541},[35,42593],{"className":42594,"style":7134},[7133],[35,42596,42598],{"className":42597},[7138,7139,7140,7141],[35,42599,5312],{"className":42600},[7104,7141],[35,42602,7756],{"className":42603},[7755],[35,42605,42607],{"className":42606},[7121],[35,42608,42610],{"className":42609,"style":7763},[7125],[35,42611],{},[35,42613,1694],{"className":42614},[7314],[35,42616,9487],{"className":42617},[7104,7145],[35,42619,7829],{"className":42620},[7104,7145],[35,42622,42624],{"className":42623},[7104],[35,42625,1694],{"className":42626},[7478,7479],[35,42628,7756],{"className":42629},[7755],[35,42631,42633],{"className":42632},[7121],[35,42634,42636],{"className":42635,"style":16272},[7125],[35,42637],{},[35,42639],{"className":42640},[7314,14311],[35,42642,2877],{"className":42643},[7492],[12,42645,32939,42646,42679,42680,42886,42887,15316,42994,43038,43039,43067,43068,5122],{},[35,42647,42649,42664],{"className":42648},[7057],[35,42650,42652],{"className":42651},[7061],[7063,42653,42654],{"xmlns":7065},[7067,42655,42656,42662],{},[7070,42657,42658,42660],{},[7076,42659,9487],{},[7076,42661,7829],{},[7083,42663,9991],{"encoding":7085},[35,42665,42667],{"className":42666,"ariaHidden":7091},[7090],[35,42668,42670,42673,42676],{"className":42669},[7095],[35,42671],{"className":42672,"style":9962},[7099],[35,42674,9487],{"className":42675},[7104,7145],[35,42677,7829],{"className":42678},[7104,7145]," is a timestep such that ",[35,42681,42683,42727],{"className":42682},[7057],[35,42684,42686],{"className":42685},[7061],[7063,42687,42688],{"xmlns":7065},[7067,42689,42690,42724],{},[7070,42691,42692,42694,42696,42698,42704,42706,42708,42710,42716,42718,42720,42722],{},[7223,42693,7225],{},[7076,42695,13081],{},[7223,42697,10205],{},[7699,42699,42700,42702],{},[7076,42701,10709],{},[7227,42703,5312],{},[7076,42705,9487],{},[7076,42707,7829],{},[7223,42709,10205],{},[7699,42711,42712,42714],{},[7076,42713,10709],{},[10694,42715,554],{},[7076,42717,9487],{},[7076,42719,7829],{},[7223,42721,10205],{},[7076,42723,13081],{},[7083,42725,42726],{"encoding":7085},"-\\pi \u003C E_0dt \u003C E_\\text{max}dt \u003C \\pi",[35,42728,42730,42751,42813,42877],{"className":42729,"ariaHidden":7091},[7090],[35,42731,42733,42736,42739,42742,42745,42748],{"className":42732},[7095],[35,42734],{"className":42735,"style":7241},[7099],[35,42737,7225],{"className":42738},[7104],[35,42740,13081],{"className":42741,"style":7310},[7104,7145],[35,42743],{"className":42744,"style":7450},[7248],[35,42746,10205],{"className":42747},[7454],[35,42749],{"className":42750,"style":7450},[7248],[35,42752,42754,42758,42798,42801,42804,42807,42810],{"className":42753},[7095],[35,42755],{"className":42756,"style":42757},[7099],"height:0.8444em;vertical-align:-0.15em;",[35,42759,42761,42764],{"className":42760},[7104],[35,42762,10709],{"className":42763,"style":10419},[7104,7145],[35,42765,42767],{"className":42766},[7113],[35,42768,42770,42790],{"className":42769},[7117,7731],[35,42771,42773,42787],{"className":42772},[7121],[35,42774,42776],{"className":42775,"style":18794},[7125],[35,42777,42778,42781],{"style":10541},[35,42779],{"className":42780,"style":7134},[7133],[35,42782,42784],{"className":42783},[7138,7139,7140,7141],[35,42785,5312],{"className":42786},[7104,7141],[35,42788,7756],{"className":42789},[7755],[35,42791,42793],{"className":42792},[7121],[35,42794,42796],{"className":42795,"style":7763},[7125],[35,42797],{},[35,42799,9487],{"className":42800},[7104,7145],[35,42802,7829],{"className":42803},[7104,7145],[35,42805],{"className":42806,"style":7450},[7248],[35,42808,10205],{"className":42809},[7454],[35,42811],{"className":42812,"style":7450},[7248],[35,42814,42816,42819,42862,42865,42868,42871,42874],{"className":42815},[7095],[35,42817],{"className":42818,"style":42757},[7099],[35,42820,42822,42825],{"className":42821},[7104],[35,42823,10709],{"className":42824,"style":10419},[7104,7145],[35,42826,42828],{"className":42827},[7113],[35,42829,42831,42854],{"className":42830},[7117,7731],[35,42832,42834,42851],{"className":42833},[7121],[35,42835,42837],{"className":42836,"style":9035},[7125],[35,42838,42839,42842],{"style":10541},[35,42840],{"className":42841,"style":7134},[7133],[35,42843,42845],{"className":42844},[7138,7139,7140,7141],[35,42846,42848],{"className":42847},[7104,4991,7141],[35,42849,554],{"className":42850},[7104,7141],[35,42852,7756],{"className":42853},[7755],[35,42855,42857],{"className":42856},[7121],[35,42858,42860],{"className":42859,"style":7763},[7125],[35,42861],{},[35,42863,9487],{"className":42864},[7104,7145],[35,42866,7829],{"className":42867},[7104,7145],[35,42869],{"className":42870,"style":7450},[7248],[35,42872,10205],{"className":42873},[7454],[35,42875],{"className":42876,"style":7450},[7248],[35,42878,42880,42883],{"className":42879},[7095],[35,42881],{"className":42882,"style":7173},[7099],[35,42884,13081],{"className":42885,"style":7310},[7104,7145],".\nNotice that ",[35,42888,42890,42918],{"className":42889},[7057],[35,42891,42893],{"className":42892},[7061],[7063,42894,42895],{"xmlns":7065},[7067,42896,42897,42915],{},[7070,42898,42899,42901,42903,42909,42911,42913],{},[7076,42900,42397],{},[7223,42902,275],{"stretchy":7289},[7699,42904,42905,42907],{},[7076,42906,10709],{},[7227,42908,5312],{},[7223,42910,1694],{"stretchy":7289},[7223,42912,242],{},[7227,42914,5312],{},[7083,42916,42917],{"encoding":7085},"g(E_0)=0",[35,42919,42921,42985],{"className":42920,"ariaHidden":7091},[7090],[35,42922,42924,42927,42930,42933,42973,42976,42979,42982],{"className":42923},[7095],[35,42925],{"className":42926,"style":7303},[7099],[35,42928,42397],{"className":42929,"style":7310},[7104,7145],[35,42931,275],{"className":42932},[7472],[35,42934,42936,42939],{"className":42935},[7104],[35,42937,10709],{"className":42938,"style":10419},[7104,7145],[35,42940,42942],{"className":42941},[7113],[35,42943,42945,42965],{"className":42944},[7117,7731],[35,42946,42948,42962],{"className":42947},[7121],[35,42949,42951],{"className":42950,"style":18794},[7125],[35,42952,42953,42956],{"style":10541},[35,42954],{"className":42955,"style":7134},[7133],[35,42957,42959],{"className":42958},[7138,7139,7140,7141],[35,42960,5312],{"className":42961},[7104,7141],[35,42963,7756],{"className":42964},[7755],[35,42966,42968],{"className":42967},[7121],[35,42969,42971],{"className":42970,"style":7763},[7125],[35,42972],{},[35,42974,1694],{"className":42975},[7314],[35,42977],{"className":42978,"style":7450},[7248],[35,42980,242],{"className":42981},[7454],[35,42983],{"className":42984,"style":7450},[7248],[35,42986,42988,42991],{"className":42987},[7095],[35,42989],{"className":42990,"style":7263},[7099],[35,42992,5312],{"className":42993},[7104],[35,42995,42997,43017],{"className":42996},[7057],[35,42998,43000],{"className":42999},[7061],[7063,43001,43002],{"xmlns":7065},[7067,43003,43004,43014],{},[7070,43005,43006,43008,43010,43012],{},[7076,43007,42397],{},[7223,43009,275],{"stretchy":7289},[7076,43011,10709],{},[7223,43013,1694],{"stretchy":7289},[7083,43015,43016],{"encoding":7085},"g(E)",[35,43018,43020],{"className":43019,"ariaHidden":7091},[7090],[35,43021,43023,43026,43029,43032,43035],{"className":43022},[7095],[35,43024],{"className":43025,"style":7303},[7099],[35,43027,42397],{"className":43028,"style":7310},[7104,7145],[35,43030,275],{"className":43031},[7472],[35,43033,10709],{"className":43034,"style":10419},[7104,7145],[35,43036,1694],{"className":43037},[7314]," grows as ",[35,43040,43042,43055],{"className":43041},[7057],[35,43043,43045],{"className":43044},[7061],[7063,43046,43047],{"xmlns":7065},[7067,43048,43049,43053],{},[7070,43050,43051],{},[7076,43052,10709],{},[7083,43054,10709],{"encoding":7085},[35,43056,43058],{"className":43057,"ariaHidden":7091},[7090],[35,43059,43061,43064],{"className":43060},[7095],[35,43062],{"className":43063,"style":7100},[7099],[35,43065,10709],{"className":43066,"style":10419},[7104,7145]," moves away from ",[35,43069,43071,43088],{"className":43070},[7057],[35,43072,43074],{"className":43073},[7061],[7063,43075,43076],{"xmlns":7065},[7067,43077,43078,43086],{},[7070,43079,43080],{},[7699,43081,43082,43084],{},[7076,43083,10709],{},[7227,43085,5312],{},[7083,43087,18766],{"encoding":7085},[35,43089,43091],{"className":43090,"ariaHidden":7091},[7090],[35,43092,43094,43097],{"className":43093},[7095],[35,43095],{"className":43096,"style":7718},[7099],[35,43098,43100,43103],{"className":43099},[7104],[35,43101,10709],{"className":43102,"style":10419},[7104,7145],[35,43104,43106],{"className":43105},[7113],[35,43107,43109,43129],{"className":43108},[7117,7731],[35,43110,43112,43126],{"className":43111},[7121],[35,43113,43115],{"className":43114,"style":18794},[7125],[35,43116,43117,43120],{"style":10541},[35,43118],{"className":43119,"style":7134},[7133],[35,43121,43123],{"className":43122},[7138,7139,7140,7141],[35,43124,5312],{"className":43125},[7104,7141],[35,43127,7756],{"className":43128},[7755],[35,43130,43132],{"className":43131},[7121],[35,43133,43135],{"className":43134,"style":7763},[7125],[35,43136],{},[12,43138,43139,43140,43214,43215,288,43344,43395],{},"Now using the polynomial ",[35,43141,43143,43167],{"className":43142},[7057],[35,43144,43146],{"className":43145},[7061],[7063,43147,43148],{"xmlns":7065},[7067,43149,43150,43164],{},[7070,43151,43152,43158,43160,43162],{},[7073,43153,43154,43156],{},[7076,43155,12],{},[7223,43157,25282],{},[7223,43159,275],{"stretchy":7289},[7076,43161,13657],{},[7223,43163,1694],{"stretchy":7289},[7083,43165,43166],{"encoding":7085},"p^*(x)",[35,43168,43170],{"className":43169,"ariaHidden":7091},[7090],[35,43171,43173,43176,43205,43208,43211],{"className":43172},[7095],[35,43174],{"className":43175,"style":7303},[7099],[35,43177,43179,43182],{"className":43178},[7104],[35,43180,12],{"className":43181},[7104,7145],[35,43183,43185],{"className":43184},[7113],[35,43186,43188],{"className":43187},[7117],[35,43189,43191],{"className":43190},[7121],[35,43192,43194],{"className":43193,"style":41067},[7125],[35,43195,43196,43199],{"style":7129},[35,43197],{"className":43198,"style":7134},[7133],[35,43200,43202],{"className":43201},[7138,7139,7140,7141],[35,43203,25282],{"className":43204},[7253,7141],[35,43206,275],{"className":43207},[7472],[35,43209,13657],{"className":43210},[7104,7145],[35,43212,1694],{"className":43213},[7314]," with the parameters a, b, d set to ",[35,43216,43218,43250],{"className":43217},[7057],[35,43219,43221],{"className":43220},[7061],[7063,43222,43223],{"xmlns":7065},[7067,43224,43225,43247],{},[7070,43226,43227,43229,43231,43233,43235,43241,43243,43245],{},[7076,43228,19],{},[7223,43230,242],{},[7076,43232,42397],{},[7223,43234,275],{"stretchy":7289},[7699,43236,43237,43239],{},[7076,43238,10709],{},[7227,43240,5312],{},[7223,43242,445],{},[7076,43244,32009],{},[7223,43246,1694],{"stretchy":7289},[7083,43248,43249],{"encoding":7085},"a = g(E_0 + \\delta)",[35,43251,43253,43271,43332],{"className":43252,"ariaHidden":7091},[7090],[35,43254,43256,43259,43262,43265,43268],{"className":43255},[7095],[35,43257],{"className":43258,"style":7173},[7099],[35,43260,19],{"className":43261},[7104,7145],[35,43263],{"className":43264,"style":7450},[7248],[35,43266,242],{"className":43267},[7454],[35,43269],{"className":43270,"style":7450},[7248],[35,43272,43274,43277,43280,43283,43323,43326,43329],{"className":43273},[7095],[35,43275],{"className":43276,"style":7303},[7099],[35,43278,42397],{"className":43279,"style":7310},[7104,7145],[35,43281,275],{"className":43282},[7472],[35,43284,43286,43289],{"className":43285},[7104],[35,43287,10709],{"className":43288,"style":10419},[7104,7145],[35,43290,43292],{"className":43291},[7113],[35,43293,43295,43315],{"className":43294},[7117,7731],[35,43296,43298,43312],{"className":43297},[7121],[35,43299,43301],{"className":43300,"style":18794},[7125],[35,43302,43303,43306],{"style":10541},[35,43304],{"className":43305,"style":7134},[7133],[35,43307,43309],{"className":43308},[7138,7139,7140,7141],[35,43310,5312],{"className":43311},[7104,7141],[35,43313,7756],{"className":43314},[7755],[35,43316,43318],{"className":43317},[7121],[35,43319,43321],{"className":43320,"style":7763},[7125],[35,43322],{},[35,43324],{"className":43325,"style":7249},[7248],[35,43327,445],{"className":43328},[7253],[35,43330],{"className":43331,"style":7249},[7248],[35,43333,43335,43338,43341],{"className":43334},[7095],[35,43336],{"className":43337,"style":7303},[7099],[35,43339,32009],{"className":43340,"style":32094},[7104,7145],[35,43342,1694],{"className":43343},[7314],[35,43345,43347,43365],{"className":43346},[7057],[35,43348,43350],{"className":43349},[7061],[7063,43351,43352],{"xmlns":7065},[7067,43353,43354,43362],{},[7070,43355,43356,43358,43360],{},[7076,43357,17617],{},[7223,43359,242],{},[7227,43361,500],{},[7083,43363,43364],{"encoding":7085},"b = 1",[35,43366,43368,43386],{"className":43367,"ariaHidden":7091},[7090],[35,43369,43371,43374,43377,43380,43383],{"className":43370},[7095],[35,43372],{"className":43373,"style":9962},[7099],[35,43375,17617],{"className":43376},[7104,7145],[35,43378],{"className":43379,"style":7450},[7248],[35,43381,242],{"className":43382},[7454],[35,43384],{"className":43385,"style":7450},[7248],[35,43387,43389,43392],{"className":43388},[7095],[35,43390],{"className":43391,"style":7263},[7099],[35,43393,500],{"className":43394},[7104],", and d = int(r\u002F2), we define the function:",[35,43397,43399],{"className":43398},[7318],[35,43400,43402,43604],{"className":43401},[7057],[35,43403,43405],{"className":43404},[7061],[7063,43406,43407],{"xmlns":7065,"display":7327},[7067,43408,43409,43601],{},[7070,43410,43411,43413,43415,43417,43419,43421,43427,43441,43443],{},[7076,43412,1675],{},[7223,43414,275],{"stretchy":7289},[7076,43416,10709],{},[7223,43418,1694],{"stretchy":7289},[7223,43420,242],{},[7073,43422,43423,43425],{},[7076,43424,12],{},[7223,43426,25282],{},[7070,43428,43429,43431,43433,43435,43437,43439],{},[7223,43430,275],{"fence":7091},[7076,43432,42397],{},[7223,43434,275],{"stretchy":7289},[7076,43436,10709],{},[7223,43438,1694],{"stretchy":7289},[7223,43440,1694],{"fence":7091},[7223,43442,242],{},[14009,43444,43445,43535],{},[7070,43446,43447,43453],{},[7699,43448,43449,43451],{},[7076,43450,41530],{},[7076,43452,9487],{},[7070,43454,43455,43457,43459,43461,43463,43533],{},[7223,43456,275],{"fence":7091},[7227,43458,500],{},[7223,43460,445],{},[7227,43462,460],{},[14009,43464,43465,43511],{},[7070,43466,43467,43469,43471,43473,43475,43477,43479,43485,43487,43489,43491,43493,43495,43497,43499,43501,43503,43505,43507,43509],{},[7076,43468,42416],{},[7223,43470,41175],{},[7223,43472,275],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7223,43474,275],{"stretchy":7289},[7076,43476,10709],{},[7223,43478,7225],{},[7699,43480,43481,43483],{},[7076,43482,10709],{},[7227,43484,5312],{},[7223,43486,1694],{"stretchy":7289},[7076,43488,9487],{},[7076,43490,7829],{},[7223,43492,1694],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7223,43494,7225],{},[7076,43496,42416],{},[7223,43498,41175],{},[7223,43500,275],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7076,43502,32009],{},[10694,43504,30173],{},[7076,43506,9487],{},[7076,43508,7829],{},[7223,43510,1694],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7070,43512,43513,43515,43517,43519,43521,43523,43525,43527,43529,43531],{},[7227,43514,500],{},[7223,43516,445],{},[7076,43518,42416],{},[7223,43520,41175],{},[7223,43522,275],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7076,43524,32009],{},[10694,43526,30173],{},[7076,43528,9487],{},[7076,43530,7829],{},[7223,43532,1694],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7223,43534,1694],{"fence":7091},[7070,43536,43537,43543],{},[7699,43538,43539,43541],{},[7076,43540,41530],{},[7076,43542,9487],{},[7070,43544,43545,43547,43549,43551,43553,43599],{},[7223,43546,275],{"fence":7091},[7227,43548,500],{},[7223,43550,445],{},[7227,43552,460],{},[14009,43554,43555,43577],{},[7070,43556,43557,43559,43561,43563,43565,43567,43569,43571,43573,43575],{},[7227,43558,500],{},[7223,43560,7225],{},[7076,43562,42416],{},[7223,43564,41175],{},[7223,43566,275],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7076,43568,32009],{},[10694,43570,30173],{},[7076,43572,9487],{},[7076,43574,7829],{},[7223,43576,1694],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7070,43578,43579,43581,43583,43585,43587,43589,43591,43593,43595,43597],{},[7227,43580,500],{},[7223,43582,445],{},[7076,43584,42416],{},[7223,43586,41175],{},[7223,43588,275],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7076,43590,32009],{},[10694,43592,30173],{},[7076,43594,9487],{},[7076,43596,7829],{},[7223,43598,1694],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7223,43600,1694],{"fence":7091},[7083,43602,43603],{"encoding":7085},"f(E) = p^* \\left( g(E) \\right) = \\frac{T_d\\left(1 + 2\\frac{\\cos\\big((E-E_0)dt\\big) - \\cos\\big(\\delta\\,dt\\big)}{1 +\\cos\\big(\\delta\\,dt\\big)}\\right)}{T_d\\left(1 + 2\\frac{1-\\cos\\big(\\delta\\,dt\\big)}{1 + 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We can see by inserting 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that ",[35,44651,44653,44672],{"className":44652},[7057],[35,44654,44656],{"className":44655},[7061],[7063,44657,44658],{"xmlns":7065},[7067,44659,44660,44670],{},[7070,44661,44662,44664,44666,44668],{},[7076,44663,1675],{},[7223,44665,275],{"stretchy":7289},[7076,44667,10709],{},[7223,44669,1694],{"stretchy":7289},[7083,44671,32298],{"encoding":7085},[35,44673,44675],{"className":44674,"ariaHidden":7091},[7090],[35,44676,44678,44681,44684,44687,44690],{"className":44677},[7095],[35,44679],{"className":44680,"style":7303},[7099],[35,44682,1675],{"className":44683,"style":31824},[7104,7145],[35,44685,275],{"className":44686},[7472],[35,44688,10709],{"className":44689,"style":10419},[7104,7145],[35,44691,1694],{"className":44692},[7314]," is a trigonometric polynomial of degree ",[35,44695,44697,44710],{"className":44696},[7057],[35,44698,44700],{"className":44699},[7061],[7063,44701,44702],{"xmlns":7065},[7067,44703,44704,44708],{},[7070,44705,44706],{},[7076,44707,9487],{},[7083,44709,9487],{"encoding":7085},[35,44711,44713],{"className":44712,"ariaHidden":7091},[7090],[35,44714,44716,44719],{"className":44715},[7095],[35,44717],{"className":44718,"style":9962},[7099],[35,44720,9487],{"className":44721},[7104,7145],", that is, a linear combination of ",[35,44724,44726,44755],{"className":44725},[7057],[35,44727,44729],{"className":44728},[7061],[7063,44730,44731],{"xmlns":7065},[7067,44732,44733,44753],{},[7070,44734,44735],{},[7073,44736,44737,44739],{},[7076,44738,7817],{},[7070,44740,44741,44743,44745,44747,44749,44751],{},[7076,44742,7824],{},[7076,44744,6920],{},[7076,44746,10709],{},[10694,44748,30173],{},[7076,44750,9487],{},[7076,44752,7829],{},[7083,44754,32353],{"encoding":7085},[35,44756,44758],{"className":44757,"ariaHidden":7091},[7090],[35,44759,44761,44764],{"className":44760},[7095],[35,44762],{"className":44763,"style":8191},[7099],[35,44765,44767,44770],{"className":44766},[7104],[35,44768,7817],{"className":44769},[7104,7145],[35,44771,44773],{"className":44772},[7113],[35,44774,44776],{"className":44775},[7117],[35,44777,44779],{"className":44778},[7121],[35,44780,44782],{"className":44781,"style":8191},[7125],[35,44783,44784,44787],{"style":7129},[35,44785],{"className":44786,"style":7134},[7133],[35,44788,44790],{"className":44789},[7138,7139,7140,7141],[35,44791,44793,44796,44799,44802,44805],{"className":44792},[7104,7141],[35,44794,32395],{"className":44795,"style":11350},[7104,7145,7141],[35,44797,10709],{"className":44798,"style":10419},[7104,7145,7141],[35,44800],{"className":44801,"style":30383},[7248,7141],[35,44803,9487],{"className":44804},[7104,7145,7141],[35,44806,7829],{"className":44807},[7104,7145,7141],[35,44809,44811,44860],{"className":44810},[7057],[35,44812,44814],{"className":44813},[7061],[7063,44815,44816],{"xmlns":7065},[7067,44817,44818,44858],{},[7070,44819,44820,44822,44824,44826,44828,44830,44832,44834,44836,44838,44840,44842,44844,44846,44848,44850,44852,44854,44856],{},[7076,44821,6920],{},[7223,44823,242],{},[7223,44825,7225],{},[7076,44827,9487],{},[7223,44829,2877],{"separator":7091},[7223,44831,7225],{},[7076,44833,9487],{},[7223,44835,445],{},[7227,44837,500],{},[7223,44839,2877],{"separator":7091},[7076,44841,5122],{"mathvariant":7282},[7076,44843,5122],{"mathvariant":7282},[7076,44845,5122],{"mathvariant":7282},[7223,44847,2877],{"separator":7091},[7076,44849,9487],{},[7223,44851,7225],{},[7227,44853,500],{},[7223,44855,2877],{"separator":7091},[7076,44857,9487],{},[7083,44859,32461],{"encoding":7085},[35,44861,44863,44881,44914,44950],{"className":44862,"ariaHidden":7091},[7090],[35,44864,44866,44869,44872,44875,44878],{"className":44865},[7095],[35,44867],{"className":44868,"style":32471},[7099],[35,44870,6920],{"className":44871,"style":11350},[7104,7145],[35,44873],{"className":44874,"style":7450},[7248],[35,44876,242],{"className":44877},[7454],[35,44879],{"className":44880,"style":7450},[7248],[35,44882,44884,44887,44890,44893,44896,44899,44902,44905,44908,44911],{"className":44883},[7095],[35,44885],{"className":44886,"style":11562},[7099],[35,44888,7225],{"className":44889},[7104],[35,44891,9487],{"className":44892},[7104,7145],[35,44894,2877],{"className":44895},[7492],[35,44897],{"className":44898,"style":7496},[7248],[35,44900,7225],{"className":44901},[7104],[35,44903,9487],{"className":44904},[7104,7145],[35,44906],{"className":44907,"style":7249},[7248],[35,44909,445],{"className":44910},[7253],[35,44912],{"className":44913,"style":7249},[7248],[35,44915,44917,44920,44923,44926,44929,44932,44935,44938,44941,44944,44947],{"className":44916},[7095],[35,44918],{"className":44919,"style":11562},[7099],[35,44921,500],{"className":44922},[7104],[35,44924,2877],{"className":44925},[7492],[35,44927],{"className":44928,"style":7496},[7248],[35,44930,7563],{"className":44931},[7104],[35,44933,2877],{"className":44934},[7492],[35,44936],{"className":44937,"style":7496},[7248],[35,44939,9487],{"className":44940},[7104,7145],[35,44942],{"className":44943,"style":7249},[7248],[35,44945,7225],{"className":44946},[7253],[35,44948],{"className":44949,"style":7249},[7248],[35,44951,44953,44956,44959,44962,44965],{"className":44952},[7095],[35,44954],{"className":44955,"style":11562},[7099],[35,44957,500],{"className":44958},[7104],[35,44960,2877],{"className":44961},[7492],[35,44963],{"className":44964,"style":7496},[7248],[35,44966,9487],{"className":44967},[7104,7145],". Furthermore, from the definition of ",[35,44970,44972,44995],{"className":44971},[7057],[35,44973,44975],{"className":44974},[7061],[7063,44976,44977],{"xmlns":7065},[7067,44978,44979,44993],{},[7070,44980,44981,44987,44989,44991],{},[7073,44982,44983,44985],{},[7076,44984,12],{},[7223,44986,25282],{},[7223,44988,275],{"stretchy":7289},[7076,44990,13657],{},[7223,44992,1694],{"stretchy":7289},[7083,44994,43166],{"encoding":7085},[35,44996,44998],{"className":44997,"ariaHidden":7091},[7090],[35,44999,45001,45004,45033,45036,45039],{"className":45000},[7095],[35,45002],{"className":45003,"style":7303},[7099],[35,45005,45007,45010],{"className":45006},[7104],[35,45008,12],{"className":45009},[7104,7145],[35,45011,45013],{"className":45012},[7113],[35,45014,45016],{"className":45015},[7117],[35,45017,45019],{"className":45018},[7121],[35,45020,45022],{"className":45021,"style":41067},[7125],[35,45023,45024,45027],{"style":7129},[35,45025],{"className":45026,"style":7134},[7133],[35,45028,45030],{"className":45029},[7138,7139,7140,7141],[35,45031,25282],{"className":45032},[7253,7141],[35,45034,275],{"className":45035},[7472],[35,45037,13657],{"className":45038},[7104,7145],[35,45040,1694],{"className":45041},[7314]," above we have that ",[35,45044,45046,45084],{"className":45045},[7057],[35,45047,45049],{"className":45048},[7061],[7063,45050,45051],{"xmlns":7065},[7067,45052,45053,45081],{},[7070,45054,45055,45057,45059,45065,45067,45069,45071,45073,45075,45077,45079],{},[7076,45056,1675],{},[7223,45058,275],{"stretchy":7289},[7699,45060,45061,45063],{},[7076,45062,10709],{},[7227,45064,5312],{},[7223,45066,1694],{"stretchy":7289},[7223,45068,242],{},[7076,45070,12],{},[7223,45072,275],{"stretchy":7289},[7227,45074,5312],{},[7223,45076,1694],{"stretchy":7289},[7223,45078,242],{},[7227,45080,500],{},[7083,45082,45083],{"encoding":7085},"f(E_0)=p(0)=1",[35,45085,45087,45151,45178],{"className":45086,"ariaHidden":7091},[7090],[35,45088,45090,45093,45096,45099,45139,45142,45145,45148],{"className":45089},[7095],[35,45091],{"className":45092,"style":7303},[7099],[35,45094,1675],{"className":45095,"style":31824},[7104,7145],[35,45097,275],{"className":45098},[7472],[35,45100,45102,45105],{"className":45101},[7104],[35,45103,10709],{"className":45104,"style":10419},[7104,7145],[35,45106,45108],{"className":45107},[7113],[35,45109,45111,45131],{"className":45110},[7117,7731],[35,45112,45114,45128],{"className":45113},[7121],[35,45115,45117],{"className":45116,"style":18794},[7125],[35,45118,45119,45122],{"style":10541},[35,45120],{"className":45121,"style":7134},[7133],[35,45123,45125],{"className":45124},[7138,7139,7140,7141],[35,45126,5312],{"className":45127},[7104,7141],[35,45129,7756],{"className":45130},[7755],[35,45132,45134],{"className":45133},[7121],[35,45135,45137],{"className":45136,"style":7763},[7125],[35,45138],{},[35,45140,1694],{"className":45141},[7314],[35,45143],{"className":45144,"style":7450},[7248],[35,45146,242],{"className":45147},[7454],[35,45149],{"className":45150,"style":7450},[7248],[35,45152,45154,45157,45160,45163,45166,45169,45172,45175],{"className":45153},[7095],[35,45155],{"className":45156,"style":7303},[7099],[35,45158,12],{"className":45159},[7104,7145],[35,45161,275],{"className":45162},[7472],[35,45164,5312],{"className":45165},[7104],[35,45167,1694],{"className":45168},[7314],[35,45170],{"className":45171,"style":7450},[7248],[35,45173,242],{"className":45174},[7454],[35,45176],{"className":45177,"style":7450},[7248],[35,45179,45181,45184],{"className":45180},[7095],[35,45182],{"className":45183,"style":7263},[7099],[35,45185,500],{"className":45186},[7104]," and for any ",[35,45189,45191,45204],{"className":45190},[7057],[35,45192,45194],{"className":45193},[7061],[7063,45195,45196],{"xmlns":7065},[7067,45197,45198,45202],{},[7070,45199,45200],{},[7076,45201,10709],{},[7083,45203,10709],{"encoding":7085},[35,45205,45207],{"className":45206,"ariaHidden":7091},[7090],[35,45208,45210,45213],{"className":45209},[7095],[35,45211],{"className":45212,"style":7100},[7099],[35,45214,10709],{"className":45215,"style":10419},[7104,7145]," in the spectral range such that ",[35,45218,45220,45250],{"className":45219},[7057],[35,45221,45223],{"className":45222},[7061],[7063,45224,45225],{"xmlns":7065},[7067,45226,45227,45247],{},[7070,45228,45229,45231,45233,45235,45241,45243,45245],{},[7076,45230,7283],{"mathvariant":7282},[7076,45232,10709],{},[7223,45234,7225],{},[7699,45236,45237,45239],{},[7076,45238,10709],{},[7227,45240,5312],{},[7076,45242,7283],{"mathvariant":7282},[7223,45244,1497],{},[7076,45246,32009],{},[7083,45248,45249],{"encoding":7085},"\\vert E-E_0 \\vert > \\delta",[35,45251,45253,45274,45332],{"className":45252,"ariaHidden":7091},[7090],[35,45254,45256,45259,45262,45265,45268,45271],{"className":45255},[7095],[35,45257],{"className":45258,"style":7303},[7099],[35,45260,7283],{"className":45261},[7104],[35,45263,10709],{"className":45264,"style":10419},[7104,7145],[35,45266],{"className":45267,"style":7249},[7248],[35,45269,7225],{"className":45270},[7253],[35,45272],{"className":45273,"style":7249},[7248],[35,45275,45277,45280,45320,45323,45326,45329],{"className":45276},[7095],[35,45278],{"className":45279,"style":7303},[7099],[35,45281,45283,45286],{"className":45282},[7104],[35,45284,10709],{"className":45285,"style":10419},[7104,7145],[35,45287,45289],{"className":45288},[7113],[35,45290,45292,45312],{"className":45291},[7117,7731],[35,45293,45295,45309],{"className":45294},[7121],[35,45296,45298],{"className":45297,"style":18794},[7125],[35,45299,45300,45303],{"style":10541},[35,45301],{"className":45302,"style":7134},[7133],[35,45304,45306],{"className":45305},[7138,7139,7140,7141],[35,45307,5312],{"className":45308},[7104,7141],[35,45310,7756],{"className":45311},[7755],[35,45313,45315],{"className":45314},[7121],[35,45316,45318],{"className":45317,"style":7763},[7125],[35,45319],{},[35,45321,7283],{"className":45322},[7104],[35,45324],{"className":45325,"style":7450},[7248],[35,45327,1497],{"className":45328},[7454],[35,45330],{"className":45331,"style":7450},[7248],[35,45333,45335,45338],{"className":45334},[7095],[35,45336],{"className":45337,"style":9962},[7099],[35,45339,32009],{"className":45340,"style":32094},[7104,7145]," we have",[35,45343,45345],{"className":45344},[7318],[35,45346,45348,45462],{"className":45347},[7057],[35,45349,45351],{"className":45350},[7061],[7063,45352,45353],{"xmlns":7065,"display":7327},[7067,45354,45355,45459],{},[7070,45356,45357,45359,45361,45363,45365,45367,45369,45371,45373,45375,45377,45379,45381,45383,45385,45387,45389,45401],{},[7076,45358,7283],{"mathvariant":7282},[7076,45360,1675],{},[7223,45362,275],{"stretchy":7289},[7076,45364,10709],{},[7223,45366,1694],{"stretchy":7289},[7076,45368,7283],{"mathvariant":7282},[7223,45370,31994],{},[7076,45372,41150],{},[7223,45374,275],{"stretchy":7289},[7076,45376,19],{},[7223,45378,2877],{"separator":7091},[7076,45380,17617],{},[7223,45382,2877],{"separator":7091},[7076,45384,9487],{},[7223,45386,1694],{"stretchy":7289},[7223,45388,242],{},[8304,45390,45391,45393,45395],{},[7076,45392,41530],{},[7076,45394,9487],{},[7070,45396,45397,45399],{},[7223,45398,7225],{},[7227,45400,500],{},[7070,45402,45403,45405,45407,45409,45411,45457],{},[7223,45404,275],{"fence":7091},[7227,45406,500],{},[7223,45408,445],{},[7227,45410,460],{},[14009,45412,45413,45435],{},[7070,45414,45415,45417,45419,45421,45423,45425,45427,45429,45431,45433],{},[7227,45416,500],{},[7223,45418,7225],{},[7076,45420,42416],{},[7223,45422,41175],{},[7223,45424,275],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7076,45426,32009],{},[10694,45428,30173],{},[7076,45430,9487],{},[7076,45432,7829],{},[7223,45434,1694],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7070,45436,45437,45439,45441,45443,45445,45447,45449,45451,45453,45455],{},[7227,45438,500],{},[7223,45440,445],{},[7076,45442,42416],{},[7223,45444,41175],{},[7223,45446,275],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7076,45448,32009],{},[10694,45450,30173],{},[7076,45452,9487],{},[7076,45454,7829],{},[7223,45456,1694],{"fence":7289,"stretchy":7091,"minsize":42421,"maxsize":42421},[7223,45458,1694],{"fence":7091},[7083,45460,45461],{"encoding":7085},"|f(E)| \\le \\beta(a, b, d) = T_d^{-1}\\left(1 + 2\\frac{1-\\cos\\big(\\delta\\,dt\\big)}{1 + \\cos\\big(\\delta\\,dt\\big)}\\right)",[35,45463,45465,45498,45543],{"className":45464,"ariaHidden":7091},[7090],[35,45466,45468,45471,45474,45477,45480,45483,45486,45489,45492,45495],{"className":45467},[7095],[35,45469],{"className":45470,"style":7303},[7099],[35,45472,7283],{"className":45473},[7104],[35,45475,1675],{"className":45476,"style":31824},[7104,7145],[35,45478,275],{"className":45479},[7472],[35,45481,10709],{"className":45482,"style":10419},[7104,7145],[35,45484,1694],{"className":45485},[7314],[35,45487,7283],{"className":45488},[7104],[35,45490],{"className":45491,"style":7450},[7248],[35,45493,31994],{"className":45494},[7454],[35,45496],{"className":45497,"style":7450},[7248],[35,45499,45501,45504,45507,45510,45513,45516,45519,45522,45525,45528,45531,45534,45537,45540],{"className":45500},[7095],[35,45502],{"className":45503,"style":7303},[7099],[35,45505,41150],{"className":45506,"style":41244},[7104,7145],[35,45508,275],{"className":45509},[7472],[35,45511,19],{"className":45512},[7104,7145],[35,45514,2877],{"className":45515},[7492],[35,45517],{"className":45518,"style":7496},[7248],[35,45520,17617],{"className":45521},[7104,7145],[35,45523,2877],{"className":45524},[7492],[35,45526],{"className":45527,"style":7496},[7248],[35,45529,9487],{"className":45530},[7104,7145],[35,45532,1694],{"className":45533},[7314],[35,45535],{"className":45536,"style":7450},[7248],[35,45538,242],{"className":45539},[7454],[35,45541],{"className":45542,"style":7450},[7248],[35,45544,45546,45550,45607,45610],{"className":45545},[7095],[35,45547],{"className":45548,"style":45549},[7099],"height:3em;vertical-align:-1.25em;",[35,45551,45553,45556],{"className":45552},[7104],[35,45554,41530],{"className":45555,"style":7751},[7104,7145],[35,45557,45559],{"className":45558},[7113],[35,45560,45562,45599],{"className":45561},[7117,7731],[35,45563,45565,45596],{"className":45564},[7121],[35,45566,45568,45579],{"className":45567,"style":7533},[7125],[35,45569,45570,45573],{"style":42173},[35,45571],{"className":45572,"style":7134},[7133],[35,45574,45576],{"className":45575},[7138,7139,7140,7141],[35,45577,9487],{"className":45578},[7104,7145,7141],[35,45580,45581,45584],{"style":7437},[35,45582],{"className":45583,"style":7134},[7133],[35,45585,45587],{"className":45586},[7138,7139,7140,7141],[35,45588,45590,45593],{"className":45589},[7104,7141],[35,45591,7225],{"className":45592},[7104,7141],[35,45594,500],{"className":45595},[7104,7141],[35,45597,7756],{"className":45598},[7755],[35,45600,45602],{"className":45601},[7121],[35,45603,45605],{"className":45604,"style":42209},[7125],[35,45606],{},[35,45608],{"className":45609,"style":7496},[7248],[35,45611,45613,45619,45622,45625,45628,45631,45634,45776],{"className":45612},[7468],[35,45614,45616],{"className":45615,"style":7474},[7472,7473],[35,45617,275],{"className":45618},[7478,17001],[35,45620,500],{"className":45621},[7104],[35,45623],{"className":45624,"style":7249},[7248],[35,45626,445],{"className":45627},[7253],[35,45629],{"className":45630,"style":7249},[7248],[35,45632,460],{"className":45633},[7104],[35,45635,45637,45640,45773],{"className":45636},[7104],[35,45638],{"className":45639},[7472,14311],[35,45641,45643],{"className":45642},[14009],[35,45644,45646,45764],{"className":45645},[7117,7731],[35,45647,45649,45761],{"className":45648},[7121],[35,45650,45652,45703,45711],{"className":45651,"style":42504},[7125],[35,45653,45655,45658],{"style":45654},"top:-2.26em;",[35,45656],{"className":45657,"style":8839},[7133],[35,45659,45661,45664,45667,45670,45673,45676,45679,45685,45688,45691,45694,45697],{"className":45660},[7104],[35,45662,500],{"className":45663},[7104],[35,45665],{"className":45666,"style":7249},[7248],[35,45668,445],{"className":45669},[7253],[35,45671],{"className":45672,"style":7249},[7248],[35,45674,42416],{"className":45675},[12096],[35,45677],{"className":45678,"style":7496},[7248],[35,45680,45682],{"className":45681},[7104],[35,45683,275],{"className":45684},[7478,7479],[35,45686,32009],{"className":45687,"style":32094},[7104,7145],[35,45689],{"className":45690,"style":7496},[7248],[35,45692,9487],{"className":45693},[7104,7145],[35,45695,7829],{"className":45696},[7104,7145],[35,45698,45700],{"className":45699},[7104],[35,45701,1694],{"className":45702},[7478,7479],[35,45704,45705,45708],{"style":14395},[35,45706],{"className":45707,"style":8839},[7133],[35,45709],{"className":45710,"style":14403},[14402],[35,45712,45713,45716],{"style":42526},[35,45714],{"className":45715,"style":8839},[7133],[35,45717,45719,45722,45725,45728,45731,45734,45737,45743,45746,45749,45752,45755],{"className":45718},[7104],[35,45720,500],{"className":45721},[7104],[35,45723],{"className":45724,"style":7249},[7248],[35,45726,7225],{"className":45727},[7253],[35,45729],{"className":45730,"style":7249},[7248],[35,45732,42416],{"className":45733},[12096],[35,45735],{"className":45736,"style":7496},[7248],[35,45738,45740],{"className":45739},[7104],[35,45741,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2\\left(1 + \\delta\\right)^{-d} = 2\\left(1 + \\delta\\right)^{-\\lfloor k\u002F2\\rfloor}.",[35,45861,45863,45875,45952],{"className":45862,"ariaHidden":7091},[7090],[35,45864,45866,45869,45872],{"className":45865},[7095],[35,45867],{"className":45868,"style":32187},[7099],[35,45870,31994],{"className":45871},[7454],[35,45873],{"className":45874,"style":7450},[7248],[35,45876,45878,45881,45884,45887,45943,45946,45949],{"className":45877},[7095],[35,45879],{"className":45880,"style":32063},[7099],[35,45882,460],{"className":45883},[7104],[35,45885],{"className":45886,"style":7496},[7248],[35,45888,45890,45914],{"className":45889},[7468],[35,45891,45893,45896,45899,45902,45905,45908,45911],{"className":45892},[7468],[35,45894,275],{"className":45895,"style":7474},[7472,7473],[35,45897,500],{"className":45898},[7104],[35,45900],{"className":45901,"style":7249},[7248],[35,45903,445],{"className":45904},[7253],[35,45906],{"className":45907,"style":7249},[7248],[35,45909,32009],{"className":45910,"style":32094},[7104,7145],[35,45912,1694],{"className":45913,"style":7474},[7314,7473],[35,45915,45917],{"className":45916},[7113],[35,45918,45920],{"className":45919},[7117],[35,45921,45923],{"className":45922},[7121],[35,45924,45926],{"className":45925,"style":32110},[7125],[35,45927,45928,45931],{"style":12371},[35,45929],{"className":45930,"style":7134},[7133],[35,45932,45934],{"className":45933},[7138,7139,7140,7141],[35,45935,45937,45940],{"className":45936},[7104,7141],[35,45938,7225],{"className":45939},[7104,7141],[35,45941,9487],{"className":45942},[7104,7145,7141],[35,45944],{"className":45945,"style":7450},[7248],[35,45947,242],{"className":45948},[7454],[35,45950],{"className":45951,"style":7450},[7248],[35,45953,45955,45959,45962,45965,46031,46034],{"className":45954},[7095],[35,45956],{"className":45957,"style":45958},[7099],"height:1.2779em;vertical-align:-0.25em;",[35,45960,460],{"className":45961},[7104],[35,45963],{"className":45964,"style":7496},[7248],[35,45966,45968,45992],{"className":45967},[7468],[35,45969,45971,45974,45977,45980,45983,45986,45989],{"className":45970},[7468],[35,45972,275],{"className":45973,"style":7474},[7472,7473],[35,45975,500],{"className":45976},[7104],[35,45978],{"className":45979,"style":7249},[7248],[35,45981,445],{"className":45982},[7253],[35,45984],{"className":45985,"style":7249},[7248],[35,45987,32009],{"className":45988,"style":32094},[7104,7145],[35,45990,1694],{"className":45991,"style":7474},[7314,7473],[35,45993,45995],{"className":45994},[7113],[35,45996,45998],{"className":45997},[7117],[35,45999,46001],{"className":46000},[7121],[35,46002,46005],{"className":46003,"style":46004},[7125],"height:1.0279em;",[35,46006,46007,46010],{"style":12371},[35,46008],{"className":46009,"style":7134},[7133],[35,46011,46013],{"className":46012},[7138,7139,7140,7141],[35,46014,46016,46019,46022,46025,46028],{"className":46015},[7104,7141],[35,46017,7225],{"className":46018},[7104,7141],[35,46020,45845],{"className":46021},[7472,7141],[35,46023,279],{"className":46024,"style":8221},[7104,7145,7141],[35,46026,30606],{"className":46027},[7104,7141],[35,46029,45854],{"className":46030},[7314,7141],[35,46032],{"className":46033,"style":7496},[7248],[35,46035,5122],{"className":46036},[7104],[5019,46038,5022],{"id":5021},[12,46040,46041,46043,46044],{},[35,46042,500],{}," N. Yoshioka, M. Amico, W. Kirby et al. \"Diagonalization of large many-body Hamiltonians on a quantum processor\". ",[19,46045,46047],{"href":46046},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.14431","arXiv:2407.14431",[12,46049,46050,46052],{},[35,46051,460],{}," Ethan N. Epperly, Lin Lin, and Yuji Nakatsukasa. \"A theory of quantum subspace diagonalization\". SIAM Journal on Matrix Analysis and Applications 43, 1263–1290 (2022).",[12,46054,46055,46057],{},[35,46056,12751],{}," Å. Björck. \"Numerical methods in matrix computations\". Texts in Applied Mathematics. Springer International Publishing. (2014).",[12,46059,46060,46062],{},[35,46061,528],{}," William Kirby. \"Analysis of quantum Krylov algorithms with errors\". Quantum 8, 1457 (2024).",[5019,46064,46066],{"id":46065},"tutorial-survey","Tutorial survey",[12,46068,46069],{},"Please take this short survey to provide feedback on this tutorial. Your insights will help us improve our content offerings and user experience.",[12,46071,46072],{},[19,46073,46075],{"href":46074},"https:\u002F\u002Fyour.feedback.ibm.com\u002Fjfe\u002Fform\u002FSV_82nennpKIjjD8rQ","Link to survey",[5019,46077,5147],{"id":5146},[12,46079,5150,46080,5122],{},[19,46081,5707],{"href":5706},[5159,46083,46084],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}",{"title":30,"searchDepth":45,"depth":45,"links":46086},[46087,46088,46089,46090,46094,46101,46106,46109,46110,46111,46112,46113],{"id":5731,"depth":45,"text":5732},{"id":5766,"depth":45,"text":5767},{"id":5790,"depth":45,"text":5791},{"id":7044,"depth":45,"text":7045,"children":46091},[46092,46093],{"id":7049,"depth":51,"text":7050},{"id":8613,"depth":51,"text":8614},{"id":18728,"depth":45,"text":18729,"children":46095},[46096,46097,46098,46099],{"id":18732,"depth":51,"text":18733},{"id":24862,"depth":51,"text":24863},{"id":25847,"depth":51,"text":25848},{"id":25944,"depth":51,"text":46100},"Template circuits for calculating matrix elements of S~\\tilde{S}S~ and H~\\tilde{H}H~ via Hadamard 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notebook provides a primer on the quantum dynamics of a driven two-level system by visualizing the excited-state probability under the rotating-wave approximation (RWA). The model treats a qubit as an ideal two-state system driven by a classical microwave tone. This formulation maps the probability of measuring the excited state as a function of drive frequency and pulse duration. The central computational object is the probability surface",[35,46155,46157],{"className":46156},[7318],[35,46158,46160,46191],{"className":46159},[7057],[35,46161,46163],{"className":46162},[7061],[7063,46164,46165],{"xmlns":7065,"display":7327},[7067,46166,46167,46188],{},[7070,46168,46169,46175,46177,46179,46181,46184,46186],{},[7699,46170,46171,46173],{},[7076,46172,7705],{},[7076,46174,7817],{},[7223,46176,275],{"stretchy":7289},[7076,46178,1675],{},[7223,46180,2877],{"separator":7091},[7076,46182,46183],{},"τ",[7223,46185,1694],{"stretchy":7289},[7223,46187,2877],{"separator":7091},[7083,46189,46190],{"encoding":7085},"P_e(f,\\tau),",[35,46192,46194],{"className":46193,"ariaHidden":7091},[7090],[35,46195,46197,46200,46240,46243,46246,46249,46252,46256,46259],{"className":46196},[7095],[35,46198],{"className":46199,"style":7303},[7099],[35,46201,46203,46206],{"className":46202},[7104],[35,46204,7705],{"className":46205,"style":7751},[7104,7145],[35,46207,46209],{"className":46208},[7113],[35,46210,46212,46232],{"className":46211},[7117,7731],[35,46213,46215,46229],{"className":46214},[7121],[35,46216,46218],{"className":46217,"style":9035},[7125],[35,46219,46220,46223],{"style":12163},[35,46221],{"className":46222,"style":7134},[7133],[35,46224,46226],{"className":46225},[7138,7139,7140,7141],[35,46227,7817],{"className":46228},[7104,7145,7141],[35,46230,7756],{"className":46231},[7755],[35,46233,46235],{"className":46234},[7121],[35,46236,46238],{"className":46237,"style":7763},[7125],[35,46239],{},[35,46241,275],{"className":46242},[7472],[35,46244,1675],{"className":46245,"style":31824},[7104,7145],[35,46247,2877],{"className":46248},[7492],[35,46250],{"className":46251,"style":7496},[7248],[35,46253,46183],{"className":46254,"style":46255},[7104,7145],"margin-right:0.1132em;",[35,46257,1694],{"className":46258},[7314],[35,46260,2877],{"className":46261},[7492],[12,46263,32939,46264,46292,46293,46322],{},[35,46265,46267,46280],{"className":46266},[7057],[35,46268,46270],{"className":46269},[7061],[7063,46271,46272],{"xmlns":7065},[7067,46273,46274,46278],{},[7070,46275,46276],{},[7076,46277,1675],{},[7083,46279,1675],{"encoding":7085},[35,46281,46283],{"className":46282,"ariaHidden":7091},[7090],[35,46284,46286,46289],{"className":46285},[7095],[35,46287],{"className":46288,"style":11562},[7099],[35,46290,1675],{"className":46291,"style":31824},[7104,7145]," denotes the applied drive frequency and ",[35,46294,46296,46310],{"className":46295},[7057],[35,46297,46299],{"className":46298},[7061],[7063,46300,46301],{"xmlns":7065},[7067,46302,46303,46307],{},[7070,46304,46305],{},[7076,46306,46183],{},[7083,46308,46309],{"encoding":7085},"\\tau",[35,46311,46313],{"className":46312,"ariaHidden":7091},[7090],[35,46314,46316,46319],{"className":46315},[7095],[35,46317],{"className":46318,"style":7173},[7099],[35,46320,46183],{"className":46321,"style":46255},[7104,7145]," denotes the microwave pulse duration. The notebook renders this probability surface with two-dimensional heatmaps, three-dimensional surfaces, one-dimensional cross sections, and Fast Fourier transform (FFT) spectrograms. Fourier analysis exposes the generalized Rabi frequency as a theoretical ridge in the frequency domain, which connects the time-domain quantum oscillation to its spectral structure.",[5019,46324,46326],{"id":46325},"nomenclature-and-definitions","Nomenclature and Definitions",[12,46328,46329],{},"The following table defines the physical and mathematical quantities that recur throughout the visualization workflow.",[46331,46332,46333,46346],"table",{},[46334,46335,46336],"thead",{},[46337,46338,46339,46343],"tr",{},[46340,46341,46342],"th",{},"Symbol or Term",[46340,46344,46345],{},"Meaning",[46347,46348,46349,46396,46442,46488,46615,46692,46770,46805,47026,47123,47160,47206,47357,47434,47659,47694,47787,47795],"tbody",{},[46337,46350,46351,46393],{},[46352,46353,46354],"td",{},[35,46355,46357,46375],{"className":46356},[7057],[35,46358,46360],{"className":46359},[7061],[7063,46361,46362],{"xmlns":7065},[7067,46363,46364,46372],{},[7070,46365,46366,46368,46370],{},[7223,46367,7283],{"stretchy":7289},[7076,46369,42397],{},[7223,46371,7290],{"stretchy":7289},[7083,46373,46374],{"encoding":7085},"\\lvert g\\rangle",[35,46376,46378],{"className":46377,"ariaHidden":7091},[7090],[35,46379,46381,46384,46387,46390],{"className":46380},[7095],[35,46382],{"className":46383,"style":7303},[7099],[35,46385,7283],{"className":46386},[7472],[35,46388,42397],{"className":46389,"style":7310},[7104,7145],[35,46391,7290],{"className":46392},[7314],[46352,46394,46395],{},"Ground state of the qubit",[46337,46397,46398,46439],{},[46352,46399,46400],{},[35,46401,46403,46421],{"className":46402},[7057],[35,46404,46406],{"className":46405},[7061],[7063,46407,46408],{"xmlns":7065},[7067,46409,46410,46418],{},[7070,46411,46412,46414,46416],{},[7223,46413,7283],{"stretchy":7289},[7076,46415,7817],{},[7223,46417,7290],{"stretchy":7289},[7083,46419,46420],{"encoding":7085},"\\lvert e\\rangle",[35,46422,46424],{"className":46423,"ariaHidden":7091},[7090],[35,46425,46427,46430,46433,46436],{"className":46426},[7095],[35,46428],{"className":46429,"style":7303},[7099],[35,46431,7283],{"className":46432},[7472],[35,46434,7817],{"className":46435},[7104,7145],[35,46437,7290],{"className":46438},[7314],[46352,46440,46441],{},"Excited state of the qubit",[46337,46443,46444,46485],{},[46352,46445,46446],{},[35,46447,46449,46467],{"className":46448},[7057],[35,46450,46452],{"className":46451},[7061],[7063,46453,46454],{"xmlns":7065},[7067,46455,46456,46464],{},[7070,46457,46458,46460,46462],{},[7223,46459,7283],{"stretchy":7289},[7076,46461,8322],{},[7223,46463,7290],{"stretchy":7289},[7083,46465,46466],{"encoding":7085},"\\lvert \\psi\\rangle",[35,46468,46470],{"className":46469,"ariaHidden":7091},[7090],[35,46471,46473,46476,46479,46482],{"className":46472},[7095],[35,46474],{"className":46475,"style":7303},[7099],[35,46477,7283],{"className":46478},[7472],[35,46480,8322],{"className":46481,"style":7310},[7104,7145],[35,46483,7290],{"className":46484},[7314],[46352,46486,46487],{},"Quantum state vector in a two-dimensional Hilbert space",[46337,46489,46490,46536],{},[46352,46491,46492],{},[35,46493,46495,46514],{"className":46494},[7057],[35,46496,46498],{"className":46497},[7061],[7063,46499,46500],{"xmlns":7065},[7067,46501,46502,46511],{},[7070,46503,46504,46507,46509],{},[7076,46505,46506],{},"α",[7223,46508,2877],{"separator":7091},[7076,46510,41150],{},[7083,46512,46513],{"encoding":7085},"\\alpha,\\beta",[35,46515,46517],{"className":46516,"ariaHidden":7091},[7090],[35,46518,46520,46523,46527,46530,46533],{"className":46519},[7095],[35,46521],{"className":46522,"style":11562},[7099],[35,46524,46506],{"className":46525,"style":46526},[7104,7145],"margin-right:0.0037em;",[35,46528,2877],{"className":46529},[7492],[35,46531],{"className":46532,"style":7496},[7248],[35,46534,41150],{"className":46535,"style":41244},[7104,7145],[46352,46537,46538,46539,15316,46577],{},"Complex probability amplitudes for ",[35,46540,46542,46559],{"className":46541},[7057],[35,46543,46545],{"className":46544},[7061],[7063,46546,46547],{"xmlns":7065},[7067,46548,46549,46557],{},[7070,46550,46551,46553,46555],{},[7223,46552,7283],{"stretchy":7289},[7076,46554,42397],{},[7223,46556,7290],{"stretchy":7289},[7083,46558,46374],{"encoding":7085},[35,46560,46562],{"className":46561,"ariaHidden":7091},[7090],[35,46563,46565,46568,46571,46574],{"className":46564},[7095],[35,46566],{"className":46567,"style":7303},[7099],[35,46569,7283],{"className":46570},[7472],[35,46572,42397],{"className":46573,"style":7310},[7104,7145],[35,46575,7290],{"className":46576},[7314],[35,46578,46580,46597],{"className":46579},[7057],[35,46581,46583],{"className":46582},[7061],[7063,46584,46585],{"xmlns":7065},[7067,46586,46587,46595],{},[7070,46588,46589,46591,46593],{},[7223,46590,7283],{"stretchy":7289},[7076,46592,7817],{},[7223,46594,7290],{"stretchy":7289},[7083,46596,46420],{"encoding":7085},[35,46598,46600],{"className":46599,"ariaHidden":7091},[7090],[35,46601,46603,46606,46609,46612],{"className":46602},[7095],[35,46604],{"className":46605,"style":7303},[7099],[35,46607,7283],{"className":46608},[7472],[35,46610,7817],{"className":46611},[7104,7145],[35,46613,7290],{"className":46614},[7314],[46337,46616,46617,46689],{},[46352,46618,46619],{},[35,46620,46622,46640],{"className":46621},[7057],[35,46623,46625],{"className":46624},[7061],[7063,46626,46627],{"xmlns":7065},[7067,46628,46629,46637],{},[7070,46630,46631],{},[7699,46632,46633,46635],{},[7076,46634,7705],{},[7076,46636,7817],{},[7083,46638,46639],{"encoding":7085},"P_e",[35,46641,46643],{"className":46642,"ariaHidden":7091},[7090],[35,46644,46646,46649],{"className":46645},[7095],[35,46647],{"className":46648,"style":7718},[7099],[35,46650,46652,46655],{"className":46651},[7104],[35,46653,7705],{"className":46654,"style":7751},[7104,7145],[35,46656,46658],{"className":46657},[7113],[35,46659,46661,46681],{"className":46660},[7117,7731],[35,46662,46664,46678],{"className":46663},[7121],[35,46665,46667],{"className":46666,"style":9035},[7125],[35,46668,46669,46672],{"style":12163},[35,46670],{"className":46671,"style":7134},[7133],[35,46673,46675],{"className":46674},[7138,7139,7140,7141],[35,46676,7817],{"className":46677},[7104,7145,7141],[35,46679,7756],{"className":46680},[7755],[35,46682,46684],{"className":46683},[7121],[35,46685,46687],{"className":46686,"style":7763},[7125],[35,46688],{},[46352,46690,46691],{},"Probability of measuring the qubit in the excited state",[46337,46693,46694,46767],{},[46352,46695,46696],{},[35,46697,46699,46717],{"className":46698},[7057],[35,46700,46702],{"className":46701},[7061],[7063,46703,46704],{"xmlns":7065},[7067,46705,46706,46714],{},[7070,46707,46708],{},[7699,46709,46710,46712],{},[7076,46711,1675],{},[7227,46713,5312],{},[7083,46715,46716],{"encoding":7085},"f_0",[35,46718,46720],{"className":46719,"ariaHidden":7091},[7090],[35,46721,46723,46726],{"className":46722},[7095],[35,46724],{"className":46725,"style":11562},[7099],[35,46727,46729,46732],{"className":46728},[7104],[35,46730,1675],{"className":46731,"style":31824},[7104,7145],[35,46733,46735],{"className":46734},[7113],[35,46736,46738,46759],{"className":46737},[7117,7731],[35,46739,46741,46756],{"className":46740},[7121],[35,46742,46744],{"className":46743,"style":18794},[7125],[35,46745,46747,46750],{"style":46746},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[35,46748],{"className":46749,"style":7134},[7133],[35,46751,46753],{"className":46752},[7138,7139,7140,7141],[35,46754,5312],{"className":46755},[7104,7141],[35,46757,7756],{"className":46758},[7755],[35,46760,46762],{"className":46761},[7121],[35,46763,46765],{"className":46764,"style":7763},[7125],[35,46766],{},[46352,46768,46769],{},"Natural transition frequency of the qubit",[46337,46771,46772,46802],{},[46352,46773,46774],{},[35,46775,46777,46790],{"className":46776},[7057],[35,46778,46780],{"className":46779},[7061],[7063,46781,46782],{"xmlns":7065},[7067,46783,46784,46788],{},[7070,46785,46786],{},[7076,46787,1675],{},[7083,46789,1675],{"encoding":7085},[35,46791,46793],{"className":46792,"ariaHidden":7091},[7090],[35,46794,46796,46799],{"className":46795},[7095],[35,46797],{"className":46798,"style":11562},[7099],[35,46800,1675],{"className":46801,"style":31824},[7104,7145],[46352,46803,46804],{},"Applied drive frequency",[46337,46806,46807,46880],{},[46352,46808,46809],{},[35,46810,46812,46831],{"className":46811},[7057],[35,46813,46815],{"className":46814},[7061],[7063,46816,46817],{"xmlns":7065},[7067,46818,46819,46828],{},[7070,46820,46821],{},[7699,46822,46823,46826],{},[7076,46824,46825],{},"ω",[7227,46827,5312],{},[7083,46829,46830],{"encoding":7085},"\\omega_0",[35,46832,46834],{"className":46833,"ariaHidden":7091},[7090],[35,46835,46837,46840],{"className":46836},[7095],[35,46838],{"className":46839,"style":29689},[7099],[35,46841,46843,46846],{"className":46842},[7104],[35,46844,46825],{"className":46845,"style":7310},[7104,7145],[35,46847,46849],{"className":46848},[7113],[35,46850,46852,46872],{"className":46851},[7117,7731],[35,46853,46855,46869],{"className":46854},[7121],[35,46856,46858],{"className":46857,"style":18794},[7125],[35,46859,46860,46863],{"style":9100},[35,46861],{"className":46862,"style":7134},[7133],[35,46864,46866],{"className":46865},[7138,7139,7140,7141],[35,46867,5312],{"className":46868},[7104,7141],[35,46870,7756],{"className":46871},[7755],[35,46873,46875],{"className":46874},[7121],[35,46876,46878],{"className":46877,"style":7763},[7125],[35,46879],{},[46352,46881,46882,46883],{},"Angular transition frequency, ",[35,46884,46886,46916],{"className":46885},[7057],[35,46887,46889],{"className":46888},[7061],[7063,46890,46891],{"xmlns":7065},[7067,46892,46893,46913],{},[7070,46894,46895,46901,46903,46905,46907],{},[7699,46896,46897,46899],{},[7076,46898,46825],{},[7227,46900,5312],{},[7223,46902,242],{},[7227,46904,460],{},[7076,46906,13081],{},[7699,46908,46909,46911],{},[7076,46910,1675],{},[7227,46912,5312],{},[7083,46914,46915],{"encoding":7085},"\\omega_0=2\\pi f_0",[35,46917,46919,46974],{"className":46918,"ariaHidden":7091},[7090],[35,46920,46922,46925,46965,46968,46971],{"className":46921},[7095],[35,46923],{"className":46924,"style":29689},[7099],[35,46926,46928,46931],{"className":46927},[7104],[35,46929,46825],{"className":46930,"style":7310},[7104,7145],[35,46932,46934],{"className":46933},[7113],[35,46935,46937,46957],{"className":46936},[7117,7731],[35,46938,46940,46954],{"className":46939},[7121],[35,46941,46943],{"className":46942,"style":18794},[7125],[35,46944,46945,46948],{"style":9100},[35,46946],{"className":46947,"style":7134},[7133],[35,46949,46951],{"className":46950},[7138,7139,7140,7141],[35,46952,5312],{"className":46953},[7104,7141],[35,46955,7756],{"className":46956},[7755],[35,46958,46960],{"className":46959},[7121],[35,46961,46963],{"className":46962,"style":7763},[7125],[35,46964],{},[35,46966],{"className":46967,"style":7450},[7248],[35,46969,242],{"className":46970},[7454],[35,46972],{"className":46973,"style":7450},[7248],[35,46975,46977,46980,46983,46986],{"className":46976},[7095],[35,46978],{"className":46979,"style":11562},[7099],[35,46981,460],{"className":46982},[7104],[35,46984,13081],{"className":46985,"style":7310},[7104,7145],[35,46987,46989,46992],{"className":46988},[7104],[35,46990,1675],{"className":46991,"style":31824},[7104,7145],[35,46993,46995],{"className":46994},[7113],[35,46996,46998,47018],{"className":46997},[7117,7731],[35,46999,47001,47015],{"className":47000},[7121],[35,47002,47004],{"className":47003,"style":18794},[7125],[35,47005,47006,47009],{"style":46746},[35,47007],{"className":47008,"style":7134},[7133],[35,47010,47012],{"className":47011},[7138,7139,7140,7141],[35,47013,5312],{"className":47014},[7104,7141],[35,47016,7756],{"className":47017},[7755],[35,47019,47021],{"className":47020},[7121],[35,47022,47024],{"className":47023,"style":7763},[7125],[35,47025],{},[46337,47027,47028,47059],{},[46352,47029,47030],{},[35,47031,47033,47047],{"className":47032},[7057],[35,47034,47036],{"className":47035},[7061],[7063,47037,47038],{"xmlns":7065},[7067,47039,47040,47044],{},[7070,47041,47042],{},[7076,47043,46825],{},[7083,47045,47046],{"encoding":7085},"\\omega",[35,47048,47050],{"className":47049,"ariaHidden":7091},[7090],[35,47051,47053,47056],{"className":47052},[7095],[35,47054],{"className":47055,"style":7173},[7099],[35,47057,46825],{"className":47058,"style":7310},[7104,7145],[46352,47060,47061,47062],{},"Angular drive frequency, ",[35,47063,47065,47087],{"className":47064},[7057],[35,47066,47068],{"className":47067},[7061],[7063,47069,47070],{"xmlns":7065},[7067,47071,47072,47084],{},[7070,47073,47074,47076,47078,47080,47082],{},[7076,47075,46825],{},[7223,47077,242],{},[7227,47079,460],{},[7076,47081,13081],{},[7076,47083,1675],{},[7083,47085,47086],{"encoding":7085},"\\omega=2\\pi f",[35,47088,47090,47108],{"className":47089,"ariaHidden":7091},[7090],[35,47091,47093,47096,47099,47102,47105],{"className":47092},[7095],[35,47094],{"className":47095,"style":7173},[7099],[35,47097,46825],{"className":47098,"style":7310},[7104,7145],[35,47100],{"className":47101,"style":7450},[7248],[35,47103,242],{"className":47104},[7454],[35,47106],{"className":47107,"style":7450},[7248],[35,47109,47111,47114,47117,47120],{"className":47110},[7095],[35,47112],{"className":47113,"style":11562},[7099],[35,47115,460],{"className":47116},[7104],[35,47118,13081],{"className":47119,"style":7310},[7104,7145],[35,47121,1675],{"className":47122,"style":31824},[7104,7145],[46337,47124,47125,47157],{},[46352,47126,47127],{},[35,47128,47130,47145],{"className":47129},[7057],[35,47131,47133],{"className":47132},[7061],[7063,47134,47135],{"xmlns":7065},[7067,47136,47137,47142],{},[7070,47138,47139],{},[7076,47140,47141],{"mathvariant":7282},"Ω",[7083,47143,47144],{"encoding":7085},"\\Omega",[35,47146,47148],{"className":47147,"ariaHidden":7091},[7090],[35,47149,47151,47154],{"className":47150},[7095],[35,47152],{"className":47153,"style":7100},[7099],[35,47155,47141],{"className":47156},[7104],[46352,47158,47159],{},"On-resonance Rabi angular frequency",[46337,47161,47162,47203],{},[46352,47163,47164],{},[35,47165,47167,47187],{"className":47166},[7057],[35,47168,47170],{"className":47169},[7061],[7063,47171,47172],{"xmlns":7065},[7067,47173,47174,47184],{},[7070,47175,47176,47178,47180,47182],{},[7076,47177,47141],{"mathvariant":7282},[7076,47179,398],{"mathvariant":7282},[7227,47181,460],{},[7076,47183,13081],{},[7083,47185,47186],{"encoding":7085},"\\Omega\u002F2\\pi",[35,47188,47190],{"className":47189,"ariaHidden":7091},[7090],[35,47191,47193,47196,47200],{"className":47192},[7095],[35,47194],{"className":47195,"style":7303},[7099],[35,47197,47199],{"className":47198},[7104],"Ω\u002F2",[35,47201,13081],{"className":47202,"style":7310},[7104,7145],[46352,47204,47205],{},"On-resonance Rabi frequency in ordinary frequency units",[46337,47207,47208,47240],{},[46352,47209,47210],{},[35,47211,47213,47228],{"className":47212},[7057],[35,47214,47216],{"className":47215},[7061],[7063,47217,47218],{"xmlns":7065},[7067,47219,47220,47225],{},[7070,47221,47222],{},[7076,47223,47224],{"mathvariant":7282},"Δ",[7083,47226,47227],{"encoding":7085},"\\Delta",[35,47229,47231],{"className":47230,"ariaHidden":7091},[7090],[35,47232,47234,47237],{"className":47233},[7095],[35,47235],{"className":47236,"style":7100},[7099],[35,47238,47224],{"className":47239},[7104],[46352,47241,47242,47243],{},"Drive detuning, ",[35,47244,47246,47272],{"className":47245},[7057],[35,47247,47249],{"className":47248},[7061],[7063,47250,47251],{"xmlns":7065},[7067,47252,47253,47269],{},[7070,47254,47255,47257,47259,47261,47263],{},[7076,47256,47224],{"mathvariant":7282},[7223,47258,242],{},[7076,47260,46825],{},[7223,47262,7225],{},[7699,47264,47265,47267],{},[7076,47266,46825],{},[7227,47268,5312],{},[7083,47270,47271],{"encoding":7085},"\\Delta=\\omega-\\omega_0",[35,47273,47275,47293,47311],{"className":47274,"ariaHidden":7091},[7090],[35,47276,47278,47281,47284,47287,47290],{"className":47277},[7095],[35,47279],{"className":47280,"style":7100},[7099],[35,47282,47224],{"className":47283},[7104],[35,47285],{"className":47286,"style":7450},[7248],[35,47288,242],{"className":47289},[7454],[35,47291],{"className":47292,"style":7450},[7248],[35,47294,47296,47299,47302,47305,47308],{"className":47295},[7095],[35,47297],{"className":47298,"style":7241},[7099],[35,47300,46825],{"className":47301,"style":7310},[7104,7145],[35,47303],{"className":47304,"style":7249},[7248],[35,47306,7225],{"className":47307},[7253],[35,47309],{"className":47310,"style":7249},[7248],[35,47312,47314,47317],{"className":47313},[7095],[35,47315],{"className":47316,"style":29689},[7099],[35,47318,47320,47323],{"className":47319},[7104],[35,47321,46825],{"className":47322,"style":7310},[7104,7145],[35,47324,47326],{"className":47325},[7113],[35,47327,47329,47349],{"className":47328},[7117,7731],[35,47330,47332,47346],{"className":47331},[7121],[35,47333,47335],{"className":47334,"style":18794},[7125],[35,47336,47337,47340],{"style":9100},[35,47338],{"className":47339,"style":7134},[7133],[35,47341,47343],{"className":47342},[7138,7139,7140,7141],[35,47344,5312],{"className":47345},[7104,7141],[35,47347,7756],{"className":47348},[7755],[35,47350,47352],{"className":47351},[7121],[35,47353,47355],{"className":47354,"style":7763},[7125],[35,47356],{},[46337,47358,47359,47431],{},[46352,47360,47361],{},[35,47362,47364,47382],{"className":47363},[7057],[35,47365,47367],{"className":47366},[7061],[7063,47368,47369],{"xmlns":7065},[7067,47370,47371,47379],{},[7070,47372,47373],{},[7699,47374,47375,47377],{},[7076,47376,47141],{"mathvariant":7282},[7076,47378,24883],{},[7083,47380,47381],{"encoding":7085},"\\Omega_R",[35,47383,47385],{"className":47384,"ariaHidden":7091},[7090],[35,47386,47388,47391],{"className":47387},[7095],[35,47389],{"className":47390,"style":7718},[7099],[35,47392,47394,47397],{"className":47393},[7104],[35,47395,47141],{"className":47396},[7104],[35,47398,47400],{"className":47399},[7113],[35,47401,47403,47423],{"className":47402},[7117,7731],[35,47404,47406,47420],{"className":47405},[7121],[35,47407,47409],{"className":47408,"style":7738},[7125],[35,47410,47411,47414],{"style":9926},[35,47412],{"className":47413,"style":7134},[7133],[35,47415,47417],{"className":47416},[7138,7139,7140,7141],[35,47418,24883],{"className":47419,"style":24908},[7104,7145,7141],[35,47421,7756],{"className":47422},[7755],[35,47424,47426],{"className":47425},[7121],[35,47427,47429],{"className":47428,"style":7763},[7125],[35,47430],{},[46352,47432,47433],{},"Generalized Rabi angular frequency",[46337,47435,47436,47511],{},[46352,47437,47438],{},[35,47439,47441,47460],{"className":47440},[7057],[35,47442,47444],{"className":47443},[7061],[7063,47445,47446],{"xmlns":7065},[7067,47447,47448,47457],{},[7070,47449,47450],{},[7699,47451,47452,47455],{},[7076,47453,47454],{},"ν",[7076,47456,24883],{},[7083,47458,47459],{"encoding":7085},"\\nu_R",[35,47461,47463],{"className":47462,"ariaHidden":7091},[7090],[35,47464,47466,47469],{"className":47465},[7095],[35,47467],{"className":47468,"style":29689},[7099],[35,47470,47472,47476],{"className":47471},[7104],[35,47473,47454],{"className":47474,"style":47475},[7104,7145],"margin-right:0.0637em;",[35,47477,47479],{"className":47478},[7113],[35,47480,47482,47503],{"className":47481},[7117,7731],[35,47483,47485,47500],{"className":47484},[7121],[35,47486,47488],{"className":47487,"style":7738},[7125],[35,47489,47491,47494],{"style":47490},"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;",[35,47492],{"className":47493,"style":7134},[7133],[35,47495,47497],{"className":47496},[7138,7139,7140,7141],[35,47498,24883],{"className":47499,"style":24908},[7104,7145,7141],[35,47501,7756],{"className":47502},[7755],[35,47504,47506],{"className":47505},[7121],[35,47507,47509],{"className":47508,"style":7763},[7125],[35,47510],{},[46352,47512,47513,47514],{},"Generalized Rabi frequency, ",[35,47515,47517,47549],{"className":47516},[7057],[35,47518,47520],{"className":47519},[7061],[7063,47521,47522],{"xmlns":7065},[7067,47523,47524,47546],{},[7070,47525,47526,47532,47534,47540,47542,47544],{},[7699,47527,47528,47530],{},[7076,47529,47454],{},[7076,47531,24883],{},[7223,47533,242],{},[7699,47535,47536,47538],{},[7076,47537,47141],{"mathvariant":7282},[7076,47539,24883],{},[7076,47541,398],{"mathvariant":7282},[7227,47543,460],{},[7076,47545,13081],{},[7083,47547,47548],{"encoding":7085},"\\nu_R=\\Omega_R\u002F2\\pi",[35,47550,47552,47607],{"className":47551,"ariaHidden":7091},[7090],[35,47553,47555,47558,47598,47601,47604],{"className":47554},[7095],[35,47556],{"className":47557,"style":29689},[7099],[35,47559,47561,47564],{"className":47560},[7104],[35,47562,47454],{"className":47563,"style":47475},[7104,7145],[35,47565,47567],{"className":47566},[7113],[35,47568,47570,47590],{"className":47569},[7117,7731],[35,47571,47573,47587],{"className":47572},[7121],[35,47574,47576],{"className":47575,"style":7738},[7125],[35,47577,47578,47581],{"style":47490},[35,47579],{"className":47580,"style":7134},[7133],[35,47582,47584],{"className":47583},[7138,7139,7140,7141],[35,47585,24883],{"className":47586,"style":24908},[7104,7145,7141],[35,47588,7756],{"className":47589},[7755],[35,47591,47593],{"className":47592},[7121],[35,47594,47596],{"className":47595,"style":7763},[7125],[35,47597],{},[35,47599],{"className":47600,"style":7450},[7248],[35,47602,242],{"className":47603},[7454],[35,47605],{"className":47606,"style":7450},[7248],[35,47608,47610,47613,47653,47656],{"className":47609},[7095],[35,47611],{"className":47612,"style":7303},[7099],[35,47614,47616,47619],{"className":47615},[7104],[35,47617,47141],{"className":47618},[7104],[35,47620,47622],{"className":47621},[7113],[35,47623,47625,47645],{"className":47624},[7117,7731],[35,47626,47628,47642],{"className":47627},[7121],[35,47629,47631],{"className":47630,"style":7738},[7125],[35,47632,47633,47636],{"style":9926},[35,47634],{"className":47635,"style":7134},[7133],[35,47637,47639],{"className":47638},[7138,7139,7140,7141],[35,47640,24883],{"className":47641,"style":24908},[7104,7145,7141],[35,47643,7756],{"className":47644},[7755],[35,47646,47648],{"className":47647},[7121],[35,47649,47651],{"className":47650,"style":7763},[7125],[35,47652],{},[35,47654,30606],{"className":47655},[7104],[35,47657,13081],{"className":47658,"style":7310},[7104,7145],[46337,47660,47661,47691],{},[46352,47662,47663],{},[35,47664,47666,47679],{"className":47665},[7057],[35,47667,47669],{"className":47668},[7061],[7063,47670,47671],{"xmlns":7065},[7067,47672,47673,47677],{},[7070,47674,47675],{},[7076,47676,46183],{},[7083,47678,46309],{"encoding":7085},[35,47680,47682],{"className":47681,"ariaHidden":7091},[7090],[35,47683,47685,47688],{"className":47684},[7095],[35,47686],{"className":47687,"style":7173},[7099],[35,47689,46183],{"className":47690,"style":46255},[7104,7145],[46352,47692,47693],{},"Pulse duration",[46337,47695,47696,47784],{},[46352,47697,47698],{},[35,47699,47701,47721],{"className":47700},[7057],[35,47702,47704],{"className":47703},[7061],[7063,47705,47706],{"xmlns":7065},[7067,47707,47708,47718],{},[7070,47709,47710],{},[8304,47711,47712,47714,47716],{},[7076,47713,41530],{},[7227,47715,460],{},[7223,47717,25282],{},[7083,47719,47720],{"encoding":7085},"T_2^\\ast",[35,47722,47724],{"className":47723,"ariaHidden":7091},[7090],[35,47725,47727,47731],{"className":47726},[7095],[35,47728],{"className":47729,"style":47730},[7099],"height:0.9368em;vertical-align:-0.2481em;",[35,47732,47734,47737],{"className":47733},[7104],[35,47735,41530],{"className":47736,"style":7751},[7104,7145],[35,47738,47740],{"className":47739},[7113],[35,47741,47743,47775],{"className":47742},[7117,7731],[35,47744,47746,47772],{"className":47745},[7121],[35,47747,47749,47761],{"className":47748,"style":41067},[7125],[35,47750,47752,47755],{"style":47751},"top:-2.4519em;margin-left:-0.1389em;margin-right:0.05em;",[35,47753],{"className":47754,"style":7134},[7133],[35,47756,47758],{"className":47757},[7138,7139,7140,7141],[35,47759,460],{"className":47760},[7104,7141],[35,47762,47763,47766],{"style":7129},[35,47764],{"className":47765,"style":7134},[7133],[35,47767,47769],{"className":47768},[7138,7139,7140,7141],[35,47770,25282],{"className":47771},[7253,7141],[35,47773,7756],{"className":47774},[7755],[35,47776,47778],{"className":47777},[7121],[35,47779,47782],{"className":47780,"style":47781},[7125],"height:0.2481em;",[35,47783],{},[46352,47785,47786],{},"Phenomenological dephasing time used as an optional damping parameter",[46337,47788,47789,47792],{},[46352,47790,47791],{},"RWA",[46352,47793,47794],{},"Rotating-wave approximation",[46337,47796,47797,47800],{},[46352,47798,47799],{},"FFT",[46352,47801,47802],{},"Fast Fourier transform",[5019,47804,47806],{"id":47805},"physical-model","Physical Model",[12,47808,47809,47810,47848,47849,47887],{},"A qubit can be modeled as a two-level quantum system with a ground state ",[35,47811,47813,47830],{"className":47812},[7057],[35,47814,47816],{"className":47815},[7061],[7063,47817,47818],{"xmlns":7065},[7067,47819,47820,47828],{},[7070,47821,47822,47824,47826],{},[7223,47823,7283],{"stretchy":7289},[7076,47825,42397],{},[7223,47827,7290],{"stretchy":7289},[7083,47829,46374],{"encoding":7085},[35,47831,47833],{"className":47832,"ariaHidden":7091},[7090],[35,47834,47836,47839,47842,47845],{"className":47835},[7095],[35,47837],{"className":47838,"style":7303},[7099],[35,47840,7283],{"className":47841},[7472],[35,47843,42397],{"className":47844,"style":7310},[7104,7145],[35,47846,7290],{"className":47847},[7314]," and an excited state ",[35,47850,47852,47869],{"className":47851},[7057],[35,47853,47855],{"className":47854},[7061],[7063,47856,47857],{"xmlns":7065},[7067,47858,47859,47867],{},[7070,47860,47861,47863,47865],{},[7223,47862,7283],{"stretchy":7289},[7076,47864,7817],{},[7223,47866,7290],{"stretchy":7289},[7083,47868,46420],{"encoding":7085},[35,47870,47872],{"className":47871,"ariaHidden":7091},[7090],[35,47873,47875,47878,47881,47884],{"className":47874},[7095],[35,47876],{"className":47877,"style":7303},[7099],[35,47879,7283],{"className":47880},[7472],[35,47882,7817],{"className":47883},[7104,7145],[35,47885,7290],{"className":47886},[7314],". In the absence of a microwave drive, the energy splitting is",[35,47889,47891],{"className":47890},[7318],[35,47892,47894,47933],{"className":47893},[7057],[35,47895,47897],{"className":47896},[7061],[7063,47898,47899],{"xmlns":7065,"display":7327},[7067,47900,47901,47930],{},[7070,47902,47903,47909,47911,47917,47919,47922,47928],{},[7699,47904,47905,47907],{},[7076,47906,10709],{},[7076,47908,7817],{},[7223,47910,7225],{},[7699,47912,47913,47915],{},[7076,47914,10709],{},[7076,47916,42397],{},[7223,47918,242],{},[7076,47920,47921],{"mathvariant":7282},"ℏ",[7699,47923,47924,47926],{},[7076,47925,46825],{},[7227,47927,5312],{},[7223,47929,2877],{"separator":7091},[7083,47931,47932],{"encoding":7085},"E_e-E_g=\\hbar\\omega_0,",[35,47934,47936,47991,48046],{"className":47935,"ariaHidden":7091},[7090],[35,47937,47939,47942,47982,47985,47988],{"className":47938},[7095],[35,47940],{"className":47941,"style":7718},[7099],[35,47943,47945,47948],{"className":47944},[7104],[35,47946,10709],{"className":47947,"style":10419},[7104,7145],[35,47949,47951],{"className":47950},[7113],[35,47952,47954,47974],{"className":47953},[7117,7731],[35,47955,47957,47971],{"className":47956},[7121],[35,47958,47960],{"className":47959,"style":9035},[7125],[35,47961,47962,47965],{"style":10541},[35,47963],{"className":47964,"style":7134},[7133],[35,47966,47968],{"className":47967},[7138,7139,7140,7141],[35,47969,7817],{"className":47970},[7104,7145,7141],[35,47972,7756],{"className":47973},[7755],[35,47975,47977],{"className":47976},[7121],[35,47978,47980],{"className":47979,"style":7763},[7125],[35,47981],{},[35,47983],{"className":47984,"style":7249},[7248],[35,47986,7225],{"className":47987},[7253],[35,47989],{"className":47990,"style":7249},[7248],[35,47992,47994,47997,48037,48040,48043],{"className":47993},[7095],[35,47995],{"className":47996,"style":12493},[7099],[35,47998,48000,48003],{"className":47999},[7104],[35,48001,10709],{"className":48002,"style":10419},[7104,7145],[35,48004,48006],{"className":48005},[7113],[35,48007,48009,48029],{"className":48008},[7117,7731],[35,48010,48012,48026],{"className":48011},[7121],[35,48013,48015],{"className":48014,"style":9035},[7125],[35,48016,48017,48020],{"style":10541},[35,48018],{"className":48019,"style":7134},[7133],[35,48021,48023],{"className":48022},[7138,7139,7140,7141],[35,48024,42397],{"className":48025,"style":7310},[7104,7145,7141],[35,48027,7756],{"className":48028},[7755],[35,48030,48032],{"className":48031},[7121],[35,48033,48035],{"className":48034,"style":11360},[7125],[35,48036],{},[35,48038],{"className":48039,"style":7450},[7248],[35,48041,242],{"className":48042},[7454],[35,48044],{"className":48045,"style":7450},[7248],[35,48047,48049,48053,48056,48096],{"className":48048},[7095],[35,48050],{"className":48051,"style":48052},[7099],"height:0.8833em;vertical-align:-0.1944em;",[35,48054,47921],{"className":48055},[7104],[35,48057,48059,48062],{"className":48058},[7104],[35,48060,46825],{"className":48061,"style":7310},[7104,7145],[35,48063,48065],{"className":48064},[7113],[35,48066,48068,48088],{"className":48067},[7117,7731],[35,48069,48071,48085],{"className":48070},[7121],[35,48072,48074],{"className":48073,"style":18794},[7125],[35,48075,48076,48079],{"style":9100},[35,48077],{"className":48078,"style":7134},[7133],[35,48080,48082],{"className":48081},[7138,7139,7140,7141],[35,48083,5312],{"className":48084},[7104,7141],[35,48086,7756],{"className":48087},[7755],[35,48089,48091],{"className":48090},[7121],[35,48092,48094],{"className":48093,"style":7763},[7125],[35,48095],{},[35,48097,2877],{"className":48098},[7492],[12,48100,48101],{},"where the angular transition frequency is",[35,48103,48105],{"className":48104},[7318],[35,48106,48108,48140],{"className":48107},[7057],[35,48109,48111],{"className":48110},[7061],[7063,48112,48113],{"xmlns":7065,"display":7327},[7067,48114,48115,48137],{},[7070,48116,48117,48123,48125,48127,48129,48135],{},[7699,48118,48119,48121],{},[7076,48120,46825],{},[7227,48122,5312],{},[7223,48124,242],{},[7227,48126,460],{},[7076,48128,13081],{},[7699,48130,48131,48133],{},[7076,48132,1675],{},[7227,48134,5312],{},[7076,48136,5122],{"mathvariant":7282},[7083,48138,48139],{"encoding":7085},"\\omega_0=2\\pi f_0.",[35,48141,48143,48198],{"className":48142,"ariaHidden":7091},[7090],[35,48144,48146,48149,48189,48192,48195],{"className":48145},[7095],[35,48147],{"className":48148,"style":29689},[7099],[35,48150,48152,48155],{"className":48151},[7104],[35,48153,46825],{"className":48154,"style":7310},[7104,7145],[35,48156,48158],{"className":48157},[7113],[35,48159,48161,48181],{"className":48160},[7117,7731],[35,48162,48164,48178],{"className":48163},[7121],[35,48165,48167],{"className":48166,"style":18794},[7125],[35,48168,48169,48172],{"style":9100},[35,48170],{"className":48171,"style":7134},[7133],[35,48173,48175],{"className":48174},[7138,7139,7140,7141],[35,48176,5312],{"className":48177},[7104,7141],[35,48179,7756],{"className":48180},[7755],[35,48182,48184],{"className":48183},[7121],[35,48185,48187],{"className":48186,"style":7763},[7125],[35,48188],{},[35,48190],{"className":48191,"style":7450},[7248],[35,48193,242],{"className":48194},[7454],[35,48196],{"className":48197,"style":7450},[7248],[35,48199,48201,48204,48207,48210,48250],{"className":48200},[7095],[35,48202],{"className":48203,"style":11562},[7099],[35,48205,460],{"className":48206},[7104],[35,48208,13081],{"className":48209,"style":7310},[7104,7145],[35,48211,48213,48216],{"className":48212},[7104],[35,48214,1675],{"className":48215,"style":31824},[7104,7145],[35,48217,48219],{"className":48218},[7113],[35,48220,48222,48242],{"className":48221},[7117,7731],[35,48223,48225,48239],{"className":48224},[7121],[35,48226,48228],{"className":48227,"style":18794},[7125],[35,48229,48230,48233],{"style":46746},[35,48231],{"className":48232,"style":7134},[7133],[35,48234,48236],{"className":48235},[7138,7139,7140,7141],[35,48237,5312],{"className":48238},[7104,7141],[35,48240,7756],{"className":48241},[7755],[35,48243,48245],{"className":48244},[7121],[35,48246,48248],{"className":48247,"style":7763},[7125],[35,48249],{},[35,48251,5122],{"className":48252},[7104],[12,48254,48255],{},"The default configuration uses a transition frequency of",[35,48257,48259],{"className":48258},[7318],[35,48260,48262,48298],{"className":48261},[7057],[35,48263,48265],{"className":48264},[7061],[7063,48266,48267],{"xmlns":7065,"display":7327},[7067,48268,48269,48295],{},[7070,48270,48271,48277,48279,48282,48284,48293],{},[7699,48272,48273,48275],{},[7076,48274,1675],{},[7227,48276,5312],{},[7223,48278,242],{},[7227,48280,48281],{},"5.000",[10694,48283,10696],{},[7070,48285,48286,48289,48291],{},[7076,48287,48288],{"mathvariant":7282},"G",[7076,48290,7669],{"mathvariant":7282},[7076,48292,300],{"mathvariant":7282},[7076,48294,5122],{"mathvariant":7282},[7083,48296,48297],{"encoding":7085},"f_0=5.000~\\mathrm{GHz}.",[35,48299,48301,48356],{"className":48300,"ariaHidden":7091},[7090],[35,48302,48304,48307,48347,48350,48353],{"className":48303},[7095],[35,48305],{"className":48306,"style":11562},[7099],[35,48308,48310,48313],{"className":48309},[7104],[35,48311,1675],{"className":48312,"style":31824},[7104,7145],[35,48314,48316],{"className":48315},[7113],[35,48317,48319,48339],{"className":48318},[7117,7731],[35,48320,48322,48336],{"className":48321},[7121],[35,48323,48325],{"className":48324,"style":18794},[7125],[35,48326,48327,48330],{"style":46746},[35,48328],{"className":48329,"style":7134},[7133],[35,48331,48333],{"className":48332},[7138,7139,7140,7141],[35,48334,5312],{"className":48335},[7104,7141],[35,48337,7756],{"className":48338},[7755],[35,48340,48342],{"className":48341},[7121],[35,48343,48345],{"className":48344,"style":7763},[7125],[35,48346],{},[35,48348],{"className":48349,"style":7450},[7248],[35,48351,242],{"className":48352},[7454],[35,48354],{"className":48355,"style":7450},[7248],[35,48357,48359,48362,48365,48369,48377],{"className":48358},[7095],[35,48360],{"className":48361,"style":7100},[7099],[35,48363,48281],{"className":48364},[7104],[35,48366,10696],{"className":48367},[7248,48368],"nobreak",[35,48370,48372],{"className":48371},[7104],[35,48373,48376],{"className":48374},[7104,48375],"mathrm","GHz",[35,48378,5122],{"className":48379},[7104],[12,48381,48382,48383,48411,48412,15316,48450,48488],{},"Applying a microwave drive with frequency ",[35,48384,48386,48399],{"className":48385},[7057],[35,48387,48389],{"className":48388},[7061],[7063,48390,48391],{"xmlns":7065},[7067,48392,48393,48397],{},[7070,48394,48395],{},[7076,48396,1675],{},[7083,48398,1675],{"encoding":7085},[35,48400,48402],{"className":48401,"ariaHidden":7091},[7090],[35,48403,48405,48408],{"className":48404},[7095],[35,48406],{"className":48407,"style":11562},[7099],[35,48409,1675],{"className":48410,"style":31824},[7104,7145]," induces coherent rotations between ",[35,48413,48415,48432],{"className":48414},[7057],[35,48416,48418],{"className":48417},[7061],[7063,48419,48420],{"xmlns":7065},[7067,48421,48422,48430],{},[7070,48423,48424,48426,48428],{},[7223,48425,7283],{"stretchy":7289},[7076,48427,42397],{},[7223,48429,7290],{"stretchy":7289},[7083,48431,46374],{"encoding":7085},[35,48433,48435],{"className":48434,"ariaHidden":7091},[7090],[35,48436,48438,48441,48444,48447],{"className":48437},[7095],[35,48439],{"className":48440,"style":7303},[7099],[35,48442,7283],{"className":48443},[7472],[35,48445,42397],{"className":48446,"style":7310},[7104,7145],[35,48448,7290],{"className":48449},[7314],[35,48451,48453,48470],{"className":48452},[7057],[35,48454,48456],{"className":48455},[7061],[7063,48457,48458],{"xmlns":7065},[7067,48459,48460,48468],{},[7070,48461,48462,48464,48466],{},[7223,48463,7283],{"stretchy":7289},[7076,48465,7817],{},[7223,48467,7290],{"stretchy":7289},[7083,48469,46420],{"encoding":7085},[35,48471,48473],{"className":48472,"ariaHidden":7091},[7090],[35,48474,48476,48479,48482,48485],{"className":48475},[7095],[35,48477],{"className":48478,"style":7303},[7099],[35,48480,7283],{"className":48481},[7472],[35,48483,7817],{"className":48484},[7104,7145],[35,48486,7290],{"className":48487},[7314],". Exact resonance occurs when",[35,48490,48492],{"className":48491},[7318],[35,48493,48495,48519],{"className":48494},[7057],[35,48496,48498],{"className":48497},[7061],[7063,48499,48500],{"xmlns":7065,"display":7327},[7067,48501,48502,48516],{},[7070,48503,48504,48506,48508,48514],{},[7076,48505,1675],{},[7223,48507,242],{},[7699,48509,48510,48512],{},[7076,48511,1675],{},[7227,48513,5312],{},[7076,48515,5122],{"mathvariant":7282},[7083,48517,48518],{"encoding":7085},"f=f_0.",[35,48520,48522,48540],{"className":48521,"ariaHidden":7091},[7090],[35,48523,48525,48528,48531,48534,48537],{"className":48524},[7095],[35,48526],{"className":48527,"style":11562},[7099],[35,48529,1675],{"className":48530,"style":31824},[7104,7145],[35,48532],{"className":48533,"style":7450},[7248],[35,48535,242],{"className":48536},[7454],[35,48538],{"className":48539,"style":7450},[7248],[35,48541,48543,48546,48586],{"className":48542},[7095],[35,48544],{"className":48545,"style":11562},[7099],[35,48547,48549,48552],{"className":48548},[7104],[35,48550,1675],{"className":48551,"style":31824},[7104,7145],[35,48553,48555],{"className":48554},[7113],[35,48556,48558,48578],{"className":48557},[7117,7731],[35,48559,48561,48575],{"className":48560},[7121],[35,48562,48564],{"className":48563,"style":18794},[7125],[35,48565,48566,48569],{"style":46746},[35,48567],{"className":48568,"style":7134},[7133],[35,48570,48572],{"className":48571},[7138,7139,7140,7141],[35,48573,5312],{"className":48574},[7104,7141],[35,48576,7756],{"className":48577},[7755],[35,48579,48581],{"className":48580},[7121],[35,48582,48584],{"className":48583,"style":7763},[7125],[35,48585],{},[35,48587,5122],{"className":48588},[7104],[12,48590,48591],{},"At resonance, the drive produces ideal Rabi oscillations. Away from resonance, detuning changes the oscillation frequency and reduces the maximum achievable excited-state probability. This same two-level control structure appears across superconducting qubits, spin qubits, trapped ions, neutral atoms, and nitrogen-vacancy centers whenever a selected transition can be treated as an effective qubit.",[5019,48593,48595],{"id":48594},"mathematical-foundation","Mathematical Foundation",[12,48597,48598],{},"The quantum state is a normalized vector in a two-dimensional Hilbert space,",[35,48600,48602],{"className":48601},[7318],[35,48603,48605,48645],{"className":48604},[7057],[35,48606,48608],{"className":48607},[7061],[7063,48609,48610],{"xmlns":7065,"display":7327},[7067,48611,48612,48642],{},[7070,48613,48614,48616,48618,48620,48622,48624,48626,48628,48630,48632,48634,48636,48638,48640],{},[7223,48615,7283],{"stretchy":7289},[7076,48617,8322],{},[7223,48619,7290],{"stretchy":7289},[7223,48621,242],{},[7076,48623,46506],{},[7223,48625,7283],{"stretchy":7289},[7076,48627,42397],{},[7223,48629,7290],{"stretchy":7289},[7223,48631,445],{},[7076,48633,41150],{},[7223,48635,7283],{"stretchy":7289},[7076,48637,7817],{},[7223,48639,7290],{"stretchy":7289},[7223,48641,2877],{"separator":7091},[7083,48643,48644],{"encoding":7085},"\\lvert \\psi\\rangle=\\alpha\\lvert g\\rangle+\\beta\\lvert e\\rangle,",[35,48646,48648,48672,48699],{"className":48647,"ariaHidden":7091},[7090],[35,48649,48651,48654,48657,48660,48663,48666,48669],{"className":48650},[7095],[35,48652],{"className":48653,"style":7303},[7099],[35,48655,7283],{"className":48656},[7472],[35,48658,8322],{"className":48659,"style":7310},[7104,7145],[35,48661,7290],{"className":48662},[7314],[35,48664],{"className":48665,"style":7450},[7248],[35,48667,242],{"className":48668},[7454],[35,48670],{"className":48671,"style":7450},[7248],[35,48673,48675,48678,48681,48684,48687,48690,48693,48696],{"className":48674},[7095],[35,48676],{"className":48677,"style":7303},[7099],[35,48679,46506],{"className":48680,"style":46526},[7104,7145],[35,48682,7283],{"className":48683},[7472],[35,48685,42397],{"className":48686,"style":7310},[7104,7145],[35,48688,7290],{"className":48689},[7314],[35,48691],{"className":48692,"style":7249},[7248],[35,48694,445],{"className":48695},[7253],[35,48697],{"className":48698,"style":7249},[7248],[35,48700,48702,48705,48708,48711,48714,48717],{"className":48701},[7095],[35,48703],{"className":48704,"style":7303},[7099],[35,48706,41150],{"className":48707,"style":41244},[7104,7145],[35,48709,7283],{"className":48710},[7472],[35,48712,7817],{"className":48713},[7104,7145],[35,48715,7290],{"className":48716},[7314],[35,48718,2877],{"className":48719},[7492],[12,48721,32939,48722,15316,48751,48780],{},[35,48723,48725,48739],{"className":48724},[7057],[35,48726,48728],{"className":48727},[7061],[7063,48729,48730],{"xmlns":7065},[7067,48731,48732,48736],{},[7070,48733,48734],{},[7076,48735,46506],{},[7083,48737,48738],{"encoding":7085},"\\alpha",[35,48740,48742],{"className":48741,"ariaHidden":7091},[7090],[35,48743,48745,48748],{"className":48744},[7095],[35,48746],{"className":48747,"style":7173},[7099],[35,48749,46506],{"className":48750,"style":46526},[7104,7145],[35,48752,48754,48768],{"className":48753},[7057],[35,48755,48757],{"className":48756},[7061],[7063,48758,48759],{"xmlns":7065},[7067,48760,48761,48765],{},[7070,48762,48763],{},[7076,48764,41150],{},[7083,48766,48767],{"encoding":7085},"\\beta",[35,48769,48771],{"className":48770,"ariaHidden":7091},[7090],[35,48772,48774,48777],{"className":48773},[7095],[35,48775],{"className":48776,"style":11562},[7099],[35,48778,41150],{"className":48779,"style":41244},[7104,7145]," are complex probability amplitudes. Normalization requires",[35,48782,48784],{"className":48783},[7318],[35,48785,48787,48826],{"className":48786},[7057],[35,48788,48790],{"className":48789},[7061],[7063,48791,48792],{"xmlns":7065,"display":7327},[7067,48793,48794,48823],{},[7070,48795,48796,48798,48800,48806,48808,48810,48812,48818,48820],{},[7076,48797,7283],{"mathvariant":7282},[7076,48799,46506],{},[7073,48801,48802,48804],{},[7076,48803,7283],{"mathvariant":7282},[7227,48805,460],{},[7223,48807,445],{},[7076,48809,7283],{"mathvariant":7282},[7076,48811,41150],{},[7073,48813,48814,48816],{},[7076,48815,7283],{"mathvariant":7282},[7227,48817,460],{},[7223,48819,242],{},[7227,48821,48822],{},"1.",[7083,48824,48825],{"encoding":7085},"|\\alpha|^2+|\\beta|^2=1.",[35,48827,48829,48880,48930],{"className":48828,"ariaHidden":7091},[7090],[35,48830,48832,48836,48839,48842,48871,48874,48877],{"className":48831},[7095],[35,48833],{"className":48834,"style":48835},[7099],"height:1.1141em;vertical-align:-0.25em;",[35,48837,7283],{"className":48838},[7104],[35,48840,46506],{"className":48841,"style":46526},[7104,7145],[35,48843,48845,48848],{"className":48844},[7104],[35,48846,7283],{"className":48847},[7104],[35,48849,48851],{"className":48850},[7113],[35,48852,48854],{"className":48853},[7117],[35,48855,48857],{"className":48856},[7121],[35,48858,48860],{"className":48859,"style":7533},[7125],[35,48861,48862,48865],{"style":7437},[35,48863],{"className":48864,"style":7134},[7133],[35,48866,48868],{"className":48867},[7138,7139,7140,7141],[35,48869,460],{"className":48870},[7104,7141],[35,48872],{"className":48873,"style":7249},[7248],[35,48875,445],{"className":48876},[7253],[35,48878],{"className":48879,"style":7249},[7248],[35,48881,48883,48886,48889,48892,48921,48924,48927],{"className":48882},[7095],[35,48884],{"className":48885,"style":48835},[7099],[35,48887,7283],{"className":48888},[7104],[35,48890,41150],{"className":48891,"style":41244},[7104,7145],[35,48893,48895,48898],{"className":48894},[7104],[35,48896,7283],{"className":48897},[7104],[35,48899,48901],{"className":48900},[7113],[35,48902,48904],{"className":48903},[7117],[35,48905,48907],{"className":48906},[7121],[35,48908,48910],{"className":48909,"style":7533},[7125],[35,48911,48912,48915],{"style":7437},[35,48913],{"className":48914,"style":7134},[7133],[35,48916,48918],{"className":48917},[7138,7139,7140,7141],[35,48919,460],{"className":48920},[7104,7141],[35,48922],{"className":48923,"style":7450},[7248],[35,48925,242],{"className":48926},[7454],[35,48928],{"className":48929,"style":7450},[7248],[35,48931,48933,48936],{"className":48932},[7095],[35,48934],{"className":48935,"style":7263},[7099],[35,48937,48822],{"className":48938},[7104],[12,48940,48941],{},"The Born rule gives the probability of measuring the excited state,",[35,48943,48945],{"className":48944},[7318],[35,48946,48948,49000],{"className":48947},[7057],[35,48949,48951],{"className":48950},[7061],[7063,48952,48953],{"xmlns":7065,"display":7327},[7067,48954,48955,48997],{},[7070,48956,48957,48963,48965,48967,48969,48971,48973,48975,48977,48983,48985,48987,48989,48995],{},[7699,48958,48959,48961],{},[7076,48960,7705],{},[7076,48962,7817],{},[7223,48964,242],{},[7076,48966,7283],{"mathvariant":7282},[7223,48968,8953],{"stretchy":7289},[7076,48970,7817],{},[7223,48972,7283],{"stretchy":7289},[7076,48974,8322],{},[7223,48976,7290],{"stretchy":7289},[7073,48978,48979,48981],{},[7076,48980,7283],{"mathvariant":7282},[7227,48982,460],{},[7223,48984,242],{},[7076,48986,7283],{"mathvariant":7282},[7076,48988,41150],{},[7073,48990,48991,48993],{},[7076,48992,7283],{"mathvariant":7282},[7227,48994,460],{},[7076,48996,5122],{"mathvariant":7282},[7083,48998,48999],{"encoding":7085},"P_e=|\\langle e\\lvert\\psi\\rangle|^2=|\\beta|^2.",[35,49001,49003,49058,49120],{"className":49002,"ariaHidden":7091},[7090],[35,49004,49006,49009,49049,49052,49055],{"className":49005},[7095],[35,49007],{"className":49008,"style":7718},[7099],[35,49010,49012,49015],{"className":49011},[7104],[35,49013,7705],{"className":49014,"style":7751},[7104,7145],[35,49016,49018],{"className":49017},[7113],[35,49019,49021,49041],{"className":49020},[7117,7731],[35,49022,49024,49038],{"className":49023},[7121],[35,49025,49027],{"className":49026,"style":9035},[7125],[35,49028,49029,49032],{"style":12163},[35,49030],{"className":49031,"style":7134},[7133],[35,49033,49035],{"className":49034},[7138,7139,7140,7141],[35,49036,7817],{"className":49037},[7104,7145,7141],[35,49039,7756],{"className":49040},[7755],[35,49042,49044],{"className":49043},[7121],[35,49045,49047],{"className":49046,"style":7763},[7125],[35,49048],{},[35,49050],{"className":49051,"style":7450},[7248],[35,49053,242],{"className":49054},[7454],[35,49056],{"className":49057,"style":7450},[7248],[35,49059,49061,49064,49067,49070,49073,49076,49079,49082,49111,49114,49117],{"className":49060},[7095],[35,49062],{"className":49063,"style":48835},[7099],[35,49065,7283],{"className":49066},[7104],[35,49068,8953],{"className":49069},[7472],[35,49071,7817],{"className":49072},[7104,7145],[35,49074,7283],{"className":49075},[7472],[35,49077,8322],{"className":49078,"style":7310},[7104,7145],[35,49080,7290],{"className":49081},[7314],[35,49083,49085,49088],{"className":49084},[7104],[35,49086,7283],{"className":49087},[7104],[35,49089,49091],{"className":49090},[7113],[35,49092,49094],{"className":49093},[7117],[35,49095,49097],{"className":49096},[7121],[35,49098,49100],{"className":49099,"style":7533},[7125],[35,49101,49102,49105],{"style":7437},[35,49103],{"className":49104,"style":7134},[7133],[35,49106,49108],{"className":49107},[7138,7139,7140,7141],[35,49109,460],{"className":49110},[7104,7141],[35,49112],{"className":49113,"style":7450},[7248],[35,49115,242],{"className":49116},[7454],[35,49118],{"className":49119,"style":7450},[7248],[35,49121,49123,49126,49129,49132,49161],{"className":49122},[7095],[35,49124],{"className":49125,"style":48835},[7099],[35,49127,7283],{"className":49128},[7104],[35,49130,41150],{"className":49131,"style":41244},[7104,7145],[35,49133,49135,49138],{"className":49134},[7104],[35,49136,7283],{"className":49137},[7104],[35,49139,49141],{"className":49140},[7113],[35,49142,49144],{"className":49143},[7117],[35,49145,49147],{"className":49146},[7121],[35,49148,49150],{"className":49149,"style":7533},[7125],[35,49151,49152,49155],{"style":7437},[35,49153],{"className":49154,"style":7134},[7133],[35,49156,49158],{"className":49157},[7138,7139,7140,7141],[35,49159,460],{"className":49160},[7104,7141],[35,49162,5122],{"className":49163},[7104],[12,49165,49166],{},"Pauli matrices then provide a natural operator basis for the two-level system,",[35,49168,49170],{"className":49169},[7318],[35,49171,49173,49337],{"className":49172},[7057],[35,49174,49176],{"className":49175},[7061],[7063,49177,49178],{"xmlns":7065,"display":7327},[7067,49179,49180,49334],{},[7070,49181,49182,49189,49191,49228,49230,49232,49238,49240,49280,49282,49284,49290,49292,49332],{},[7699,49183,49184,49187],{},[7076,49185,49186],{},"σ",[7076,49188,13657],{},[7223,49190,242],{},[7070,49192,49193,49195,49226],{},[7223,49194,1254],{"fence":7091},[13970,49196,49198,49212],{"rowspacing":13972,"columnalign":49197,"columnspacing":13973},"center 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-1\\end{bmatrix}.",[35,49338,49340,49395,49585,49772],{"className":49339,"ariaHidden":7091},[7090],[35,49341,49343,49346,49386,49389,49392],{"className":49342},[7095],[35,49344],{"className":49345,"style":29689},[7099],[35,49347,49349,49352],{"className":49348},[7104],[35,49350,49186],{"className":49351,"style":7310},[7104,7145],[35,49353,49355],{"className":49354},[7113],[35,49356,49358,49378],{"className":49357},[7117,7731],[35,49359,49361,49375],{"className":49360},[7121],[35,49362,49364],{"className":49363,"style":9035},[7125],[35,49365,49366,49369],{"style":9100},[35,49367],{"className":49368,"style":7134},[7133],[35,49370,49372],{"className":49371},[7138,7139,7140,7141],[35,49373,13657],{"className":49374},[7104,7145,7141],[35,49376,7756],{"className":49377},[7755],[35,49379,49381],{"className":49380},[7121],[35,49382,49384],{"className":49383,"style":7763},[7125],[35,49385],{},[35,49387],{"className":49388,"style":7450},[7248],[35,49390,242],{"className":49391},[7454],[35,49393],{"className":49394,"style":7450},[7248],[35,49396,49398,49401,49524,49527,49530,49533,49536,49576,49579,49582],{"className":49397},[7095],[35,49399],{"className":49400,"style":42152},[7099],[35,49402,49404,49410,49518],{"className":49403},[7468],[35,49405,49407],{"className":49406,"style":7474},[7472,7473],[35,49408,1254],{"className":49409},[7478,7140],[35,49411,49413],{"className":49412},[7104],[35,49414,49416,49465,49470,49473],{"className":49415},[13970],[35,49417,49419],{"className":49418},[14229],[35,49420,49422,49456],{"className":49421},[7117,7731],[35,49423,49425,49453],{"className":49424},[7121],[35,49426,49429,49441],{"className":49427,"style":49428},[7125],"height:1.45em;",[35,49430,49432,49435],{"style":49431},"top:-3.61em;",[35,49433],{"className":49434,"style":8839},[7133],[35,49436,49438],{"className":49437},[7104],[35,49439,5312],{"className":49440},[7104],[35,49442,49444,49447],{"style":49443},"top:-2.41em;",[35,49445],{"className":49446,"style":8839},[7133],[35,49448,49450],{"className":49449},[7104],[35,49451,500],{"className":49452},[7104],[35,49454,7756],{"className":49455},[7755],[35,49457,49459],{"className":49458},[7121],[35,49460,49463],{"className":49461,"style":49462},[7125],"height:0.95em;",[35,49464],{},[35,49466],{"className":49467,"style":49469},[49468],"arraycolsep","width:0.5em;",[35,49471],{"className":49472,"style":49469},[49468],[35,49474,49476],{"className":49475},[14229],[35,49477,49479,49510],{"className":49478},[7117,7731],[35,49480,49482,49507],{"className":49481},[7121],[35,49483,49485,49496],{"className":49484,"style":49428},[7125],[35,49486,49487,49490],{"style":49431},[35,49488],{"className":49489,"style":8839},[7133],[35,49491,49493],{"className":49492},[7104],[35,49494,500],{"className":49495},[7104],[35,49497,49498,49501],{"style":49443},[35,49499],{"className":49500,"style":8839},[7133],[35,49502,49504],{"className":49503},[7104],[35,49505,5312],{"className":49506},[7104],[35,49508,7756],{"className":49509},[7755],[35,49511,49513],{"className":49512},[7121],[35,49514,49516],{"className":49515,"style":49462},[7125],[35,49517],{},[35,49519,49521],{"className":49520,"style":7474},[7314,7473],[35,49522,16788],{"className":49523},[7478,7140],[35,49525],{"className":49526,"style":7496},[7248],[35,49528,2877],{"className":49529},[7492],[35,49531],{"className":49532,"style":23142},[7248],[35,49534],{"className":49535,"style":7496},[7248],[35,49537,49539,49542],{"className":49538},[7104],[35,49540,49186],{"className":49541,"style":7310},[7104,7145],[35,49543,49545],{"className":49544},[7113],[35,49546,49548,49568],{"className":49547},[7117,7731],[35,49549,49551,49565],{"className":49550},[7121],[35,49552,49554],{"className":49553,"style":9035},[7125],[35,49555,49556,49559],{"style":9100},[35,49557],{"className":49558,"style":7134},[7133],[35,49560,49562],{"className":49561},[7138,7139,7140,7141],[35,49563,24968],{"className":49564,"style":7310},[7104,7145,7141],[35,49566,7756],{"className":49567},[7755],[35,49569,49571],{"className":49570},[7121],[35,49572,49574],{"className":49573,"style":11360},[7125],[35,49575],{},[35,49577],{"className":49578,"style":7450},[7248],[35,49580,242],{"className":49581},[7454],[35,49583],{"className":49584,"style":7450},[7248],[35,49586,49588,49591,49711,49714,49717,49720,49723,49763,49766,49769],{"className":49587},[7095],[35,49589],{"className":49590,"style":42152},[7099],[35,49592,49594,49600,49705],{"className":49593},[7468],[35,49595,49597],{"className":49596,"style":7474},[7472,7473],[35,49598,1254],{"className":49599},[7478,7140],[35,49601,49603],{"className":49602},[7104],[35,49604,49606,49651,49654,49657],{"className":49605},[13970],[35,49607,49609],{"className":49608},[14229],[35,49610,49612,49643],{"className":49611},[7117,7731],[35,49613,49615,49640],{"className":49614},[7121],[35,49616,49618,49629],{"className":49617,"style":49428},[7125],[35,49619,49620,49623],{"style":49431},[35,49621],{"className":49622,"style":8839},[7133],[35,49624,4962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The drive strength supplies the transverse control component. Applied detuning supplies the longitudinal control component.",[5019,49942,49944],{"id":49943},"rotating-frame-hamiltonian","Rotating-Frame Hamiltonian",[12,49946,49947],{},"In the laboratory frame, a minimal semiclassical Hamiltonian for the driven qubit is",[35,49949,49951],{"className":49950},[7318],[35,49952,49954,50022],{"className":49953},[7057],[35,49955,49957],{"className":49956},[7061],[7063,49958,49959],{"xmlns":7065,"display":7327},[7067,49960,49961,50019],{},[7070,49962,49963,49965,49967,49969,49971,49973,49987,49993,49995,49997,49999,50001,50003,50005,50007,50009,50011,50017],{},[7076,49964,7669],{},[7223,49966,275],{"stretchy":7289},[7076,49968,7829],{},[7223,49970,1694],{"stretchy":7289},[7223,49972,242],{},[14009,49974,49975,49985],{},[7070,49976,49977,49979],{},[7076,49978,47921],{"mathvariant":7282},[7699,49980,49981,49983],{},[7076,49982,46825],{},[7227,49984,5312],{},[7227,49986,460],{},[7699,49988,49989,49991],{},[7076,49990,49186],{},[7076,49992,300],{},[7223,49994,445],{},[7076,49996,47921],{"mathvariant":7282},[7076,49998,47141],{"mathvariant":7282},[7076,50000,42416],{},[7223,50002,41175],{},[7223,50004,275],{"stretchy":7289},[7076,50006,46825],{},[7076,50008,7829],{},[7223,50010,1694],{"stretchy":7289},[7699,50012,50013,50015],{},[7076,50014,49186],{},[7076,50016,13657],{},[7223,50018,2877],{"separator":7091},[7083,50020,50021],{"encoding":7085},"H(t)=\\frac{\\hbar\\omega_0}{2}\\sigma_z+\\hbar\\Omega\\cos(\\omega t)\\sigma_x,",[35,50023,50025,50052,50211],{"className":50024,"ariaHidden":7091},[7090],[35,50026,50028,50031,50034,50037,50040,50043,50046,50049],{"className":50027},[7095],[35,50029],{"className":50030,"style":7303},[7099],[35,50032,7669],{"className":50033,"style":7684},[7104,7145],[35,50035,275],{"className":50036},[7472],[35,50038,7829],{"className":50039},[7104,7145],[35,50041,1694],{"className":50042},[7314],[35,50044],{"className":50045,"style":7450},[7248],[35,50047,242],{"className":50048},[7454],[35,50050],{"className":50051,"style":7450},[7248],[35,50053,50055,50059,50162,50202,50205,50208],{"className":50054},[7095],[35,50056],{"className":50057,"style":50058},[7099],"height:2.0519em;vertical-align:-0.686em;",[35,50060,50062,50065,50159],{"className":50061},[7104],[35,50063],{"className":50064},[7472,14311],[35,50066,50068],{"className":50067},[14009],[35,50069,50071,50151],{"className":50070},[7117,7731],[35,50072,50074,50148],{"className":50073},[7121],[35,50075,50078,50089,50097],{"className":50076,"style":50077},[7125],"height:1.3659em;",[35,50079,50080,50083],{"style":16234},[35,50081],{"className":50082,"style":8839},[7133],[35,50084,50086],{"className":50085},[7104],[35,50087,460],{"className":50088},[7104],[35,50090,50091,50094],{"style":14395},[35,50092],{"className":50093,"style":8839},[7133],[35,50095],{"className":50096,"style":14403},[14402],[35,50098,50099,50102],{"style":14406},[35,50100],{"className":50101,"style":8839},[7133],[35,50103,50105,50108],{"className":50104},[7104],[35,50106,47921],{"className":50107},[7104],[35,50109,50111,50114],{"className":50110},[7104],[35,50112,46825],{"className":50113,"style":7310},[7104,7145],[35,50115,50117],{"className":50116},[7113],[35,50118,50120,50140],{"className":50119},[7117,7731],[35,50121,50123,50137],{"className":50122},[7121],[35,50124,50126],{"className":50125,"style":18794},[7125],[35,50127,50128,50131],{"style":9100},[35,50129],{"className":50130,"style":7134},[7133],[35,50132,50134],{"className":50133},[7138,7139,7140,7141],[35,50135,5312],{"className":50136},[7104,7141],[35,50138,7756],{"className":50139},[7755],[35,50141,50143],{"className":50142},[7121],[35,50144,50146],{"className":50145,"style":7763},[7125],[35,50147],{},[35,50149,7756],{"className":50150},[7755],[35,50152,50154],{"className":50153},[7121],[35,50155,50157],{"className":50156,"style":16272},[7125],[35,50158],{},[35,50160],{"className":50161},[7314,14311],[35,50163,50165,50168],{"className":50164},[7104],[35,50166,49186],{"className":50167,"style":7310},[7104,7145],[35,50169,50171],{"className":50170},[7113],[35,50172,50174,50194],{"className":50173},[7117,7731],[35,50175,50177,50191],{"className":50176},[7121],[35,50178,50180],{"className":50179,"style":9035},[7125],[35,50181,50182,50185],{"style":9100},[35,50183],{"className":50184,"style":7134},[7133],[35,50186,50188],{"className":50187},[7138,7139,7140,7141],[35,50189,300],{"className":50190,"style":25095},[7104,7145,7141],[35,50192,7756],{"className":50193},[7755],[35,50195,50197],{"className":50196},[7121],[35,50198,50200],{"className":50199,"style":7763},[7125],[35,50201],{},[35,50203],{"className":50204,"style":7249},[7248],[35,50206,445],{"className":50207},[7253],[35,50209],{"className":50210,"style":7249},[7248],[35,50212,50214,50217,50221,50224,50227,50230,50233,50236,50239,50279],{"className":50213},[7095],[35,50215],{"className":50216,"style":7303},[7099],[35,50218,50220],{"className":50219},[7104],"ℏΩ",[35,50222],{"className":50223,"style":7496},[7248],[35,50225,42416],{"className":50226},[12096],[35,50228,275],{"className":50229},[7472],[35,50231,46825],{"className":50232,"style":7310},[7104,7145],[35,50234,7829],{"className":50235},[7104,7145],[35,50237,1694],{"className":50238},[7314],[35,50240,50242,50245],{"className":50241},[7104],[35,50243,49186],{"className":50244,"style":7310},[7104,7145],[35,50246,50248],{"className":50247},[7113],[35,50249,50251,50271],{"className":50250},[7117,7731],[35,50252,50254,50268],{"className":50253},[7121],[35,50255,50257],{"className":50256,"style":9035},[7125],[35,50258,50259,50262],{"style":9100},[35,50260],{"className":50261,"style":7134},[7133],[35,50263,50265],{"className":50264},[7138,7139,7140,7141],[35,50266,13657],{"className":50267},[7104,7145,7141],[35,50269,7756],{"className":50270},[7755],[35,50272,50274],{"className":50273},[7121],[35,50275,50277],{"className":50276,"style":7763},[7125],[35,50278],{},[35,50280,2877],{"className":50281},[7492],[12,50283,32939,50284,50344,50345,50373,50374,50411],{},[35,50285,50287,50308],{"className":50286},[7057],[35,50288,50290],{"className":50289},[7061],[7063,50291,50292],{"xmlns":7065},[7067,50293,50294,50306],{},[7070,50295,50296,50298,50300,50302,50304],{},[7076,50297,46825],{},[7223,50299,242],{},[7227,50301,460],{},[7076,50303,13081],{},[7076,50305,1675],{},[7083,50307,47086],{"encoding":7085},[35,50309,50311,50329],{"className":50310,"ariaHidden":7091},[7090],[35,50312,50314,50317,50320,50323,50326],{"className":50313},[7095],[35,50315],{"className":50316,"style":7173},[7099],[35,50318,46825],{"className":50319,"style":7310},[7104,7145],[35,50321],{"className":50322,"style":7450},[7248],[35,50324,242],{"className":50325},[7454],[35,50327],{"className":50328,"style":7450},[7248],[35,50330,50332,50335,50338,50341],{"className":50331},[7095],[35,50333],{"className":50334,"style":11562},[7099],[35,50336,460],{"className":50337},[7104],[35,50339,13081],{"className":50340,"style":7310},[7104,7145],[35,50342,1675],{"className":50343,"style":31824},[7104,7145]," is the drive angular frequency and ",[35,50346,50348,50361],{"className":50347},[7057],[35,50349,50351],{"className":50350},[7061],[7063,50352,50353],{"xmlns":7065},[7067,50354,50355,50359],{},[7070,50356,50357],{},[7076,50358,47141],{"mathvariant":7282},[7083,50360,47144],{"encoding":7085},[35,50362,50364],{"className":50363,"ariaHidden":7091},[7090],[35,50365,50367,50370],{"className":50366},[7095],[35,50368],{"className":50369,"style":7100},[7099],[35,50371,47141],{"className":50372},[7104]," is the on-resonance Rabi angular frequency. The computational model specifies the drive strength using ",[35,50375,50377,50396],{"className":50376},[7057],[35,50378,50380],{"className":50379},[7061],[7063,50381,50382],{"xmlns":7065},[7067,50383,50384,50394],{},[7070,50385,50386,50388,50390,50392],{},[7076,50387,47141],{"mathvariant":7282},[7076,50389,398],{"mathvariant":7282},[7227,50391,460],{},[7076,50393,13081],{},[7083,50395,47186],{"encoding":7085},[35,50397,50399],{"className":50398,"ariaHidden":7091},[7090],[35,50400,50402,50405,50408],{"className":50401},[7095],[35,50403],{"className":50404,"style":7303},[7099],[35,50406,47199],{"className":50407},[7104],[35,50409,13081],{"className":50410,"style":7310},[7104,7145]," in MHz. Transforming into a frame rotating at the drive frequency and applying the rotating-wave approximation gives the effective time-independent Hamiltonian",[35,50413,50415],{"className":50414},[7318],[35,50416,50418,50477],{"className":50417},[7057],[35,50419,50421],{"className":50420},[7061],[7063,50422,50423],{"xmlns":7065,"display":7327},[7067,50424,50425,50474],{},[7070,50426,50427,50440,50442,50448,50472],{},[7699,50428,50429,50431],{},[7076,50430,7669],{},[7070,50432,50433,50435,50438],{},[7076,50434,24883],{"mathvariant":7282},[7076,50436,50437],{"mathvariant":7282},"W",[7076,50439,7192],{"mathvariant":7282},[7223,50441,242],{},[14009,50443,50444,50446],{},[7076,50445,47921],{"mathvariant":7282},[7227,50447,460],{},[7070,50449,50450,50452,50454,50460,50462,50464,50470],{},[7223,50451,275],{"fence":7091},[7076,50453,47224],{"mathvariant":7282},[7699,50455,50456,50458],{},[7076,50457,49186],{},[7076,50459,300],{},[7223,50461,445],{},[7076,50463,47141],{"mathvariant":7282},[7699,50465,50466,50468],{},[7076,50467,49186],{},[7076,50469,13657],{},[7223,50471,1694],{"fence":7091},[7223,50473,2877],{"separator":7091},[7083,50475,50476],{"encoding":7085},"H_{\\mathrm{RWA}}=\\frac{\\hbar}{2}\\left(\\Delta\\sigma_z+\\Omega\\sigma_x\\right),",[35,50478,50480,50541],{"className":50479,"ariaHidden":7091},[7090],[35,50481,50483,50486,50532,50535,50538],{"className":50482},[7095],[35,50484],{"className":50485,"style":7718},[7099],[35,50487,50489,50492],{"className":50488},[7104],[35,50490,7669],{"className":50491,"style":7684},[7104,7145],[35,50493,50495],{"className":50494},[7113],[35,50496,50498,50524],{"className":50497},[7117,7731],[35,50499,50501,50521],{"className":50500},[7121],[35,50502,50504],{"className":50503,"style":7738},[7125],[35,50505,50506,50509],{"style":9038},[35,50507],{"className":50508,"style":7134},[7133],[35,50510,50512],{"className":50511},[7138,7139,7140,7141],[35,50513,50515],{"className":50514},[7104,7141],[35,50516,50518],{"className":50517},[7104,7141],[35,50519,47791],{"className":50520},[7104,48375,7141],[35,50522,7756],{"className":50523},[7755],[35,50525,50527],{"className":50526},[7121],[35,50528,50530],{"className":50529,"style":7763},[7125],[35,50531],{},[35,50533],{"className":50534,"style":7450},[7248],[35,50536,242],{"className":50537},[7454],[35,50539],{"className":50540,"style":7450},[7248],[35,50542,50544,50547,50609,50612,50716,50719],{"className":50543},[7095],[35,50545],{"className":50546,"style":50058},[7099],[35,50548,50550,50553,50606],{"className":50549},[7104],[35,50551],{"className":50552},[7472,14311],[35,50554,50556],{"className":50555},[14009],[35,50557,50559,50598],{"className":50558},[7117,7731],[35,50560,50562,50595],{"className":50561},[7121],[35,50563,50565,50576,50584],{"className":50564,"style":50077},[7125],[35,50566,50567,50570],{"style":16234},[35,50568],{"className":50569,"style":8839},[7133],[35,50571,50573],{"className":50572},[7104],[35,50574,460],{"className":50575},[7104],[35,50577,50578,50581],{"style":14395},[35,50579],{"className":50580,"style":8839},[7133],[35,50582],{"className":50583,"style":14403},[14402],[35,50585,50586,50589],{"style":14406},[35,50587],{"className":50588,"style":8839},[7133],[35,50590,50592],{"className":50591},[7104],[35,50593,47921],{"className":50594},[7104],[35,50596,7756],{"className":50597},[7755],[35,50599,50601],{"className":50600},[7121],[35,50602,50604],{"className":50603,"style":16272},[7125],[35,50605],{},[35,50607],{"className":50608},[7314,14311],[35,50610],{"className":50611,"style":7496},[7248],[35,50613,50615,50618,50621,50661,50664,50667,50670,50673,50713],{"className":50614},[7468],[35,50616,275],{"className":50617,"style":7474},[7472,7473],[35,50619,47224],{"className":50620},[7104],[35,50622,50624,50627],{"className":50623},[7104],[35,50625,49186],{"className":50626,"style":7310},[7104,7145],[35,50628,50630],{"className":50629},[7113],[35,50631,50633,50653],{"className":50632},[7117,7731],[35,50634,50636,50650],{"className":50635},[7121],[35,50637,50639],{"className":50638,"style":9035},[7125],[35,50640,50641,50644],{"style":9100},[35,50642],{"className":50643,"style":7134},[7133],[35,50645,50647],{"className":50646},[7138,7139,7140,7141],[35,50648,300],{"className":50649,"style":25095},[7104,7145,7141],[35,50651,7756],{"className":50652},[7755],[35,50654,50656],{"className":50655},[7121],[35,50657,50659],{"className":50658,"style":7763},[7125],[35,50660],{},[35,50662],{"className":50663,"style":7249},[7248],[35,50665,445],{"className":50666},[7253],[35,50668],{"className":50669,"style":7249},[7248],[35,50671,47141],{"className":50672},[7104],[35,50674,50676,50679],{"className":50675},[7104],[35,50677,49186],{"className":50678,"style":7310},[7104,7145],[35,50680,50682],{"className":50681},[7113],[35,50683,50685,50705],{"className":50684},[7117,7731],[35,50686,50688,50702],{"className":50687},[7121],[35,50689,50691],{"className":50690,"style":9035},[7125],[35,50692,50693,50696],{"style":9100},[35,50694],{"className":50695,"style":7134},[7133],[35,50697,50699],{"className":50698},[7138,7139,7140,7141],[35,50700,13657],{"className":50701},[7104,7145,7141],[35,50703,7756],{"className":50704},[7755],[35,50706,50708],{"className":50707},[7121],[35,50709,50711],{"className":50710,"style":7763},[7125],[35,50712],{},[35,50714,1694],{"className":50715,"style":7474},[7314,7473],[35,50717],{"className":50718,"style":7496},[7248],[35,50720,2877],{"className":50721},[7492],[12,50723,50724],{},"with detuning",[35,50726,50728],{"className":50727},[7318],[35,50729,50731,50779],{"className":50730},[7057],[35,50732,50734],{"className":50733},[7061],[7063,50735,50736],{"xmlns":7065,"display":7327},[7067,50737,50738,50776],{},[7070,50739,50740,50742,50744,50746,50748,50754,50756,50758,50760,50762,50764,50766,50772,50774],{},[7076,50741,47224],{"mathvariant":7282},[7223,50743,242],{},[7076,50745,46825],{},[7223,50747,7225],{},[7699,50749,50750,50752],{},[7076,50751,46825],{},[7227,50753,5312],{},[7223,50755,242],{},[7227,50757,460],{},[7076,50759,13081],{},[7223,50761,275],{"stretchy":7289},[7076,50763,1675],{},[7223,50765,7225],{},[7699,50767,50768,50770],{},[7076,50769,1675],{},[7227,50771,5312],{},[7223,50773,1694],{"stretchy":7289},[7076,50775,5122],{"mathvariant":7282},[7083,50777,50778],{"encoding":7085},"\\Delta=\\omega-\\omega_0=2\\pi(f-f_0).",[35,50780,50782,50800,50818,50873,50900],{"className":50781,"ariaHidden":7091},[7090],[35,50783,50785,50788,50791,50794,50797],{"className":50784},[7095],[35,50786],{"className":50787,"style":7100},[7099],[35,50789,47224],{"className":50790},[7104],[35,50792],{"className":50793,"style":7450},[7248],[35,50795,242],{"className":50796},[7454],[35,50798],{"className":50799,"style":7450},[7248],[35,50801,50803,50806,50809,50812,50815],{"className":50802},[7095],[35,50804],{"className":50805,"style":7241},[7099],[35,50807,46825],{"className":50808,"style":7310},[7104,7145],[35,50810],{"className":50811,"style":7249},[7248],[35,50813,7225],{"className":50814},[7253],[35,50816],{"className":50817,"style":7249},[7248],[35,50819,50821,50824,50864,50867,50870],{"className":50820},[7095],[35,50822],{"className":50823,"style":29689},[7099],[35,50825,50827,50830],{"className":50826},[7104],[35,50828,46825],{"className":50829,"style":7310},[7104,7145],[35,50831,50833],{"className":50832},[7113],[35,50834,50836,50856],{"className":50835},[7117,7731],[35,50837,50839,50853],{"className":50838},[7121],[35,50840,50842],{"className":50841,"style":18794},[7125],[35,50843,50844,50847],{"style":9100},[35,50845],{"className":50846,"style":7134},[7133],[35,50848,50850],{"className":50849},[7138,7139,7140,7141],[35,50851,5312],{"className":50852},[7104,7141],[35,50854,7756],{"className":50855},[7755],[35,50857,50859],{"className":50858},[7121],[35,50860,50862],{"className":50861,"style":7763},[7125],[35,50863],{},[35,50865],{"className":50866,"style":7450},[7248],[35,50868,242],{"className":50869},[7454],[35,50871],{"className":50872,"style":7450},[7248],[35,50874,50876,50879,50882,50885,50888,50891,50894,50897],{"className":50875},[7095],[35,50877],{"className":50878,"style":7303},[7099],[35,50880,460],{"className":50881},[7104],[35,50883,13081],{"className":50884,"style":7310},[7104,7145],[35,50886,275],{"className":50887},[7472],[35,50889,1675],{"className":50890,"style":31824},[7104,7145],[35,50892],{"className":50893,"style":7249},[7248],[35,50895,7225],{"className":50896},[7253],[35,50898],{"className":50899,"style":7249},[7248],[35,50901,50903,50906,50946,50949],{"className":50902},[7095],[35,50904],{"className":50905,"style":7303},[7099],[35,50907,50909,50912],{"className":50908},[7104],[35,50910,1675],{"className":50911,"style":31824},[7104,7145],[35,50913,50915],{"className":50914},[7113],[35,50916,50918,50938],{"className":50917},[7117,7731],[35,50919,50921,50935],{"className":50920},[7121],[35,50922,50924],{"className":50923,"style":18794},[7125],[35,50925,50926,50929],{"style":46746},[35,50927],{"className":50928,"style":7134},[7133],[35,50930,50932],{"className":50931},[7138,7139,7140,7141],[35,50933,5312],{"className":50934},[7104,7141],[35,50936,7756],{"className":50937},[7755],[35,50939,50941],{"className":50940},[7121],[35,50942,50944],{"className":50943,"style":7763},[7125],[35,50945],{},[35,50947,1694],{"className":50948},[7314],[35,50950,5122],{"className":50951},[7104],[12,50953,50954,50955,15316,50983,51011,51012,51040,51041,51069],{},"This Hamiltonian describes precession around an effective control axis with components proportional to ",[35,50956,50958,50971],{"className":50957},[7057],[35,50959,50961],{"className":50960},[7061],[7063,50962,50963],{"xmlns":7065},[7067,50964,50965,50969],{},[7070,50966,50967],{},[7076,50968,47141],{"mathvariant":7282},[7083,50970,47144],{"encoding":7085},[35,50972,50974],{"className":50973,"ariaHidden":7091},[7090],[35,50975,50977,50980],{"className":50976},[7095],[35,50978],{"className":50979,"style":7100},[7099],[35,50981,47141],{"className":50982},[7104],[35,50984,50986,50999],{"className":50985},[7057],[35,50987,50989],{"className":50988},[7061],[7063,50990,50991],{"xmlns":7065},[7067,50992,50993,50997],{},[7070,50994,50995],{},[7076,50996,47224],{"mathvariant":7282},[7083,50998,47227],{"encoding":7085},[35,51000,51002],{"className":51001,"ariaHidden":7091},[7090],[35,51003,51005,51008],{"className":51004},[7095],[35,51006],{"className":51007,"style":7100},[7099],[35,51009,47224],{"className":51010},[7104],". The transverse component ",[35,51013,51015,51028],{"className":51014},[7057],[35,51016,51018],{"className":51017},[7061],[7063,51019,51020],{"xmlns":7065},[7067,51021,51022,51026],{},[7070,51023,51024],{},[7076,51025,47141],{"mathvariant":7282},[7083,51027,47144],{"encoding":7085},[35,51029,51031],{"className":51030,"ariaHidden":7091},[7090],[35,51032,51034,51037],{"className":51033},[7095],[35,51035],{"className":51036,"style":7100},[7099],[35,51038,47141],{"className":51039},[7104]," drives population transfer. A longitudinal component ",[35,51042,51044,51057],{"className":51043},[7057],[35,51045,51047],{"className":51046},[7061],[7063,51048,51049],{"xmlns":7065},[7067,51050,51051,51055],{},[7070,51052,51053],{},[7076,51054,47224],{"mathvariant":7282},[7083,51056,47227],{"encoding":7085},[35,51058,51060],{"className":51059,"ariaHidden":7091},[7090],[35,51061,51063,51066],{"className":51062},[7095],[35,51064],{"className":51065,"style":7100},[7099],[35,51067,47224],{"className":51068},[7104]," tilts the rotation axis away from the resonant direction.",[5019,51071,51073],{"id":51072},"excited-state-probability","Excited-State Probability",[12,51075,51076,51077,51115,51116,51144],{},"The simulation assumes that the qubit begins in the ground state ",[35,51078,51080,51097],{"className":51079},[7057],[35,51081,51083],{"className":51082},[7061],[7063,51084,51085],{"xmlns":7065},[7067,51086,51087,51095],{},[7070,51088,51089,51091,51093],{},[7223,51090,7283],{"stretchy":7289},[7076,51092,42397],{},[7223,51094,7290],{"stretchy":7289},[7083,51096,46374],{"encoding":7085},[35,51098,51100],{"className":51099,"ariaHidden":7091},[7090],[35,51101,51103,51106,51109,51112],{"className":51102},[7095],[35,51104],{"className":51105,"style":7303},[7099],[35,51107,7283],{"className":51108},[7472],[35,51110,42397],{"className":51111,"style":7310},[7104,7145],[35,51113,7290],{"className":51114},[7314],". Evolving under the RWA Hamiltonian, the probability of occupying the excited state after pulse duration ",[35,51117,51119,51132],{"className":51118},[7057],[35,51120,51122],{"className":51121},[7061],[7063,51123,51124],{"xmlns":7065},[7067,51125,51126,51130],{},[7070,51127,51128],{},[7076,51129,46183],{},[7083,51131,46309],{"encoding":7085},[35,51133,51135],{"className":51134,"ariaHidden":7091},[7090],[35,51136,51138,51141],{"className":51137},[7095],[35,51139],{"className":51140,"style":7173},[7099],[35,51142,46183],{"className":51143,"style":46255},[7104,7145]," is",[35,51146,51148],{"className":51147},[7318],[35,51149,51151,51230],{"className":51150},[7057],[35,51152,51154],{"className":51153},[7061],[7063,51155,51156],{"xmlns":7065,"display":7327},[7067,51157,51158,51227],{},[7070,51159,51160,51166,51168,51170,51172,51174,51176,51178,51194,51205,51225],{},[7699,51161,51162,51164],{},[7076,51163,7705],{},[7076,51165,7817],{},[7223,51167,275],{"stretchy":7289},[7076,51169,1675],{},[7223,51171,2877],{"separator":7091},[7076,51173,46183],{},[7223,51175,1694],{"stretchy":7289},[7223,51177,242],{},[14009,51179,51180,51186],{},[7073,51181,51182,51184],{},[7076,51183,47141],{"mathvariant":7282},[7227,51185,460],{},[8304,51187,51188,51190,51192],{},[7076,51189,47141],{"mathvariant":7282},[7076,51191,24883],{},[7227,51193,460],{},[7073,51195,51196,51203],{},[7070,51197,51198,51201],{},[7076,51199,51200],{},"sin",[7223,51202,41175],{},[7227,51204,460],{},[7070,51206,51207,51209,51223],{},[7223,51208,275],{"fence":7091},[14009,51210,51211,51221],{},[7070,51212,51213,51219],{},[7699,51214,51215,51217],{},[7076,51216,47141],{"mathvariant":7282},[7076,51218,24883],{},[7076,51220,46183],{},[7227,51222,460],{},[7223,51224,1694],{"fence":7091},[7223,51226,2877],{"separator":7091},[7083,51228,51229],{"encoding":7085},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right),",[35,51231,51233,51306],{"className":51232,"ariaHidden":7091},[7090],[35,51234,51236,51239,51279,51282,51285,51288,51291,51294,51297,51300,51303],{"className":51235},[7095],[35,51237],{"className":51238,"style":7303},[7099],[35,51240,51242,51245],{"className":51241},[7104],[35,51243,7705],{"className":51244,"style":7751},[7104,7145],[35,51246,51248],{"className":51247},[7113],[35,51249,51251,51271],{"className":51250},[7117,7731],[35,51252,51254,51268],{"className":51253},[7121],[35,51255,51257],{"className":51256,"style":9035},[7125],[35,51258,51259,51262],{"style":12163},[35,51260],{"className":51261,"style":7134},[7133],[35,51263,51265],{"className":51264},[7138,7139,7140,7141],[35,51266,7817],{"className":51267},[7104,7145,7141],[35,51269,7756],{"className":51270},[7755],[35,51272,51274],{"className":51273},[7121],[35,51275,51277],{"className":51276,"style":7763},[7125],[35,51278],{},[35,51280,275],{"className":51281},[7472],[35,51283,1675],{"className":51284,"style":31824},[7104,7145],[35,51286,2877],{"className":51287},[7492],[35,51289],{"className":51290,"style":7496},[7248],[35,51292,46183],{"className":51293,"style":46255},[7104,7145],[35,51295,1694],{"className":51296},[7314],[35,51298],{"className":51299,"style":7450},[7248],[35,51301,242],{"className":51302},[7454],[35,51304],{"className":51305,"style":7450},[7248],[35,51307,51309,51313,51455,51458,51489,51492,51610,51613],{"className":51308},[7095],[35,51310],{"className":51311,"style":51312},[7099],"height:2.4706em;vertical-align:-0.9795em;",[35,51314,51316,51319,51452],{"className":51315},[7104],[35,51317],{"className":51318},[7472,14311],[35,51320,51322],{"className":51321},[14009],[35,51323,51325,51443],{"className":51324},[7117,7731],[35,51326,51328,51440],{"className":51327},[7121],[35,51329,51332,51395,51403],{"className":51330,"style":51331},[7125],"height:1.4911em;",[35,51333,51334,51337],{"style":16234},[35,51335],{"className":51336,"style":8839},[7133],[35,51338,51340],{"className":51339},[7104],[35,51341,51343,51346],{"className":51342},[7104],[35,51344,47141],{"className":51345},[7104],[35,51347,51349],{"className":51348},[7113],[35,51350,51352,51386],{"className":51351},[7117,7731],[35,51353,51355,51383],{"className":51354},[7121],[35,51356,51359,51371],{"className":51357,"style":51358},[7125],"height:0.7959em;",[35,51360,51362,51365],{"style":51361},"top:-2.4065em;margin-left:0em;margin-right:0.05em;",[35,51363],{"className":51364,"style":7134},[7133],[35,51366,51368],{"className":51367},[7138,7139,7140,7141],[35,51369,24883],{"className":51370,"style":24908},[7104,7145,7141],[35,51372,51374,51377],{"style":51373},"top:-3.0448em;margin-right:0.05em;",[35,51375],{"className":51376,"style":7134},[7133],[35,51378,51380],{"className":51379},[7138,7139,7140,7141],[35,51381,460],{"className":51382},[7104,7141],[35,51384,7756],{"className":51385},[7755],[35,51387,51389],{"className":51388},[7121],[35,51390,51393],{"className":51391,"style":51392},[7125],"height:0.2935em;",[35,51394],{},[35,51396,51397,51400],{"style":14395},[35,51398],{"className":51399,"style":8839},[7133],[35,51401],{"className":51402,"style":14403},[14402],[35,51404,51405,51408],{"style":14406},[35,51406],{"className":51407,"style":8839},[7133],[35,51409,51411],{"className":51410},[7104],[35,51412,51414,51417],{"className":51413},[7104],[35,51415,47141],{"className":51416},[7104],[35,51418,51420],{"className":51419},[7113],[35,51421,51423],{"className":51422},[7117],[35,51424,51426],{"className":51425},[7121],[35,51427,51429],{"className":51428,"style":17739},[7125],[35,51430,51431,51434],{"style":7129},[35,51432],{"className":51433,"style":7134},[7133],[35,51435,51437],{"className":51436},[7138,7139,7140,7141],[35,51438,460],{"className":51439},[7104,7141],[35,51441,7756],{"className":51442},[7755],[35,51444,51446],{"className":51445},[7121],[35,51447,51450],{"className":51448,"style":51449},[7125],"height:0.9795em;",[35,51451],{},[35,51453],{"className":51454},[7314,14311],[35,51456],{"className":51457,"style":7496},[7248],[35,51459,51461,51464],{"className":51460},[12096],[35,51462,51200],{"className":51463},[12096],[35,51465,51467],{"className":51466},[7113],[35,51468,51470],{"className":51469},[7117],[35,51471,51473],{"className":51472},[7121],[35,51474,51477],{"className":51475,"style":51476},[7125],"height:0.8719em;",[35,51478,51480,51483],{"style":51479},"top:-3.1208em;margin-right:0.05em;",[35,51481],{"className":51482,"style":7134},[7133],[35,51484,51486],{"className":51485},[7138,7139,7140,7141],[35,51487,460],{"className":51488},[7104,7141],[35,51490],{"className":51491,"style":7496},[7248],[35,51493,51495,51501,51604],{"className":51494},[7468],[35,51496,51498],{"className":51497,"style":7474},[7472,7473],[35,51499,275],{"className":51500},[7478,7140],[35,51502,51504,51507,51601],{"className":51503},[7104],[35,51505],{"className":51506},[7472,14311],[35,51508,51510],{"className":51509},[14009],[35,51511,51513,51593],{"className":51512},[7117,7731],[35,51514,51516,51590],{"className":51515},[7121],[35,51517,51520,51531,51539],{"className":51518,"style":51519},[7125],"height:1.3603em;",[35,51521,51522,51525],{"style":16234},[35,51523],{"className":51524,"style":8839},[7133],[35,51526,51528],{"className":51527},[7104],[35,51529,460],{"className":51530},[7104],[35,51532,51533,51536],{"style":14395},[35,51534],{"className":51535,"style":8839},[7133],[35,51537],{"className":51538,"style":14403},[14402],[35,51540,51541,51544],{"style":14406},[35,51542],{"className":51543,"style":8839},[7133],[35,51545,51547,51587],{"className":51546},[7104],[35,51548,51550,51553],{"className":51549},[7104],[35,51551,47141],{"className":51552},[7104],[35,51554,51556],{"className":51555},[7113],[35,51557,51559,51579],{"className":51558},[7117,7731],[35,51560,51562,51576],{"className":51561},[7121],[35,51563,51565],{"className":51564,"style":7738},[7125],[35,51566,51567,51570],{"style":9926},[35,51568],{"className":51569,"style":7134},[7133],[35,51571,51573],{"className":51572},[7138,7139,7140,7141],[35,51574,24883],{"className":51575,"style":24908},[7104,7145,7141],[35,51577,7756],{"className":51578},[7755],[35,51580,51582],{"className":51581},[7121],[35,51583,51585],{"className":51584,"style":7763},[7125],[35,51586],{},[35,51588,46183],{"className":51589,"style":46255},[7104,7145],[35,51591,7756],{"className":51592},[7755],[35,51594,51596],{"className":51595},[7121],[35,51597,51599],{"className":51598,"style":16272},[7125],[35,51600],{},[35,51602],{"className":51603},[7314,14311],[35,51605,51607],{"className":51606,"style":7474},[7314,7473],[35,51608,1694],{"className":51609},[7478,7140],[35,51611],{"className":51612,"style":7496},[7248],[35,51614,2877],{"className":51615},[7492],[12,51617,51618],{},"where",[35,51620,51622],{"className":51621},[7318],[35,51623,51625,51663],{"className":51624},[7057],[35,51626,51628],{"className":51627},[7061],[7063,51629,51630],{"xmlns":7065,"display":7327},[7067,51631,51632,51660],{},[7070,51633,51634,51640,51642],{},[7699,51635,51636,51638],{},[7076,51637,47141],{"mathvariant":7282},[7076,51639,24883],{},[7223,51641,242],{},[14014,51643,51644],{},[7070,51645,51646,51652,51654],{},[7073,51647,51648,51650],{},[7076,51649,47141],{"mathvariant":7282},[7227,51651,460],{},[7223,51653,445],{},[7073,51655,51656,51658],{},[7076,51657,47224],{"mathvariant":7282},[7227,51659,460],{},[7083,51661,51662],{"encoding":7085},"\\Omega_R=\\sqrt{\\Omega^2+\\Delta^2}",[35,51664,51666,51721],{"className":51665,"ariaHidden":7091},[7090],[35,51667,51669,51672,51712,51715,51718],{"className":51668},[7095],[35,51670],{"className":51671,"style":7718},[7099],[35,51673,51675,51678],{"className":51674},[7104],[35,51676,47141],{"className":51677},[7104],[35,51679,51681],{"className":51680},[7113],[35,51682,51684,51704],{"className":51683},[7117,7731],[35,51685,51687,51701],{"className":51686},[7121],[35,51688,51690],{"className":51689,"style":7738},[7125],[35,51691,51692,51695],{"style":9926},[35,51693],{"className":51694,"style":7134},[7133],[35,51696,51698],{"className":51697},[7138,7139,7140,7141],[35,51699,24883],{"className":51700,"style":24908},[7104,7145,7141],[35,51702,7756],{"className":51703},[7755],[35,51705,51707],{"className":51706},[7121],[35,51708,51710],{"className":51709,"style":7763},[7125],[35,51711],{},[35,51713],{"className":51714,"style":7450},[7248],[35,51716,242],{"className":51717},[7454],[35,51719],{"className":51720,"style":7450},[7248],[35,51722,51724,51728],{"className":51723},[7095],[35,51725],{"className":51726,"style":51727},[7099],"height:1.24em;vertical-align:-0.1777em;",[35,51729,51731],{"className":51730},[7104,490],[35,51732,51734,51840],{"className":51733},[7117,7731],[35,51735,51737,51837],{"className":51736},[7121],[35,51738,51741,51820],{"className":51739,"style":51740},[7125],"height:1.0623em;",[35,51742,51745,51749],{"className":51743,"style":51744},[14350],"top:-3.2em;",[35,51746],{"className":51747,"style":51748},[7133],"height:3.2em;",[35,51750,51753,51782,51785,51788,51791],{"className":51751,"style":51752},[7104],"padding-left:1em;",[35,51754,51756,51759],{"className":51755},[7104],[35,51757,47141],{"className":51758},[7104],[35,51760,51762],{"className":51761},[7113],[35,51763,51765],{"className":51764},[7117],[35,51766,51768],{"className":51767},[7121],[35,51769,51771],{"className":51770,"style":33894},[7125],[35,51772,51773,51776],{"style":33897},[35,51774],{"className":51775,"style":7134},[7133],[35,51777,51779],{"className":51778},[7138,7139,7140,7141],[35,51780,460],{"className":51781},[7104,7141],[35,51783],{"className":51784,"style":7249},[7248],[35,51786,445],{"className":51787},[7253],[35,51789],{"className":51790,"style":7249},[7248],[35,51792,51794,51797],{"className":51793},[7104],[35,51795,47224],{"className":51796},[7104],[35,51798,51800],{"className":51799},[7113],[35,51801,51803],{"className":51802},[7117],[35,51804,51806],{"className":51805},[7121],[35,51807,51809],{"className":51808,"style":33894},[7125],[35,51810,51811,51814],{"style":33897},[35,51812],{"className":51813,"style":7134},[7133],[35,51815,51817],{"className":51816},[7138,7139,7140,7141],[35,51818,460],{"className":51819},[7104,7141],[35,51821,51823,51826],{"style":51822},"top:-3.0223em;",[35,51824],{"className":51825,"style":51748},[7133],[35,51827,51830],{"className":51828,"style":51829},[14370],"min-width:1.02em;height:1.28em;",[10808,51831,51834],{"xmlns":10810,"width":14374,"height":51832,"viewBox":51833,"preserveAspectRatio":14377},"1.28em","0 0 400000 1296",[10817,51835],{"d":51836},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[35,51838,7756],{"className":51839},[7755],[35,51841,51843],{"className":51842},[7121],[35,51844,51847],{"className":51845,"style":51846},[7125],"height:0.1777em;",[35,51848],{},[12,51850,51851],{},"is the generalized Rabi angular frequency. The amplitude coefficient",[35,51853,51855],{"className":51854},[7318],[35,51856,51858,51886],{"className":51857},[7057],[35,51859,51861],{"className":51860},[7061],[7063,51862,51863],{"xmlns":7065,"display":7327},[7067,51864,51865,51883],{},[7070,51866,51867],{},[14009,51868,51869,51875],{},[7073,51870,51871,51873],{},[7076,51872,47141],{"mathvariant":7282},[7227,51874,460],{},[8304,51876,51877,51879,51881],{},[7076,51878,47141],{"mathvariant":7282},[7076,51880,24883],{},[7227,51882,460],{},[7083,51884,51885],{"encoding":7085},"\\frac{\\Omega^2}{\\Omega_R^2}",[35,51887,51889],{"className":51888,"ariaHidden":7091},[7090],[35,51890,51892,51895],{"className":51891},[7095],[35,51893],{"className":51894,"style":51312},[7099],[35,51896,51898,51901,52028],{"className":51897},[7104],[35,51899],{"className":51900},[7472,14311],[35,51902,51904],{"className":51903},[14009],[35,51905,51907,52020],{"className":51906},[7117,7731],[35,51908,51910,52017],{"className":51909},[7121],[35,51911,51913,51972,51980],{"className":51912,"style":51331},[7125],[35,51914,51915,51918],{"style":16234},[35,51916],{"className":51917,"style":8839},[7133],[35,51919,51921],{"className":51920},[7104],[35,51922,51924,51927],{"className":51923},[7104],[35,51925,47141],{"className":51926},[7104],[35,51928,51930],{"className":51929},[7113],[35,51931,51933,51964],{"className":51932},[7117,7731],[35,51934,51936,51961],{"className":51935},[7121],[35,51937,51939,51950],{"className":51938,"style":51358},[7125],[35,51940,51941,51944],{"style":51361},[35,51942],{"className":51943,"style":7134},[7133],[35,51945,51947],{"className":51946},[7138,7139,7140,7141],[35,51948,24883],{"className":51949,"style":24908},[7104,7145,7141],[35,51951,51952,51955],{"style":51373},[35,51953],{"className":51954,"style":7134},[7133],[35,51956,51958],{"className":51957},[7138,7139,7140,7141],[35,51959,460],{"className":51960},[7104,7141],[35,51962,7756],{"className":51963},[7755],[35,51965,51967],{"className":51966},[7121],[35,51968,51970],{"className":51969,"style":51392},[7125],[35,51971],{},[35,51973,51974,51977],{"style":14395},[35,51975],{"className":51976,"style":8839},[7133],[35,51978],{"className":51979,"style":14403},[14402],[35,51981,51982,51985],{"style":14406},[35,51983],{"className":51984,"style":8839},[7133],[35,51986,51988],{"className":51987},[7104],[35,51989,51991,51994],{"className":51990},[7104],[35,51992,47141],{"className":51993},[7104],[35,51995,51997],{"className":51996},[7113],[35,51998,52000],{"className":51999},[7117],[35,52001,52003],{"className":52002},[7121],[35,52004,52006],{"className":52005,"style":17739},[7125],[35,52007,52008,52011],{"style":7129},[35,52009],{"className":52010,"style":7134},[7133],[35,52012,52014],{"className":52013},[7138,7139,7140,7141],[35,52015,460],{"className":52016},[7104,7141],[35,52018,7756],{"className":52019},[7755],[35,52021,52023],{"className":52022},[7121],[35,52024,52026],{"className":52025,"style":51449},[7125],[35,52027],{},[35,52029],{"className":52030},[7314,14311],[12,52032,52033,52034,52085],{},"sets the maximum possible population transfer. At exact resonance, ",[35,52035,52037,52055],{"className":52036},[7057],[35,52038,52040],{"className":52039},[7061],[7063,52041,52042],{"xmlns":7065},[7067,52043,52044,52052],{},[7070,52045,52046,52048,52050],{},[7076,52047,47224],{"mathvariant":7282},[7223,52049,242],{},[7227,52051,5312],{},[7083,52053,52054],{"encoding":7085},"\\Delta=0",[35,52056,52058,52076],{"className":52057,"ariaHidden":7091},[7090],[35,52059,52061,52064,52067,52070,52073],{"className":52060},[7095],[35,52062],{"className":52063,"style":7100},[7099],[35,52065,47224],{"className":52066},[7104],[35,52068],{"className":52069,"style":7450},[7248],[35,52071,242],{"className":52072},[7454],[35,52074],{"className":52075,"style":7450},[7248],[35,52077,52079,52082],{"className":52078},[7095],[35,52080],{"className":52081,"style":7263},[7099],[35,52083,5312],{"className":52084},[7104],", so",[35,52087,52089],{"className":52088},[7318],[35,52090,52092,52124],{"className":52091},[7057],[35,52093,52095],{"className":52094},[7061],[7063,52096,52097],{"xmlns":7065,"display":7327},[7067,52098,52099,52121],{},[7070,52100,52101,52117,52119],{},[14009,52102,52103,52109],{},[7073,52104,52105,52107],{},[7076,52106,47141],{"mathvariant":7282},[7227,52108,460],{},[8304,52110,52111,52113,52115],{},[7076,52112,47141],{"mathvariant":7282},[7076,52114,24883],{},[7227,52116,460],{},[7223,52118,242],{},[7227,52120,48822],{},[7083,52122,52123],{"encoding":7085},"\\frac{\\Omega^2}{\\Omega_R^2}=1.",[35,52125,52127,52278],{"className":52126,"ariaHidden":7091},[7090],[35,52128,52130,52133,52269,52272,52275],{"className":52129},[7095],[35,52131],{"className":52132,"style":51312},[7099],[35,52134,52136,52139,52266],{"className":52135},[7104],[35,52137],{"className":52138},[7472,14311],[35,52140,52142],{"className":52141},[14009],[35,52143,52145,52258],{"className":52144},[7117,7731],[35,52146,52148,52255],{"className":52147},[7121],[35,52149,52151,52210,52218],{"className":52150,"style":51331},[7125],[35,52152,52153,52156],{"style":16234},[35,52154],{"className":52155,"style":8839},[7133],[35,52157,52159],{"className":52158},[7104],[35,52160,52162,52165],{"className":52161},[7104],[35,52163,47141],{"className":52164},[7104],[35,52166,52168],{"className":52167},[7113],[35,52169,52171,52202],{"className":52170},[7117,7731],[35,52172,52174,52199],{"className":52173},[7121],[35,52175,52177,52188],{"className":52176,"style":51358},[7125],[35,52178,52179,52182],{"style":51361},[35,52180],{"className":52181,"style":7134},[7133],[35,52183,52185],{"className":52184},[7138,7139,7140,7141],[35,52186,24883],{"className":52187,"style":24908},[7104,7145,7141],[35,52189,52190,52193],{"style":51373},[35,52191],{"className":52192,"style":7134},[7133],[35,52194,52196],{"className":52195},[7138,7139,7140,7141],[35,52197,460],{"className":52198},[7104,7141],[35,52200,7756],{"className":52201},[7755],[35,52203,52205],{"className":52204},[7121],[35,52206,52208],{"className":52207,"style":51392},[7125],[35,52209],{},[35,52211,52212,52215],{"style":14395},[35,52213],{"className":52214,"style":8839},[7133],[35,52216],{"className":52217,"style":14403},[14402],[35,52219,52220,52223],{"style":14406},[35,52221],{"className":52222,"style":8839},[7133],[35,52224,52226],{"className":52225},[7104],[35,52227,52229,52232],{"className":52228},[7104],[35,52230,47141],{"className":52231},[7104],[35,52233,52235],{"className":52234},[7113],[35,52236,52238],{"className":52237},[7117],[35,52239,52241],{"className":52240},[7121],[35,52242,52244],{"className":52243,"style":17739},[7125],[35,52245,52246,52249],{"style":7129},[35,52247],{"className":52248,"style":7134},[7133],[35,52250,52252],{"className":52251},[7138,7139,7140,7141],[35,52253,460],{"className":52254},[7104,7141],[35,52256,7756],{"className":52257},[7755],[35,52259,52261],{"className":52260},[7121],[35,52262,52264],{"className":52263,"style":51449},[7125],[35,52265],{},[35,52267],{"className":52268},[7314,14311],[35,52270],{"className":52271,"style":7450},[7248],[35,52273,242],{"className":52274},[7454],[35,52276],{"className":52277,"style":7450},[7248],[35,52279,52281,52284],{"className":52280},[7095],[35,52282],{"className":52283,"style":7263},[7099],[35,52285,48822],{"className":52286},[7104],[12,52288,52289],{},"Full population inversion is achievable in the ideal resonant model. For any nonzero detuning, the coefficient falls below one, so the state generally cannot reach unit excited-state probability.",[12,52291,52292],{},"At resonance, the excited-state probability simplifies to",[35,52294,52296],{"className":52295},[7318],[35,52297,52299,52353],{"className":52298},[7057],[35,52300,52302],{"className":52301},[7061],[7063,52303,52304],{"xmlns":7065,"display":7327},[7067,52305,52306,52350],{},[7070,52307,52308,52314,52316,52318,52320,52322,52332,52348],{},[7699,52309,52310,52312],{},[7076,52311,7705],{},[7076,52313,7817],{},[7223,52315,275],{"stretchy":7289},[7076,52317,46183],{},[7223,52319,1694],{"stretchy":7289},[7223,52321,242],{},[7073,52323,52324,52330],{},[7070,52325,52326,52328],{},[7076,52327,51200],{},[7223,52329,41175],{},[7227,52331,460],{},[7070,52333,52334,52336,52346],{},[7223,52335,275],{"fence":7091},[14009,52337,52338,52344],{},[7070,52339,52340,52342],{},[7076,52341,47141],{"mathvariant":7282},[7076,52343,46183],{},[7227,52345,460],{},[7223,52347,1694],{"fence":7091},[7076,52349,5122],{"mathvariant":7282},[7083,52351,52352],{"encoding":7085},"P_e(\\tau)=\\sin^2\\left(\\frac{\\Omega\\tau}{2}\\right).",[35,52354,52356,52420],{"className":52355,"ariaHidden":7091},[7090],[35,52357,52359,52362,52402,52405,52408,52411,52414,52417],{"className":52358},[7095],[35,52360],{"className":52361,"style":7303},[7099],[35,52363,52365,52368],{"className":52364},[7104],[35,52366,7705],{"className":52367,"style":7751},[7104,7145],[35,52369,52371],{"className":52370},[7113],[35,52372,52374,52394],{"className":52373},[7117,7731],[35,52375,52377,52391],{"className":52376},[7121],[35,52378,52380],{"className":52379,"style":9035},[7125],[35,52381,52382,52385],{"style":12163},[35,52383],{"className":52384,"style":7134},[7133],[35,52386,52388],{"className":52387},[7138,7139,7140,7141],[35,52389,7817],{"className":52390},[7104,7145,7141],[35,52392,7756],{"className":52393},[7755],[35,52395,52397],{"className":52396},[7121],[35,52398,52400],{"className":52399,"style":7763},[7125],[35,52401],{},[35,52403,275],{"className":52404},[7472],[35,52406,46183],{"className":52407,"style":46255},[7104,7145],[35,52409,1694],{"className":52410},[7314],[35,52412],{"className":52413,"style":7450},[7248],[35,52415,242],{"className":52416},[7454],[35,52418],{"className":52419,"style":7450},[7248],[35,52421,52423,52426,52455,52458,52538,52541],{"className":52422},[7095],[35,52424],{"className":52425,"style":42152},[7099],[35,52427,52429,52432],{"className":52428},[12096],[35,52430,51200],{"className":52431},[12096],[35,52433,52435],{"className":52434},[7113],[35,52436,52438],{"className":52437},[7117],[35,52439,52441],{"className":52440},[7121],[35,52442,52444],{"className":52443,"style":51476},[7125],[35,52445,52446,52449],{"style":51479},[35,52447],{"className":52448,"style":7134},[7133],[35,52450,52452],{"className":52451},[7138,7139,7140,7141],[35,52453,460],{"className":52454},[7104,7141],[35,52456],{"className":52457,"style":7496},[7248],[35,52459,52461,52467,52532],{"className":52460},[7468],[35,52462,52464],{"className":52463,"style":7474},[7472,7473],[35,52465,275],{"className":52466},[7478,7140],[35,52468,52470,52473,52529],{"className":52469},[7104],[35,52471],{"className":52472},[7472,14311],[35,52474,52476],{"className":52475},[14009],[35,52477,52479,52521],{"className":52478},[7117,7731],[35,52480,52482,52518],{"className":52481},[7121],[35,52483,52485,52496,52504],{"className":52484,"style":51519},[7125],[35,52486,52487,52490],{"style":16234},[35,52488],{"className":52489,"style":8839},[7133],[35,52491,52493],{"className":52492},[7104],[35,52494,460],{"className":52495},[7104],[35,52497,52498,52501],{"style":14395},[35,52499],{"className":52500,"style":8839},[7133],[35,52502],{"className":52503,"style":14403},[14402],[35,52505,52506,52509],{"style":14406},[35,52507],{"className":52508,"style":8839},[7133],[35,52510,52512,52515],{"className":52511},[7104],[35,52513,47141],{"className":52514},[7104],[35,52516,46183],{"className":52517,"style":46255},[7104,7145],[35,52519,7756],{"className":52520},[7755],[35,52522,52524],{"className":52523},[7121],[35,52525,52527],{"className":52526,"style":16272},[7125],[35,52528],{},[35,52530],{"className":52531},[7314,14311],[35,52533,52535],{"className":52534,"style":7474},[7314,7473],[35,52536,1694],{"className":52537},[7478,7140],[35,52539],{"className":52540,"style":7496},[7248],[35,52542,5122],{"className":52543},[7104],[12,52545,52546,52547,52576],{},"The pulse duration required for ideal population inversion defines the standard ",[35,52548,52550,52564],{"className":52549},[7057],[35,52551,52553],{"className":52552},[7061],[7063,52554,52555],{"xmlns":7065},[7067,52556,52557,52561],{},[7070,52558,52559],{},[7076,52560,13081],{},[7083,52562,52563],{"encoding":7085},"\\pi",[35,52565,52567],{"className":52566,"ariaHidden":7091},[7090],[35,52568,52570,52573],{"className":52569},[7095],[35,52571],{"className":52572,"style":7173},[7099],[35,52574,13081],{"className":52575,"style":7310},[7104,7145],"-pulse time,",[35,52578,52580],{"className":52579},[7318],[35,52581,52583,52633],{"className":52582},[7057],[35,52584,52586],{"className":52585},[7061],[7063,52587,52588],{"xmlns":7065,"display":7327},[7067,52589,52590,52630],{},[7070,52591,52592,52598,52600,52606,52608,52628],{},[7699,52593,52594,52596],{},[7076,52595,46183],{},[7076,52597,13081],{},[7223,52599,242],{},[14009,52601,52602,52604],{},[7076,52603,13081],{},[7076,52605,47141],{"mathvariant":7282},[7223,52607,242],{},[14009,52609,52610,52612],{},[7227,52611,500],{},[7070,52613,52614,52616,52618,52620,52622,52624,52626],{},[7227,52615,460],{},[7223,52617,275],{"stretchy":7289},[7076,52619,47141],{"mathvariant":7282},[7076,52621,398],{"mathvariant":7282},[7227,52623,460],{},[7076,52625,13081],{},[7223,52627,1694],{"stretchy":7289},[7076,52629,5122],{"mathvariant":7282},[7083,52631,52632],{"encoding":7085},"\\tau_\\pi=\\frac{\\pi}{\\Omega}=\\frac{1}{2(\\Omega\u002F2\\pi)}.",[35,52634,52636,52692,52771],{"className":52635,"ariaHidden":7091},[7090],[35,52637,52639,52642,52683,52686,52689],{"className":52638},[7095],[35,52640],{"className":52641,"style":29689},[7099],[35,52643,52645,52648],{"className":52644},[7104],[35,52646,46183],{"className":52647,"style":46255},[7104,7145],[35,52649,52651],{"className":52650},[7113],[35,52652,52654,52675],{"className":52653},[7117,7731],[35,52655,52657,52672],{"className":52656},[7121],[35,52658,52660],{"className":52659,"style":9035},[7125],[35,52661,52663,52666],{"style":52662},"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;",[35,52664],{"className":52665,"style":7134},[7133],[35,52667,52669],{"className":52668},[7138,7139,7140,7141],[35,52670,13081],{"className":52671,"style":7310},[7104,7145,7141],[35,52673,7756],{"className":52674},[7755],[35,52676,52678],{"className":52677},[7121],[35,52679,52681],{"className":52680,"style":7763},[7125],[35,52682],{},[35,52684],{"className":52685,"style":7450},[7248],[35,52687,242],{"className":52688},[7454],[35,52690],{"className":52691,"style":7450},[7248],[35,52693,52695,52699,52762,52765,52768],{"className":52694},[7095],[35,52696],{"className":52697,"style":52698},[7099],"height:1.7936em;vertical-align:-0.686em;",[35,52700,52702,52705,52759],{"className":52701},[7104],[35,52703],{"className":52704},[7472,14311],[35,52706,52708],{"className":52707},[14009],[35,52709,52711,52751],{"className":52710},[7117,7731],[35,52712,52714,52748],{"className":52713},[7121],[35,52715,52718,52729,52737],{"className":52716,"style":52717},[7125],"height:1.1076em;",[35,52719,52720,52723],{"style":16234},[35,52721],{"className":52722,"style":8839},[7133],[35,52724,52726],{"className":52725},[7104],[35,52727,47141],{"className":52728},[7104],[35,52730,52731,52734],{"style":14395},[35,52732],{"className":52733,"style":8839},[7133],[35,52735],{"className":52736,"style":14403},[14402],[35,52738,52739,52742],{"style":14406},[35,52740],{"className":52741,"style":8839},[7133],[35,52743,52745],{"className":52744},[7104],[35,52746,13081],{"className":52747,"style":7310},[7104,7145],[35,52749,7756],{"className":52750},[7755],[35,52752,52754],{"className":52753},[7121],[35,52755,52757],{"className":52756,"style":16272},[7125],[35,52758],{},[35,52760],{"className":52761},[7314,14311],[35,52763],{"className":52764,"style":7450},[7248],[35,52766,242],{"className":52767},[7454],[35,52769],{"className":52770,"style":7450},[7248],[35,52772,52774,52777,52851],{"className":52773},[7095],[35,52775],{"className":52776,"style":40323},[7099],[35,52778,52780,52783,52848],{"className":52779},[7104],[35,52781],{"className":52782},[7472,14311],[35,52784,52786],{"className":52785},[14009],[35,52787,52789,52840],{"className":52788},[7117,7731],[35,52790,52792,52837],{"className":52791},[7121],[35,52793,52795,52818,52826],{"className":52794,"style":14324},[7125],[35,52796,52797,52800],{"style":16234},[35,52798],{"className":52799,"style":8839},[7133],[35,52801,52803,52806,52809,52812,52815],{"className":52802},[7104],[35,52804,460],{"className":52805},[7104],[35,52807,275],{"className":52808},[7472],[35,52810,47199],{"className":52811},[7104],[35,52813,13081],{"className":52814,"style":7310},[7104,7145],[35,52816,1694],{"className":52817},[7314],[35,52819,52820,52823],{"style":14395},[35,52821],{"className":52822,"style":8839},[7133],[35,52824],{"className":52825,"style":14403},[14402],[35,52827,52828,52831],{"style":14406},[35,52829],{"className":52830,"style":8839},[7133],[35,52832,52834],{"className":52833},[7104],[35,52835,500],{"className":52836},[7104],[35,52838,7756],{"className":52839},[7755],[35,52841,52843],{"className":52842},[7121],[35,52844,52846],{"className":52845,"style":34041},[7125],[35,52847],{},[35,52849],{"className":52850},[7314,14311],[35,52852,5122],{"className":52853},[7104],[12,52855,52856],{},"Given the default drive strength",[35,52858,52860],{"className":52859},[7318],[35,52861,52863,52903],{"className":52862},[7057],[35,52864,52866],{"className":52865},[7061],[7063,52867,52868],{"xmlns":7065,"display":7327},[7067,52869,52870,52900],{},[7070,52871,52872,52882,52884,52887,52889,52898],{},[14009,52873,52874,52876],{},[7076,52875,47141],{"mathvariant":7282},[7070,52877,52878,52880],{},[7227,52879,460],{},[7076,52881,13081],{},[7223,52883,242],{},[7227,52885,52886],{},"20",[10694,52888,10696],{},[7070,52890,52891,52894,52896],{},[7076,52892,52893],{"mathvariant":7282},"M",[7076,52895,7669],{"mathvariant":7282},[7076,52897,300],{"mathvariant":7282},[7223,52899,2877],{"separator":7091},[7083,52901,52902],{"encoding":7085},"\\frac{\\Omega}{2\\pi}=20~\\mathrm{MHz},",[35,52904,52906,52987],{"className":52905,"ariaHidden":7091},[7090],[35,52907,52909,52913,52978,52981,52984],{"className":52908},[7095],[35,52910],{"className":52911,"style":52912},[7099],"height:2.0463em;vertical-align:-0.686em;",[35,52914,52916,52919,52975],{"className":52915},[7104],[35,52917],{"className":52918},[7472,14311],[35,52920,52922],{"className":52921},[14009],[35,52923,52925,52967],{"className":52924},[7117,7731],[35,52926,52928,52964],{"className":52927},[7121],[35,52929,52931,52945,52953],{"className":52930,"style":51519},[7125],[35,52932,52933,52936],{"style":16234},[35,52934],{"className":52935,"style":8839},[7133],[35,52937,52939,52942],{"className":52938},[7104],[35,52940,460],{"className":52941},[7104],[35,52943,13081],{"className":52944,"style":7310},[7104,7145],[35,52946,52947,52950],{"style":14395},[35,52948],{"className":52949,"style":8839},[7133],[35,52951],{"className":52952,"style":14403},[14402],[35,52954,52955,52958],{"style":14406},[35,52956],{"className":52957,"style":8839},[7133],[35,52959,52961],{"className":52960},[7104],[35,52962,47141],{"className":52963},[7104],[35,52965,7756],{"className":52966},[7755],[35,52968,52970],{"className":52969},[7121],[35,52971,52973],{"className":52972,"style":16272},[7125],[35,52974],{},[35,52976],{"className":52977},[7314,14311],[35,52979],{"className":52980,"style":7450},[7248],[35,52982,242],{"className":52983},[7454],[35,52985],{"className":52986,"style":7450},[7248],[35,52988,52990,52994,52997,53000,53007],{"className":52989},[7095],[35,52991],{"className":52992,"style":52993},[7099],"height:0.8778em;vertical-align:-0.1944em;",[35,52995,52886],{"className":52996},[7104],[35,52998,10696],{"className":52999},[7248,48368],[35,53001,53003],{"className":53002},[7104],[35,53004,53006],{"className":53005},[7104,48375],"MHz",[35,53008,2877],{"className":53009},[7492],[12,53011,53012,53013,53041],{},"the ideal ",[35,53014,53016,53029],{"className":53015},[7057],[35,53017,53019],{"className":53018},[7061],[7063,53020,53021],{"xmlns":7065},[7067,53022,53023,53027],{},[7070,53024,53025],{},[7076,53026,13081],{},[7083,53028,52563],{"encoding":7085},[35,53030,53032],{"className":53031,"ariaHidden":7091},[7090],[35,53033,53035,53038],{"className":53034},[7095],[35,53036],{"className":53037,"style":7173},[7099],[35,53039,13081],{"className":53040,"style":7310},[7104,7145],"-pulse time is",[35,53043,53045],{"className":53044},[7318],[35,53046,53048,53080],{"className":53047},[7057],[35,53049,53051],{"className":53050},[7061],[7063,53052,53053],{"xmlns":7065,"display":7327},[7067,53054,53055,53077],{},[7070,53056,53057,53063,53065,53067,53069,53075],{},[7699,53058,53059,53061],{},[7076,53060,46183],{},[7076,53062,13081],{},[7223,53064,242],{},[7227,53066,2680],{},[10694,53068,10696],{},[7070,53070,53071,53073],{},[7076,53072,291],{"mathvariant":7282},[7076,53074,19046],{"mathvariant":7282},[7076,53076,5122],{"mathvariant":7282},[7083,53078,53079],{"encoding":7085},"\\tau_\\pi=25~\\mathrm{ns}.",[35,53081,53083,53138],{"className":53082,"ariaHidden":7091},[7090],[35,53084,53086,53089,53129,53132,53135],{"className":53085},[7095],[35,53087],{"className":53088,"style":29689},[7099],[35,53090,53092,53095],{"className":53091},[7104],[35,53093,46183],{"className":53094,"style":46255},[7104,7145],[35,53096,53098],{"className":53097},[7113],[35,53099,53101,53121],{"className":53100},[7117,7731],[35,53102,53104,53118],{"className":53103},[7121],[35,53105,53107],{"className":53106,"style":9035},[7125],[35,53108,53109,53112],{"style":52662},[35,53110],{"className":53111,"style":7134},[7133],[35,53113,53115],{"className":53114},[7138,7139,7140,7141],[35,53116,13081],{"className":53117,"style":7310},[7104,7145,7141],[35,53119,7756],{"className":53120},[7755],[35,53122,53124],{"className":53123},[7121],[35,53125,53127],{"className":53126,"style":7763},[7125],[35,53128],{},[35,53130],{"className":53131,"style":7450},[7248],[35,53133,242],{"className":53134},[7454],[35,53136],{"className":53137,"style":7450},[7248],[35,53139,53141,53144,53147,53150,53157],{"className":53140},[7095],[35,53142],{"className":53143,"style":7263},[7099],[35,53145,2680],{"className":53146},[7104],[35,53148,10696],{"className":53149},[7248,48368],[35,53151,53153],{"className":53152},[7104],[35,53154,53156],{"className":53155},[7104,48375],"ns",[35,53158,5122],{"className":53159},[7104],[12,53161,53162],{},"A full period of the resonant excited-state probability oscillation is",[35,53164,53166],{"className":53165},[7318],[35,53167,53169,53217],{"className":53168},[7057],[35,53170,53172],{"className":53171},[7061],[7063,53173,53174],{"xmlns":7065,"display":7327},[7067,53175,53176,53214],{},[7070,53177,53178,53184,53186,53196,53198,53212],{},[7699,53179,53180,53182],{},[7076,53181,41530],{},[7076,53183,24883],{},[7223,53185,242],{},[14009,53187,53188,53194],{},[7070,53189,53190,53192],{},[7227,53191,460],{},[7076,53193,13081],{},[7076,53195,47141],{"mathvariant":7282},[7223,53197,242],{},[14009,53199,53200,53202],{},[7227,53201,500],{},[7070,53203,53204,53206,53208,53210],{},[7076,53205,47141],{"mathvariant":7282},[7076,53207,398],{"mathvariant":7282},[7227,53209,460],{},[7076,53211,13081],{},[7076,53213,5122],{"mathvariant":7282},[7083,53215,53216],{"encoding":7085},"T_R=\\frac{2\\pi}{\\Omega}=\\frac{1}{\\Omega\u002F2\\pi}.",[35,53218,53220,53275,53356],{"className":53219,"ariaHidden":7091},[7090],[35,53221,53223,53226,53266,53269,53272],{"className":53222},[7095],[35,53224],{"className":53225,"style":7718},[7099],[35,53227,53229,53232],{"className":53228},[7104],[35,53230,41530],{"className":53231,"style":7751},[7104,7145],[35,53233,53235],{"className":53234},[7113],[35,53236,53238,53258],{"className":53237},[7117,7731],[35,53239,53241,53255],{"className":53240},[7121],[35,53242,53244],{"className":53243,"style":7738},[7125],[35,53245,53246,53249],{"style":12163},[35,53247],{"className":53248,"style":7134},[7133],[35,53250,53252],{"className":53251},[7138,7139,7140,7141],[35,53253,24883],{"className":53254,"style":24908},[7104,7145,7141],[35,53256,7756],{"className":53257},[7755],[35,53259,53261],{"className":53260},[7121],[35,53262,53264],{"className":53263,"style":7763},[7125],[35,53265],{},[35,53267],{"className":53268,"style":7450},[7248],[35,53270,242],{"className":53271},[7454],[35,53273],{"className":53274,"style":7450},[7248],[35,53276,53278,53282,53347,53350,53353],{"className":53277},[7095],[35,53279],{"className":53280,"style":53281},[7099],"height:2.0074em;vertical-align:-0.686em;",[35,53283,53285,53288,53344],{"className":53284},[7104],[35,53286],{"className":53287},[7472,14311],[35,53289,53291],{"className":53290},[14009],[35,53292,53294,53336],{"className":53293},[7117,7731],[35,53295,53297,53333],{"className":53296},[7121],[35,53298,53300,53311,53319],{"className":53299,"style":14324},[7125],[35,53301,53302,53305],{"style":16234},[35,53303],{"className":53304,"style":8839},[7133],[35,53306,53308],{"className":53307},[7104],[35,53309,47141],{"className":53310},[7104],[35,53312,53313,53316],{"style":14395},[35,53314],{"className":53315,"style":8839},[7133],[35,53317],{"className":53318,"style":14403},[14402],[35,53320,53321,53324],{"style":14406},[35,53322],{"className":53323,"style":8839},[7133],[35,53325,53327,53330],{"className":53326},[7104],[35,53328,460],{"className":53329},[7104],[35,53331,13081],{"className":53332,"style":7310},[7104,7145],[35,53334,7756],{"className":53335},[7755],[35,53337,53339],{"className":53338},[7121],[35,53340,53342],{"className":53341,"style":16272},[7125],[35,53343],{},[35,53345],{"className":53346},[7314,14311],[35,53348],{"className":53349,"style":7450},[7248],[35,53351,242],{"className":53352},[7454],[35,53354],{"className":53355,"style":7450},[7248],[35,53357,53359,53362,53427],{"className":53358},[7095],[35,53360],{"className":53361,"style":40323},[7099],[35,53363,53365,53368,53424],{"className":53364},[7104],[35,53366],{"className":53367},[7472,14311],[35,53369,53371],{"className":53370},[14009],[35,53372,53374,53416],{"className":53373},[7117,7731],[35,53375,53377,53413],{"className":53376},[7121],[35,53378,53380,53394,53402],{"className":53379,"style":14324},[7125],[35,53381,53382,53385],{"style":16234},[35,53383],{"className":53384,"style":8839},[7133],[35,53386,53388,53391],{"className":53387},[7104],[35,53389,47199],{"className":53390},[7104],[35,53392,13081],{"className":53393,"style":7310},[7104,7145],[35,53395,53396,53399],{"style":14395},[35,53397],{"className":53398,"style":8839},[7133],[35,53400],{"className":53401,"style":14403},[14402],[35,53403,53404,53407],{"style":14406},[35,53405],{"className":53406,"style":8839},[7133],[35,53408,53410],{"className":53409},[7104],[35,53411,500],{"className":53412},[7104],[35,53414,7756],{"className":53415},[7755],[35,53417,53419],{"className":53418},[7121],[35,53420,53422],{"className":53421,"style":34041},[7125],[35,53423],{},[35,53425],{"className":53426},[7314,14311],[35,53428,5122],{"className":53429},[7104],[12,53431,53432],{},"For the same default drive strength, the full temporal period is",[35,53434,53436],{"className":53435},[7318],[35,53437,53439,53471],{"className":53438},[7057],[35,53440,53442],{"className":53441},[7061],[7063,53443,53444],{"xmlns":7065,"display":7327},[7067,53445,53446,53468],{},[7070,53447,53448,53454,53456,53458,53460,53466],{},[7699,53449,53450,53452],{},[7076,53451,41530],{},[7076,53453,24883],{},[7223,53455,242],{},[7227,53457,4877],{},[10694,53459,10696],{},[7070,53461,53462,53464],{},[7076,53463,291],{"mathvariant":7282},[7076,53465,19046],{"mathvariant":7282},[7076,53467,5122],{"mathvariant":7282},[7083,53469,53470],{"encoding":7085},"T_R=50~\\mathrm{ns}.",[35,53472,53474,53529],{"className":53473,"ariaHidden":7091},[7090],[35,53475,53477,53480,53520,53523,53526],{"className":53476},[7095],[35,53478],{"className":53479,"style":7718},[7099],[35,53481,53483,53486],{"className":53482},[7104],[35,53484,41530],{"className":53485,"style":7751},[7104,7145],[35,53487,53489],{"className":53488},[7113],[35,53490,53492,53512],{"className":53491},[7117,7731],[35,53493,53495,53509],{"className":53494},[7121],[35,53496,53498],{"className":53497,"style":7738},[7125],[35,53499,53500,53503],{"style":12163},[35,53501],{"className":53502,"style":7134},[7133],[35,53504,53506],{"className":53505},[7138,7139,7140,7141],[35,53507,24883],{"className":53508,"style":24908},[7104,7145,7141],[35,53510,7756],{"className":53511},[7755],[35,53513,53515],{"className":53514},[7121],[35,53516,53518],{"className":53517,"style":7763},[7125],[35,53519],{},[35,53521],{"className":53522,"style":7450},[7248],[35,53524,242],{"className":53525},[7454],[35,53527],{"className":53528,"style":7450},[7248],[35,53530,53532,53535,53538,53541,53547],{"className":53531},[7095],[35,53533],{"className":53534,"style":7263},[7099],[35,53536,4877],{"className":53537},[7104],[35,53539,10696],{"className":53540},[7248,48368],[35,53542,53544],{"className":53543},[7104],[35,53545,53156],{"className":53546},[7104,48375],[35,53548,5122],{"className":53549},[7104],[5019,53551,53553],{"id":53552},"phenomenological-dephasing","Phenomenological Dephasing",[12,53555,53556],{},"The computational model includes an optional phenomenological damping factor,",[35,53558,53560],{"className":53559},[7318],[35,53561,53563,53635],{"className":53562},[7057],[35,53564,53566],{"className":53565},[7061],[7063,53567,53568],{"xmlns":7065,"display":7327},[7067,53569,53570,53632],{},[7070,53571,53572,53578,53580,53582,53584,53586,53588,53590,53596,53598,53600,53602,53604,53606,53608,53610,53630],{},[7699,53573,53574,53576],{},[7076,53575,7705],{},[7076,53577,7817],{},[7223,53579,275],{"stretchy":7289},[7076,53581,1675],{},[7223,53583,2877],{"separator":7091},[7076,53585,46183],{},[7223,53587,1694],{"stretchy":7289},[7223,53589,19447],{},[7699,53591,53592,53594],{},[7076,53593,7705],{},[7076,53595,7817],{},[7223,53597,275],{"stretchy":7289},[7076,53599,1675],{},[7223,53601,2877],{"separator":7091},[7076,53603,46183],{},[7223,53605,1694],{"stretchy":7289},[7076,53607,28563],{},[7223,53609,41175],{},[7070,53611,53612,53614,53616,53628],{},[7223,53613,275],{"fence":7091},[7223,53615,7225],{},[14009,53617,53618,53620],{},[7076,53619,46183],{},[8304,53621,53622,53624,53626],{},[7076,53623,41530],{},[7227,53625,460],{},[7223,53627,25282],{},[7223,53629,1694],{"fence":7091},[7076,53631,5122],{"mathvariant":7282},[7083,53633,53634],{"encoding":7085},"P_e(f,\\tau)\\rightarrow P_e(f,\\tau)\\exp\\left(-\\frac{\\tau}{T_2^\\ast}\\right).",[35,53636,53638,53711],{"className":53637,"ariaHidden":7091},[7090],[35,53639,53641,53644,53684,53687,53690,53693,53696,53699,53702,53705,53708],{"className":53640},[7095],[35,53642],{"className":53643,"style":7303},[7099],[35,53645,53647,53650],{"className":53646},[7104],[35,53648,7705],{"className":53649,"style":7751},[7104,7145],[35,53651,53653],{"className":53652},[7113],[35,53654,53656,53676],{"className":53655},[7117,7731],[35,53657,53659,53673],{"className":53658},[7121],[35,53660,53662],{"className":53661,"style":9035},[7125],[35,53663,53664,53667],{"style":12163},[35,53665],{"className":53666,"style":7134},[7133],[35,53668,53670],{"className":53669},[7138,7139,7140,7141],[35,53671,7817],{"className":53672},[7104,7145,7141],[35,53674,7756],{"className":53675},[7755],[35,53677,53679],{"className":53678},[7121],[35,53680,53682],{"className":53681,"style":7763},[7125],[35,53683],{},[35,53685,275],{"className":53686},[7472],[35,53688,1675],{"className":53689,"style":31824},[7104,7145],[35,53691,2877],{"className":53692},[7492],[35,53694],{"className":53695,"style":7496},[7248],[35,53697,46183],{"className":53698,"style":46255},[7104,7145],[35,53700,1694],{"className":53701},[7314],[35,53703],{"className":53704,"style":7450},[7248],[35,53706,19447],{"className":53707},[7454],[35,53709],{"className":53710,"style":7450},[7248],[35,53712,53714,53718,53758,53761,53764,53767,53770,53773,53776,53779,53782,53785,53917,53920],{"className":53713},[7095],[35,53715],{"className":53716,"style":53717},[7099],"height:2.4023em;vertical-align:-0.9523em;",[35,53719,53721,53724],{"className":53720},[7104],[35,53722,7705],{"className":53723,"style":7751},[7104,7145],[35,53725,53727],{"className":53726},[7113],[35,53728,53730,53750],{"className":53729},[7117,7731],[35,53731,53733,53747],{"className":53732},[7121],[35,53734,53736],{"className":53735,"style":9035},[7125],[35,53737,53738,53741],{"style":12163},[35,53739],{"className":53740,"style":7134},[7133],[35,53742,53744],{"className":53743},[7138,7139,7140,7141],[35,53745,7817],{"className":53746},[7104,7145,7141],[35,53748,7756],{"className":53749},[7755],[35,53751,53753],{"className":53752},[7121],[35,53754,53756],{"className":53755,"style":7763},[7125],[35,53757],{},[35,53759,275],{"className":53760},[7472],[35,53762,1675],{"className":53763,"style":31824},[7104,7145],[35,53765,2877],{"className":53766},[7492],[35,53768],{"className":53769,"style":7496},[7248],[35,53771,46183],{"className":53772,"style":46255},[7104,7145],[35,53774,1694],{"className":53775},[7314],[35,53777],{"className":53778,"style":7496},[7248],[35,53780,28563],{"className":53781},[12096],[35,53783],{"className":53784,"style":7496},[7248],[35,53786,53788,53794,53797,53911],{"className":53787},[7468],[35,53789,53791],{"className":53790,"style":7474},[7472,7473],[35,53792,275],{"className":53793},[7478,7140],[35,53795,7225],{"className":53796},[7104],[35,53798,53800,53803,53908],{"className":53799},[7104],[35,53801],{"className":53802},[7472,14311],[35,53804,53806],{"className":53805},[14009],[35,53807,53809,53899],{"className":53808},[7117,7731],[35,53810,53812,53896],{"className":53811},[7121],[35,53813,53815,53877,53885],{"className":53814,"style":52717},[7125],[35,53816,53817,53820],{"style":16234},[35,53818],{"className":53819,"style":8839},[7133],[35,53821,53823],{"className":53822},[7104],[35,53824,53826,53829],{"className":53825},[7104],[35,53827,41530],{"className":53828,"style":7751},[7104,7145],[35,53830,53832],{"className":53831},[7113],[35,53833,53835,53868],{"className":53834},[7117,7731],[35,53836,53838,53865],{"className":53837},[7121],[35,53839,53842,53854],{"className":53840,"style":53841},[7125],"height:0.6705em;",[35,53843,53845,53848],{"style":53844},"top:-2.4337em;margin-left:-0.1389em;margin-right:0.05em;",[35,53846],{"className":53847,"style":7134},[7133],[35,53849,53851],{"className":53850},[7138,7139,7140,7141],[35,53852,460],{"className":53853},[7104,7141],[35,53855,53856,53859],{"style":51373},[35,53857],{"className":53858,"style":7134},[7133],[35,53860,53862],{"className":53861},[7138,7139,7140,7141],[35,53863,25282],{"className":53864},[7253,7141],[35,53866,7756],{"className":53867},[7755],[35,53869,53871],{"className":53870},[7121],[35,53872,53875],{"className":53873,"style":53874},[7125],"height:0.2663em;",[35,53876],{},[35,53878,53879,53882],{"style":14395},[35,53880],{"className":53881,"style":8839},[7133],[35,53883],{"className":53884,"style":14403},[14402],[35,53886,53887,53890],{"style":14406},[35,53888],{"className":53889,"style":8839},[7133],[35,53891,53893],{"className":53892},[7104],[35,53894,46183],{"className":53895,"style":46255},[7104,7145],[35,53897,7756],{"className":53898},[7755],[35,53900,53902],{"className":53901},[7121],[35,53903,53906],{"className":53904,"style":53905},[7125],"height:0.9523em;",[35,53907],{},[35,53909],{"className":53910},[7314,14311],[35,53912,53914],{"className":53913,"st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exponential envelope attenuates the plotted Rabi fringes as pulse duration increases. The factor is a compact visualization device for contrast decay. A rigorous open-quantum-system treatment would require a density matrix model with relaxation channels, dephasing channels, finite temperature effects, shaped pulse envelopes, and stochastic dynamics. The default dephasing parameter is",[35,53927,53929],{"className":53928},[7318],[35,53930,53932,53967],{"className":53931},[7057],[35,53933,53935],{"className":53934},[7061],[7063,53936,53937],{"xmlns":7065,"display":7327},[7067,53938,53939,53964],{},[7070,53940,53941,53949,53951,53954,53956,53962],{},[8304,53942,53943,53945,53947],{},[7076,53944,41530],{},[7227,53946,460],{},[7223,53948,25282],{},[7223,53950,242],{},[7227,53952,53953],{},"500",[10694,53955,10696],{},[7070,53957,53958,53960],{},[7076,53959,291],{"mathvariant":7282},[7076,53961,19046],{"mathvariant":7282},[7076,53963,5122],{"mathvariant":7282},[7083,53965,53966],{"encoding":7085},"T_2^\\ast=500~\\mathrm{ns}.",[35,53968,53970,54038],{"className":53969,"ariaHidden":7091},[7090],[35,53971,53973,53977,54029,54032,54035],{"className":53972},[7095],[35,53974],{"className":53975,"style":53976},[7099],"height:0.9857em;vertical-align:-0.247em;",[35,53978,53980,53983],{"className":53979},[7104],[35,53981,41530],{"className":53982,"style":7751},[7104,7145],[35,53984,53986],{"className":53985},[7113],[35,53987,53989,54021],{"className":53988},[7117,7731],[35,53990,53992,54018],{"className":53991},[7121],[35,53993,53995,54007],{"className":53994,"style":41629},[7125],[35,53996,53998,54001],{"style":53997},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[35,53999],{"className":54000,"style":7134},[7133],[35,54002,54004],{"className":54003},[7138,7139,7140,7141],[35,54005,460],{"className":54006},[7104,7141],[35,54008,54009,54012],{"style":7437},[35,54010],{"className":54011,"style":7134},[7133],[35,54013,54015],{"className":54014},[7138,7139,7140,7141],[35,54016,25282],{"className":54017},[7253,7141],[35,54019,7756],{"className":54020},[7755],[35,54022,54024],{"className":54023},[7121],[35,54025,54027],{"className":54026,"style":8442},[7125],[35,54028],{},[35,54030],{"className":54031,"style":7450},[7248],[35,54033,242],{"className":54034},[7454],[35,54036],{"className":54037,"style":7450},[7248],[35,54039,54041,54044,54047,54050,54056],{"className":54040},[7095],[35,54042],{"className":54043,"style":7263},[7099],[35,54045,53953],{"className":54046},[7104],[35,54048,10696],{"className":54049},[7248,48368],[35,54051,54053],{"className":54052},[7104],[35,54054,53156],{"className":54055},[7104,48375],[35,54057,5122],{"className":54058},[7104],[12,54060,54061,54062,54065],{},"Setting ",[32,54063,54064],{},"T2STAR_NS = None"," disables the phenomenological damping factor.",[5019,54067,54069],{"id":54068},"detuning-and-generalized-rabi-frequency","Detuning and Generalized Rabi Frequency",[12,54071,54072],{},"Detuning quantifies the spectral mismatch between the applied drive and the natural qubit 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At resonance, the generalized frequency equals the applied drive strength,",[35,54764,54766],{"className":54765},[7318],[35,54767,54769,54811],{"className":54768},[7057],[35,54770,54772],{"className":54771},[7061],[7063,54773,54774],{"xmlns":7065,"display":7327},[7067,54775,54776,54808],{},[7070,54777,54778,54784,54786,54792,54794,54796,54806],{},[7699,54779,54780,54782],{},[7076,54781,47454],{},[7076,54783,24883],{},[7223,54785,275],{"stretchy":7289},[7699,54787,54788,54790],{},[7076,54789,1675],{},[7227,54791,5312],{},[7223,54793,1694],{"stretchy":7289},[7223,54795,242],{},[14009,54797,54798,54800],{},[7076,54799,47141],{"mathvariant":7282},[7070,54801,54802,54804],{},[7227,54803,460],{},[7076,54805,13081],{},[7076,54807,5122],{"mathvariant":7282},[7083,54809,54810],{"encoding":7085},"\\nu_R(f_0)=\\frac{\\Omega}{2\\pi}.",[35,54812,54814,54915],{"className":54813,"ariaHidden":7091},[7090],[35,54815,54817,54820,54860,54863,54903,54906,54909,54912],{"className":54816},[7095],[35,54818],{"className":54819,"style":7303},[7099],[35,54821,54823,54826],{"className":54822},[7104],[35,54824,47454],{"className":54825,"style":47475},[7104,7145],[35,54827,54829],{"className":54828},[7113],[35,54830,54832,54852],{"className":54831},[7117,7731],[35,54833,54835,54849],{"className":54834},[7121],[35,54836,54838],{"className":54837,"style":7738},[7125],[35,54839,54840,54843],{"style":47490},[35,54841],{"className":54842,"style":7134},[7133],[35,54844,54846],{"className":54845},[7138,7139,7140,7141],[35,54847,24883],{"className":54848,"style":24908},[7104,7145,7141],[35,54850,7756],{"className":54851},[7755],[35,54853,54855],{"className":54854},[7121],[35,54856,54858],{"className":54857,"style":7763},[7125],[35,54859],{},[35,54861,275],{"className":54862},[7472],[35,54864,54866,54869],{"className":54865},[7104],[35,54867,1675],{"className":54868,"style":31824},[7104,7145],[35,54870,54872],{"className":54871},[7113],[35,54873,54875,54895],{"className":54874},[7117,7731],[35,54876,54878,54892],{"className":54877},[7121],[35,54879,54881],{"className":54880,"style":18794},[7125],[35,54882,54883,54886],{"style":46746},[35,54884],{"className":54885,"style":7134},[7133],[35,54887,54889],{"className":54888},[7138,7139,7140,7141],[35,54890,5312],{"className":54891},[7104,7141],[35,54893,7756],{"className":54894},[7755],[35,54896,54898],{"className":54897},[7121],[35,54899,54901],{"className":54900,"style":7763},[7125],[35,54902],{},[35,54904,1694],{"className":54905},[7314],[35,54907],{"className":54908,"style":7450},[7248],[35,54910,242],{"className":54911},[7454],[35,54913],{"className":54914,"style":7450},[7248],[35,54916,54918,54921,54986],{"className":54917},[7095],[35,54919],{"className":54920,"style":52912},[7099],[35,54922,54924,54927,54983],{"className":54923},[7104],[35,54925],{"className":54926},[7472,14311],[35,54928,54930],{"className":54929},[14009],[35,54931,54933,54975],{"className":54932},[7117,7731],[35,54934,54936,54972],{"className":54935},[7121],[35,54937,54939,54953,54961],{"className":54938,"style":51519},[7125],[35,54940,54941,54944],{"style":16234},[35,54942],{"className":54943,"style":8839},[7133],[35,54945,54947,54950],{"className":54946},[7104],[35,54948,460],{"className":54949},[7104],[35,54951,13081],{"className":54952,"style":7310},[7104,7145],[35,54954,54955,54958],{"style":14395},[35,54956],{"className":54957,"style":8839},[7133],[35,54959],{"className":54960,"style":14403},[14402],[35,54962,54963,54966],{"style":14406},[35,54964],{"className":54965,"style":8839},[7133],[35,54967,54969],{"className":54968},[7104],[35,54970,47141],{"className":54971},[7104],[35,54973,7756],{"className":54974},[7755],[35,54976,54978],{"className":54977},[7121],[35,54979,54981],{"className":54980,"style":16272},[7125],[35,54982],{},[35,54984],{"className":54985},[7314,14311],[35,54987,5122],{"className":54988},[7104],[12,54990,54991,54992,55061,55062,55164],{},"As the system moves away from resonance, ",[35,54993,54995,55012],{"className":54994},[7057],[35,54996,54998],{"className":54997},[7061],[7063,54999,55000],{"xmlns":7065},[7067,55001,55002,55010],{},[7070,55003,55004],{},[7699,55005,55006,55008],{},[7076,55007,47454],{},[7076,55009,24883],{},[7083,55011,47459],{"encoding":7085},[35,55013,55015],{"className":55014,"ariaHidden":7091},[7090],[35,55016,55018,55021],{"className":55017},[7095],[35,55019],{"className":55020,"style":29689},[7099],[35,55022,55024,55027],{"className":55023},[7104],[35,55025,47454],{"className":55026,"style":47475},[7104,7145],[35,55028,55030],{"className":55029},[7113],[35,55031,55033,55053],{"className":55032},[7117,7731],[35,55034,55036,55050],{"className":55035},[7121],[35,55037,55039],{"className":55038,"style":7738},[7125],[35,55040,55041,55044],{"style":47490},[35,55042],{"className":55043,"style":7134},[7133],[35,55045,55047],{"className":55046},[7138,7139,7140,7141],[35,55048,24883],{"className":55049,"style":24908},[7104,7145,7141],[35,55051,7756],{"className":55052},[7755],[35,55054,55056],{"className":55055},[7121],[35,55057,55059],{"className":55058,"style":7763},[7125],[35,55060],{}," increases symmetrically with ",[35,55063,55065,55091],{"className":55064},[7057],[35,55066,55068],{"className":55067},[7061],[7063,55069,55070],{"xmlns":7065},[7067,55071,55072,55088],{},[7070,55073,55074,55076,55078,55080,55086],{},[7223,55075,7283],{"stretchy":7289},[7076,55077,1675],{},[7223,55079,7225],{},[7699,55081,55082,55084],{},[7076,55083,1675],{},[7227,55085,5312],{},[7223,55087,7283],{"stretchy":7289},[7083,55089,55090],{"encoding":7085},"\\lvert f-f_0\\rvert",[35,55092,55094,55115],{"className":55093,"ariaHidden":7091},[7090],[35,55095,55097,55100,55103,55106,55109,55112],{"className":55096},[7095],[35,55098],{"className":55099,"style":7303},[7099],[35,55101,7283],{"className":55102},[7472],[35,55104,1675],{"className":55105,"style":31824},[7104,7145],[35,55107],{"className":55108,"style":7249},[7248],[35,55110,7225],{"className":55111},[7253],[35,55113],{"className":55114,"style":7249},[7248],[35,55116,55118,55121,55161],{"className":55117},[7095],[35,55119],{"className":55120,"style":7303},[7099],[35,55122,55124,55127],{"className":55123},[7104],[35,55125,1675],{"className":55126,"style":31824},[7104,7145],[35,55128,55130],{"className":55129},[7113],[35,55131,55133,55153],{"className":55132},[7117,7731],[35,55134,55136,55150],{"className":55135},[7121],[35,55137,55139],{"className":55138,"style":18794},[7125],[35,55140,55141,55144],{"style":46746},[35,55142],{"className":55143,"style":7134},[7133],[35,55145,55147],{"className":55146},[7138,7139,7140,7141],[35,55148,5312],{"className":55149},[7104,7141],[35,55151,7756],{"className":55152},[7755],[35,55154,55156],{"className":55155},[7121],[35,55157,55159],{"className":55158,"style":7763},[7125],[35,55160],{},[35,55162,7283],{"className":55163},[7314],". The oscillations accelerate along the pulse-duration axis, and the maximum achievable excited-state probability decreases. This tradeoff is the defining visual signature of detuned Rabi dynamics.",[5019,55166,55168],{"id":55167},"quantum-information-interpretation","Quantum Information Interpretation",[12,55170,55171,55172,55242],{},"In quantum information language, a calibrated resonant Rabi pulse implements a single-qubit rotation. At exact resonance, the effective Hamiltonian is proportional to ",[35,55173,55175,55193],{"className":55174},[7057],[35,55176,55178],{"className":55177},[7061],[7063,55179,55180],{"xmlns":7065},[7067,55181,55182,55190],{},[7070,55183,55184],{},[7699,55185,55186,55188],{},[7076,55187,49186],{},[7076,55189,13657],{},[7083,55191,55192],{"encoding":7085},"\\sigma_x",[35,55194,55196],{"className":55195,"ariaHidden":7091},[7090],[35,55197,55199,55202],{"className":55198},[7095],[35,55200],{"className":55201,"style":29689},[7099],[35,55203,55205,55208],{"className":55204},[7104],[35,55206,49186],{"className":55207,"style":7310},[7104,7145],[35,55209,55211],{"className":55210},[7113],[35,55212,55214,55234],{"className":55213},[7117,7731],[35,55215,55217,55231],{"className":55216},[7121],[35,55218,55220],{"className":55219,"style":9035},[7125],[35,55221,55222,55225],{"style":9100},[35,55223],{"className":55224,"style":7134},[7133],[35,55226,55228],{"className":55227},[7138,7139,7140,7141],[35,55229,13657],{"className":55230},[7104,7145,7141],[35,55232,7756],{"className":55233},[7755],[35,55235,55237],{"className":55236},[7121],[35,55238,55240],{"className":55239,"style":7763},[7125],[35,55241],{},", so the microwave pulse implements",[35,55244,55246],{"className":55245},[7318],[35,55247,55249,55306],{"className":55248},[7057],[35,55250,55252],{"className":55251},[7061],[7063,55253,55254],{"xmlns":7065,"display":7327},[7067,55255,55256,55303],{},[7070,55257,55258,55264,55266,55269,55271,55273,55275,55277,55301],{},[7699,55259,55260,55262],{},[7076,55261,24883],{},[7076,55263,13657],{},[7223,55265,275],{"stretchy":7289},[7076,55267,55268],{},"θ",[7223,55270,1694],{"stretchy":7289},[7223,55272,242],{},[7076,55274,28563],{},[7223,55276,41175],{},[7070,55278,55279,55281,55283,55299],{},[7223,55280,275],{"fence":7091},[7223,55282,7225],{},[14009,55284,55285,55297],{},[7070,55286,55287,55289,55291],{},[7076,55288,7824],{},[7076,55290,55268],{},[7699,55292,55293,55295],{},[7076,55294,49186],{},[7076,55296,13657],{},[7227,55298,460],{},[7223,55300,1694],{"fence":7091},[7223,55302,2877],{"separator":7091},[7083,55304,55305],{"encoding":7085},"R_x(\\theta)=\\exp\\left(-\\frac{i\\theta\\sigma_x}{2}\\right),",[35,55307,55309,55373],{"className":55308,"ariaHidden":7091},[7090],[35,55310,55312,55315,55355,55358,55361,55364,55367,55370],{"className":55311},[7095],[35,55313],{"className":55314,"style":7303},[7099],[35,55316,55318,55321],{"className":55317},[7104],[35,55319,24883],{"className":55320,"style":24908},[7104,7145],[35,55322,55324],{"className":55323},[7113],[35,55325,55327,55347],{"className":55326},[7117,7731],[35,55328,55330,55344],{"className":55329},[7121],[35,55331,55333],{"className":55332,"style":9035},[7125],[35,55334,55335,55338],{"style":24923},[35,55336],{"className":55337,"style":7134},[7133],[35,55339,55341],{"className":55340},[7138,7139,7140,7141],[35,55342,13657],{"className":55343},[7104,7145,7141],[35,55345,7756],{"className":55346},[7755],[35,55348,55350],{"className":55349},[7121],[35,55351,55353],{"className":55352,"style":7763},[7125],[35,55354],{},[35,55356,275],{"className":55357},[7472],[35,55359,55268],{"className":55360,"style":7146},[7104,7145],[35,55362,1694],{"className":55363},[7314],[35,55365],{"className":55366,"style":7450},[7248],[35,55368,242],{"className":55369},[7454],[35,55371],{"className":55372,"style":7450},[7248],[35,55374,55376,55379,55382,55385,55508,55511],{"className":55375},[7095],[35,55377],{"className":55378,"style":42152},[7099],[35,55380,28563],{"className":55381},[12096],[35,55383],{"className":55384,"style":7496},[7248],[35,55386,55388,55394,55397,55502],{"className":55387},[7468],[35,55389,55391],{"className":55390,"style":7474},[7472,7473],[35,55392,275],{"className":55393},[7478,7140],[35,55395,7225],{"className":55396},[7104],[35,55398,55400,55403,55499],{"className":55399},[7104],[35,55401],{"className":55402},[7472,14311],[35,55404,55406],{"className":55405},[14009],[35,55407,55409,55491],{"className":55408},[7117,7731],[35,55410,55412,55488],{"className":55411},[7121],[35,55413,55415,55426,55434],{"className":55414,"style":42242},[7125],[35,55416,55417,55420],{"style":16234},[35,55418],{"className":55419,"style":8839},[7133],[35,55421,55423],{"className":55422},[7104],[35,55424,460],{"className":55425},[7104],[35,55427,55428,55431],{"style":14395},[35,55429],{"className":55430,"style":8839},[7133],[35,55432],{"className":55433,"style":14403},[14402],[35,55435,55436,55439],{"style":14406},[35,55437],{"className":55438,"style":8839},[7133],[35,55440,55442,55445,55448],{"className":55441},[7104],[35,55443,7824],{"className":55444},[7104,7145],[35,55446,55268],{"className":55447,"style":7146},[7104,7145],[35,55449,55451,55454],{"className":55450},[7104],[35,55452,49186],{"className":55453,"style":7310},[7104,7145],[35,55455,55457],{"className":55456},[7113],[35,55458,55460,55480],{"className":55459},[7117,7731],[35,55461,55463,55477],{"className":55462},[7121],[35,55464,55466],{"className":55465,"style":9035},[7125],[35,55467,55468,55471],{"style":9100},[35,55469],{"className":55470,"style":7134},[7133],[35,55472,55474],{"className":55473},[7138,7139,7140,7141],[35,55475,13657],{"className":55476},[7104,7145,7141],[35,55478,7756],{"className":55479},[7755],[35,55481,55483],{"className":55482},[7121],[35,55484,55486],{"className":55485,"style":7763},[7125],[35,55487],{},[35,55489,7756],{"className":55490},[7755],[35,55492,55494],{"className":55493},[7121],[35,55495,55497],{"className":55496,"style":16272},[7125],[35,55498],{},[35,55500],{"className":55501},[7314,14311],[35,55503,55505],{"className":55504,"style":7474},[7314,7473],[35,55506,1694],{"className":55507},[7478,7140],[35,55509],{"className":55510,"style":7496},[7248],[35,55512,2877],{"className":55513},[7492],[12,55515,51618],{},[35,55517,55519],{"className":55518},[7318],[35,55520,55522,55544],{"className":55521},[7057],[35,55523,55525],{"className":55524},[7061],[7063,55526,55527],{"xmlns":7065,"display":7327},[7067,55528,55529,55541],{},[7070,55530,55531,55533,55535,55537,55539],{},[7076,55532,55268],{},[7223,55534,242],{},[7076,55536,47141],{"mathvariant":7282},[7076,55538,46183],{},[7076,55540,5122],{"mathvariant":7282},[7083,55542,55543],{"encoding":7085},"\\theta=\\Omega\\tau.",[35,55545,55547,55565],{"className":55546,"ariaHidden":7091},[7090],[35,55548,55550,55553,55556,55559,55562],{"className":55549},[7095],[35,55551],{"className":55552,"style":9962},[7099],[35,55554,55268],{"className":55555,"style":7146},[7104,7145],[35,55557],{"className":55558,"style":7450},[7248],[35,55560,242],{"className":55561},[7454],[35,55563],{"className":55564,"style":7450},[7248],[35,55566,55568,55571,55574,55577],{"className":55567},[7095],[35,55569],{"className":55570,"style":7100},[7099],[35,55572,47141],{"className":55573},[7104],[35,55575,46183],{"className":55576,"style":46255},[7104,7145],[35,55578,5122],{"className":55579},[7104],[12,55581,55582,55583,55611],{},"A calibrated resonant ",[35,55584,55586,55599],{"className":55585},[7057],[35,55587,55589],{"className":55588},[7061],[7063,55590,55591],{"xmlns":7065},[7067,55592,55593,55597],{},[7070,55594,55595],{},[7076,55596,13081],{},[7083,55598,52563],{"encoding":7085},[35,55600,55602],{"className":55601,"ariaHidden":7091},[7090],[35,55603,55605,55608],{"className":55604},[7095],[35,55606],{"className":55607,"style":7173},[7099],[35,55609,13081],{"className":55610,"style":7310},[7104,7145],"-pulse enacts",[35,55613,55615],{"className":55614},[7318],[35,55616,55618,55656],{"className":55617},[7057],[35,55619,55621],{"className":55620},[7061],[7063,55622,55623],{"xmlns":7065,"display":7327},[7067,55624,55625,55653],{},[7070,55626,55627,55633,55635,55637,55639,55641,55643,55645,55651],{},[7699,55628,55629,55631],{},[7076,55630,24883],{},[7076,55632,13657],{},[7223,55634,275],{"stretchy":7289},[7076,55636,13081],{},[7223,55638,1694],{"stretchy":7289},[7223,55640,242],{},[7223,55642,7225],{},[7076,55644,7824],{},[7699,55646,55647,55649],{},[7076,55648,49186],{},[7076,55650,13657],{},[7223,55652,2877],{"separator":7091},[7083,55654,55655],{"encoding":7085},"R_x(\\pi)=-i\\sigma_x,",[35,55657,55659,55723],{"className":55658,"ariaHidden":7091},[7090],[35,55660,55662,55665,55705,55708,55711,55714,55717,55720],{"className":55661},[7095],[35,55663],{"className":55664,"style":7303},[7099],[35,55666,55668,55671],{"className":55667},[7104],[35,55669,24883],{"className":55670,"style":24908},[7104,7145],[35,55672,55674],{"className":55673},[7113],[35,55675,55677,55697],{"className":55676},[7117,7731],[35,55678,55680,55694],{"className":55679},[7121],[35,55681,55683],{"className":55682,"style":9035},[7125],[35,55684,55685,55688],{"style":24923},[35,55686],{"className":55687,"style":7134},[7133],[35,55689,55691],{"className":55690},[7138,7139,7140,7141],[35,55692,13657],{"className":55693},[7104,7145,7141],[35,55695,7756],{"className":55696},[7755],[35,55698,55700],{"className":55699},[7121],[35,55701,55703],{"className":55702,"style":7763},[7125],[35,55704],{},[35,55706,275],{"className":55707},[7472],[35,55709,13081],{"className":55710,"style":7310},[7104,7145],[35,55712,1694],{"className":55713},[7314],[35,55715],{"className":55716,"style":7450},[7248],[35,55718,242],{"className":55719},[7454],[35,55721],{"className":55722,"style":7450},[7248],[35,55724,55726,55729,55732,55735,55775],{"className":55725},[7095],[35,55727],{"className":55728,"style":32471},[7099],[35,55730,7225],{"className":55731},[7104],[35,55733,7824],{"className":55734},[7104,7145],[35,55736,55738,55741],{"className":55737},[7104],[35,55739,49186],{"className":55740,"style":7310},[7104,7145],[35,55742,55744],{"className":55743},[7113],[35,55745,55747,55767],{"className":55746},[7117,7731],[35,55748,55750,55764],{"className":55749},[7121],[35,55751,55753],{"className":55752,"style":9035},[7125],[35,55754,55755,55758],{"style":9100},[35,55756],{"className":55757,"style":7134},[7133],[35,55759,55761],{"className":55760},[7138,7139,7140,7141],[35,55762,13657],{"className":55763},[7104,7145,7141],[35,55765,7756],{"className":55766},[7755],[35,55768,55770],{"className":55769},[7121],[35,55771,55773],{"className":55772,"style":7763},[7125],[35,55774],{},[35,55776,2877],{"className":55777},[7492],[12,55779,55780,55781,55809,55810,55846],{},"which corresponds to a quantum ",[35,55782,55784,55797],{"className":55783},[7057],[35,55785,55787],{"className":55786},[7061],[7063,55788,55789],{"xmlns":7065},[7067,55790,55791,55795],{},[7070,55792,55793],{},[7076,55794,12257],{},[7083,55796,12257],{"encoding":7085},[35,55798,55800],{"className":55799,"ariaHidden":7091},[7090],[35,55801,55803,55806],{"className":55802},[7095],[35,55804],{"className":55805,"style":7100},[7099],[35,55807,12257],{"className":55808,"style":12402},[7104,7145]," gate up to a global phase. A resonant ",[35,55811,55813,55831],{"className":55812},[7057],[35,55814,55816],{"className":55815},[7061],[7063,55817,55818],{"xmlns":7065},[7067,55819,55820,55828],{},[7070,55821,55822,55824,55826],{},[7076,55823,13081],{},[7076,55825,398],{"mathvariant":7282},[7227,55827,460],{},[7083,55829,55830],{"encoding":7085},"\\pi\u002F2",[35,55832,55834],{"className":55833,"ariaHidden":7091},[7090],[35,55835,55837,55840,55843],{"className":55836},[7095],[35,55838],{"className":55839,"style":7303},[7099],[35,55841,13081],{"className":55842,"style":7310},[7104,7145],[35,55844,30606],{"className":55845},[7104],"-pulse prepares a balanced coherent superposition,",[35,55848,55850],{"className":55849},[7318],[35,55851,55853,55899],{"className":55852},[7057],[35,55854,55856],{"className":55855},[7061],[7063,55857,55858],{"xmlns":7065,"display":7327},[7067,55859,55860,55896],{},[7070,55861,55862,55864,55866,55868,55870,55894],{},[7223,55863,7283],{"stretchy":7289},[7076,55865,42397],{},[7223,55867,7290],{"stretchy":7289},[7223,55869,19447],{},[14009,55871,55872,55890],{},[7070,55873,55874,55876,55878,55880,55882,55884,55886,55888],{},[7223,55875,7283],{"stretchy":7289},[7076,55877,42397],{},[7223,55879,7290],{"stretchy":7289},[7223,55881,7225],{},[7076,55883,7824],{},[7223,55885,7283],{"stretchy":7289},[7076,55887,7817],{},[7223,55889,7290],{"stretchy":7289},[14014,55891,55892],{},[7227,55893,460],{},[7223,55895,2877],{"separator":7091},[7083,55897,55898],{"encoding":7085},"\\lvert g\\rangle\\rightarrow\\frac{\\lvert g\\rangle-i\\lvert e\\rangle}{\\sqrt{2}},",[35,55900,55902,55926],{"className":55901,"ariaHidden":7091},[7090],[35,55903,55905,55908,55911,55914,55917,55920,55923],{"className":55904},[7095],[35,55906],{"className":55907,"style":7303},[7099],[35,55909,7283],{"className":55910},[7472],[35,55912,42397],{"className":55913,"style":7310},[7104,7145],[35,55915,7290],{"className":55916},[7314],[35,55918],{"className":55919,"style":7450},[7248],[35,55921,19447],{"className":55922},[7454],[35,55924],{"className":55925,"style":7450},[7248],[35,55927,55929,55933,56066],{"className":55928},[7095],[35,55930],{"className":55931,"style":55932},[7099],"height:2.357em;vertical-align:-0.93em;",[35,55934,55936,55939,56063],{"className":55935},[7104],[35,55937],{"className":55938},[7472,14311],[35,55940,55942],{"className":55941},[14009],[35,55943,55945,56055],{"className":55944},[7117,7731],[35,55946,55948,56052],{"className":55947},[7121],[35,55949,55951,56006,56014],{"className":55950,"style":33849},[7125],[35,55952,55953,55956],{"style":14327},[35,55954],{"className":55955,"style":8839},[7133],[35,55957,55959],{"className":55958},[7104],[35,55960,55962],{"className":55961},[7104,490],[35,55963,55965,55998],{"className":55964},[7117,7731],[35,55966,55968,55995],{"className":55967},[7121],[35,55969,55971,55983],{"className":55970,"style":14346},[7125],[35,55972,55974,55977],{"className":55973,"style":8835},[14350],[35,55975],{"className":55976,"style":8839},[7133],[35,55978,55980],{"className":55979,"style":14357},[7104],[35,55981,460],{"className":55982},[7104],[35,55984,55985,55988],{"style":14363},[35,55986],{"className":55987,"style":8839},[7133],[35,55989,55991],{"className":55990,"style":14371},[14370],[10808,55992,55993],{"xmlns":10810,"width":14374,"height":14375,"viewBox":14376,"preserveAspectRatio":14377},[10817,55994],{"d":14380},[35,55996,7756],{"className":55997},[7755],[35,55999,56001],{"className":56000},[7121],[35,56002,56004],{"className":56003,"style":14390},[7125],[35,56005],{},[35,56007,56008,56011],{"style":14395},[35,56009],{"className":56010,"style":8839},[7133],[35,56012],{"className":56013,"style":14403},[14402],[35,56015,56016,56019],{"style":14406},[35,56017],{"className":56018,"style":8839},[7133],[35,56020,56022,56025,56028,56031,56034,56037,56040,56043,56046,56049],{"className":56021},[7104],[35,56023,7283],{"className":56024},[7472],[35,56026,42397],{"className":56027,"style":7310},[7104,7145],[35,56029,7290],{"className":56030},[7314],[35,56032],{"className":56033,"style":7249},[7248],[35,56035,7225],{"className":56036},[7253],[35,56038],{"className":56039,"style":7249},[7248],[35,56041,7824],{"className":56042},[7104,7145],[35,56044,7283],{"className":56045},[7472],[35,56047,7817],{"className":56048},[7104,7145],[35,56050,7290],{"className":56051},[7314],[35,56053,7756],{"className":56054},[7755],[35,56056,56058],{"className":56057},[7121],[35,56059,56061],{"className":56060,"style":14425},[7125],[35,56062],{},[35,56064],{"className":56065},[7314,14311],[35,56067,2877],{"className":56068},[7492],[12,56070,56071],{},"with a phase convention set by the microwave drive phase.",[12,56073,56074],{},"Detuning changes the rotation axis so it is no longer purely transverse. The unitary can be written as",[35,56076,56078],{"className":56077},[7318],[35,56079,56081,56151],{"className":56080},[7057],[35,56082,56084],{"className":56083},[7061],[7063,56085,56086],{"xmlns":7065,"display":7327},[7067,56087,56088,56148],{},[7070,56089,56090,56101,56103,56105,56107,56109,56111,56113,56146],{},[7699,56091,56092,56094],{},[7076,56093,24883],{},[8800,56095,56096,56098],{"accent":7091},[7076,56097,291],{},[7223,56099,56100],{},"^",[7223,56102,275],{"stretchy":7289},[7076,56104,55268],{},[7223,56106,1694],{"stretchy":7289},[7223,56108,242],{},[7076,56110,28563],{},[7223,56112,41175],{},[7070,56114,56115,56117,56119,56129,56135,56138,56144],{},[7223,56116,275],{"fence":7091},[7223,56118,7225],{},[14009,56120,56121,56127],{},[7070,56122,56123,56125],{},[7076,56124,7824],{},[7076,56126,55268],{},[7227,56128,460],{},[8800,56130,56131,56133],{"accent":7091},[7076,56132,291],{},[7223,56134,56100],{},[7223,56136,56137],{},"⋅",[8800,56139,56140,56142],{"accent":7091},[7076,56141,49186],{},[7223,56143,10704],{},[7223,56145,1694],{"fence":7091},[7223,56147,2877],{"separator":7091},[7083,56149,56150],{"encoding":7085},"R_{\\hat{n}}(\\theta)=\\exp\\left(-\\frac{i\\theta}{2}\\hat{n}\\cdot\\vec{\\sigma}\\right),",[35,56152,56154,56251],{"className":56153,"ariaHidden":7091},[7090],[35,56155,56157,56160,56233,56236,56239,56242,56245,56248],{"className":56156},[7095],[35,56158],{"className":56159,"style":7303},[7099],[35,56161,56163,56166],{"className":56162},[7104],[35,56164,24883],{"className":56165,"style":24908},[7104,7145],[35,56167,56169],{"className":56168},[7113],[35,56170,56172,56225],{"className":56171},[7117,7731],[35,56173,56175,56222],{"className":56174},[7121],[35,56176,56178],{"className":56177,"style":15760},[7125],[35,56179,56180,56183],{"style":24923},[35,56181],{"className":56182,"style":7134},[7133],[35,56184,56186],{"className":56185},[7138,7139,7140,7141],[35,56187,56189],{"className":56188},[7104,7141],[35,56190,56192],{"className":56191},[7104,8823,7141],[35,56193,56195],{"className":56194},[7117],[35,56196,56198],{"className":56197},[7121],[35,56199,56201,56210],{"className":56200,"style":9962},[7125],[35,56202,56204,56207],{"style":56203},"top:-2.7em;",[35,56205],{"className":56206,"style":7134},[7133],[35,56208,291],{"className":56209},[7104,7145,7141],[35,56211,56212,56215],{"style":56203},[35,56213],{"className":56214,"style":7134},[7133],[35,56216,56219],{"className":56217,"style":56218},[8852],"left:-0.25em;",[35,56220,56100],{"className":56221},[7104,7141],[35,56223,7756],{"className":56224},[7755],[35,56226,56228],{"className":56227},[7121],[35,56229,56231],{"className":56230,"style":7763},[7125],[35,56232],{},[35,56234,275],{"className":56235},[7472],[35,56237,55268],{"className":56238,"style":7146},[7104,7145],[35,56240,1694],{"className":56241},[7314],[35,56243],{"className":56244,"style":7450},[7248],[35,56246,242],{"className":56247},[7454],[35,56249],{"className":56250,"style":7450},[7248],[35,56252,56254,56257,56260,56263,56422,56425],{"className":56253},[7095],[35,56255],{"className":56256,"style":42152},[7099],[35,56258,28563],{"className":56259},[12096],[35,56261],{"className":56262,"style":7496},[7248],[35,56264,56266,56272,56275,56340,56371,56374,56377,56380,56416],{"className":56265},[7468],[35,56267,56269],{"className":56268,"style":7474},[7472,7473],[35,56270,275],{"className":56271},[7478,7140],[35,56273,7225],{"className":56274},[7104],[35,56276,56278,56281,56337],{"className":56277},[7104],[35,56279],{"className":56280},[7472,14311],[35,56282,56284],{"className":56283},[14009],[35,56285,56287,56329],{"className":56286},[7117,7731],[35,56288,56290,56326],{"className":56289},[7121],[35,56291,56293,56304,56312],{"className":56292,"style":42242},[7125],[35,56294,56295,56298],{"style":16234},[35,56296],{"className":56297,"style":8839},[7133],[35,56299,56301],{"className":56300},[7104],[35,56302,460],{"className":56303},[7104],[35,56305,56306,56309],{"style":14395},[35,56307],{"className":56308,"style":8839},[7133],[35,56310],{"className":56311,"style":14403},[14402],[35,56313,56314,56317],{"style":14406},[35,56315],{"className":56316,"style":8839},[7133],[35,56318,56320,56323],{"className":56319},[7104],[35,56321,7824],{"className":56322},[7104,7145],[35,56324,55268],{"className":56325,"style":7146},[7104,7145],[35,56327,7756],{"className":56328},[7755],[35,56330,56332],{"className":56331},[7121],[35,56333,56335],{"className":56334,"style":16272},[7125],[35,56336],{},[35,56338],{"className":56339},[7314,14311],[35,56341,56343],{"className":56342},[7104,8823],[35,56344,56346],{"className":56345},[7117],[35,56347,56349],{"className":56348},[7121],[35,56350,56352,56360],{"className":56351,"style":9962},[7125],[35,56353,56354,56357],{"style":8835},[35,56355],{"className":56356,"style":8839},[7133],[35,56358,291],{"className":56359},[7104,7145],[35,56361,56362,56365],{"style":8835},[35,56363],{"className":56364,"style":8839},[7133],[35,56366,56368],{"className":56367,"style":56218},[8852],[35,56369,56100],{"className":56370},[7104],[35,56372],{"className":56373,"style":7249},[7248],[35,56375,56137],{"className":56376},[7253],[35,56378],{"className":56379,"style":7249},[7248],[35,56381,56383],{"className":56382},[7104,8823],[35,56384,56386],{"className":56385},[7117],[35,56387,56389],{"className":56388},[7121],[35,56390,56392,56400],{"className":56391,"style":10784},[7125],[35,56393,56394,56397],{"style":8835},[35,56395],{"className":56396,"style":8839},[7133],[35,56398,49186],{"className":56399,"style":7310},[7104,7145],[35,56401,56402,56405],{"style":8835},[35,56403],{"className":56404,"style":8839},[7133],[35,56406,56409],{"className":56407,"style":56408},[8852],"left:-0.2355em;",[35,56410,56412],{"className":56411,"style":10806},[10805],[10808,56413,56414],{"xmlns":10810,"width":10811,"height":10812,"style":10813,"viewBox":10814,"preserveAspectRatio":10815},[10817,56415],{"d":10819},[35,56417,56419],{"className":56418,"style":7474},[7314,7473],[35,56420,1694],{"className":56421},[7478,7140],[35,56423],{"className":56424,"style":7496},[7248],[35,56426,2877],{"className":56427},[7492],[12,56429,51618],{},[35,56431,56433],{"className":56432},[7318],[35,56434,56436,56498],{"className":56435},[7057],[35,56437,56439],{"className":56438},[7061],[7063,56440,56441],{"xmlns":7065,"display":7327},[7067,56442,56443,56495],{},[7070,56444,56445,56451,56453,56463,56465,56467,56469,56471,56473,56475,56477,56479,56481,56483,56485,56491,56493],{},[8800,56446,56447,56449],{"accent":7091},[7076,56448,291],{},[7223,56450,56100],{},[7223,56452,242],{},[14009,56454,56455,56457],{},[7227,56456,500],{},[7699,56458,56459,56461],{},[7076,56460,47141],{"mathvariant":7282},[7076,56462,24883],{},[7223,56464,275],{"stretchy":7289},[7076,56466,47141],{"mathvariant":7282},[7223,56468,2877],{"separator":7091},[7227,56470,5312],{},[7223,56472,2877],{"separator":7091},[7076,56474,47224],{"mathvariant":7282},[7223,56476,1694],{"stretchy":7289},[7223,56478,2877],{"separator":7091},[7248,56480],{"width":22524},[7076,56482,55268],{},[7223,56484,242],{},[7699,56486,56487,56489],{},[7076,56488,47141],{"mathvariant":7282},[7076,56490,24883],{},[7076,56492,46183],{},[7076,56494,5122],{"mathvariant":7282},[7083,56496,56497],{"encoding":7085},"\\hat{n}=\\frac{1}{\\Omega_R}(\\Omega,0,\\Delta), \\qquad \\theta=\\Omega_R\\tau.",[35,56499,56501,56547,56702],{"className":56500,"ariaHidden":7091},[7090],[35,56502,56504,56507,56538,56541,56544],{"className":56503},[7095],[35,56505],{"className":56506,"style":9962},[7099],[35,56508,56510],{"className":56509},[7104,8823],[35,56511,56513],{"className":56512},[7117],[35,56514,56516],{"className":56515},[7121],[35,56517,56519,56527],{"className":56518,"style":9962},[7125],[35,56520,56521,56524],{"style":8835},[35,56522],{"className":56523,"style":8839},[7133],[35,56525,291],{"className":56526},[7104,7145],[35,56528,56529,56532],{"style":8835},[35,56530],{"className":56531,"style":8839},[7133],[35,56533,56535],{"className":56534,"style":56218},[8852],[35,56536,56100],{"className":56537},[7104],[35,56539],{"className":56540,"style":7450},[7248],[35,56542,242],{"className":56543},[7454],[35,56545],{"className":56546,"style":7450},[7248],[35,56548,56550,56554,56654,56657,56660,56663,56666,56669,56672,56675,56678,56681,56684,56687,56690,56693,56696,56699],{"className":56549},[7095],[35,56551],{"className":56552,"style":56553},[7099],"height:2.1574em;vertical-align:-0.836em;",[35,56555,56557,56560,56651],{"className":56556},[7104],[35,56558],{"className":56559},[7472,14311],[35,56561,56563],{"className":56562},[14009],[35,56564,56566,56642],{"className":56565},[7117,7731],[35,56567,56569,56639],{"className":56568},[7121],[35,56570,56572,56620,56628],{"className":56571,"style":14324},[7125],[35,56573,56574,56577],{"style":16234},[35,56575],{"className":56576,"style":8839},[7133],[35,56578,56580],{"className":56579},[7104],[35,56581,56583,56586],{"className":56582},[7104],[35,56584,47141],{"className":56585},[7104],[35,56587,56589],{"className":56588},[7113],[35,56590,56592,56612],{"className":56591},[7117,7731],[35,56593,56595,56609],{"className":56594},[7121],[35,56596,56598],{"className":56597,"style":7738},[7125],[35,56599,56600,56603],{"style":9926},[35,56601],{"className":56602,"style":7134},[7133],[35,56604,56606],{"className":56605},[7138,7139,7140,7141],[35,56607,24883],{"className":56608,"style":24908},[7104,7145,7141],[35,56610,7756],{"className":56611},[7755],[35,56613,56615],{"className":56614},[7121],[35,56616,56618],{"className":56617,"style":7763},[7125],[35,56619],{},[35,56621,56622,56625],{"style":14395},[35,56623],{"className":56624,"style":8839},[7133],[35,56626],{"className":56627,"style":14403},[14402],[35,56629,56630,56633],{"style":14406},[35,56631],{"className":56632,"style":8839},[7133],[35,56634,56636],{"className":56635},[7104],[35,56637,500],{"className":56638},[7104],[35,56640,7756],{"className":56641},[7755],[35,56643,56645],{"className":56644},[7121],[35,56646,56649],{"className":56647,"style":56648},[7125],"height:0.836em;",[35,56650],{},[35,56652],{"className":56653},[7314,14311],[35,56655,275],{"className":56656},[7472],[35,56658,47141],{"className":56659},[7104],[35,56661,2877],{"className":56662},[7492],[35,56664],{"className":56665,"style":7496},[7248],[35,56667,5312],{"className":56668},[7104],[35,56670,2877],{"className":56671},[7492],[35,56673],{"className":56674,"style":7496},[7248],[35,56676,47224],{"className":56677},[7104],[35,56679,1694],{"className":56680},[7314],[35,56682,2877],{"className":56683},[7492],[35,56685],{"className":56686,"style":23142},[7248],[35,56688],{"className":56689,"style":7496},[7248],[35,56691,55268],{"className":56692,"style":7146},[7104,7145],[35,56694],{"className":56695,"style":7450},[7248],[35,56697,242],{"className":56698},[7454],[35,56700],{"className":56701,"style":7450},[7248],[35,56703,56705,56708,56748,56751],{"className":56704},[7095],[35,56706],{"className":56707,"style":7718},[7099],[35,56709,56711,56714],{"className":56710},[7104],[35,56712,47141],{"className":56713},[7104],[35,56715,56717],{"className":56716},[7113],[35,56718,56720,56740],{"className":56719},[7117,7731],[35,56721,56723,56737],{"className":56722},[7121],[35,56724,56726],{"className":56725,"style":7738},[7125],[35,56727,56728,56731],{"style":9926},[35,56729],{"className":56730,"style":7134},[7133],[35,56732,56734],{"className":56733},[7138,7139,7140,7141],[35,56735,24883],{"className":56736,"style":24908},[7104,7145,7141],[35,56738,7756],{"className":56739},[7755],[35,56741,56743],{"className":56742},[7121],[35,56744,56746],{"className":56745,"style":7763},[7125],[35,56747],{},[35,56749,46183],{"className":56750,"style":46255},[7104,7145],[35,56752,5122],{"className":56753},[7104],[12,56755,56756],{},"This representation shows that detuning increases the rotation rate and tilts the control axis. Under this idealized model, the qubit still evolves coherently, although the trajectory generally misses the pure excited-state pole.",[5019,56758,56760],{"id":56759},"bloch-sphere-interpretation","Bloch-Sphere Interpretation",[12,56762,56763],{},"A Bloch sphere maps any pure qubit state to a unit vector in three-dimensional space. Ground and excited states occupy opposite poles. Equal-amplitude coherent superpositions lie on the equator.",[12,56765,56766,56767,56805,56806,56834,56835,56870],{},"A resonant microwave drive rotates the Bloch vector around a transverse axis. Starting from ",[35,56768,56770,56787],{"className":56769},[7057],[35,56771,56773],{"className":56772},[7061],[7063,56774,56775],{"xmlns":7065},[7067,56776,56777,56785],{},[7070,56778,56779,56781,56783],{},[7223,56780,7283],{"stretchy":7289},[7076,56782,42397],{},[7223,56784,7290],{"stretchy":7289},[7083,56786,46374],{"encoding":7085},[35,56788,56790],{"className":56789,"ariaHidden":7091},[7090],[35,56791,56793,56796,56799,56802],{"className":56792},[7095],[35,56794],{"className":56795,"style":7303},[7099],[35,56797,7283],{"className":56798},[7472],[35,56800,42397],{"className":56801,"style":7310},[7104,7145],[35,56803,7290],{"className":56804},[7314],", a properly timed ",[35,56807,56809,56822],{"className":56808},[7057],[35,56810,56812],{"className":56811},[7061],[7063,56813,56814],{"xmlns":7065},[7067,56815,56816,56820],{},[7070,56817,56818],{},[7076,56819,13081],{},[7083,56821,52563],{"encoding":7085},[35,56823,56825],{"className":56824,"ariaHidden":7091},[7090],[35,56826,56828,56831],{"className":56827},[7095],[35,56829],{"className":56830,"style":7173},[7099],[35,56832,13081],{"className":56833,"style":7310},[7104,7145],"-pulse moves the state to the excited-state pole. A ",[35,56836,56838,56855],{"className":56837},[7057],[35,56839,56841],{"className":56840},[7061],[7063,56842,56843],{"xmlns":7065},[7067,56844,56845,56853],{},[7070,56846,56847,56849,56851],{},[7076,56848,13081],{},[7076,56850,398],{"mathvariant":7282},[7227,56852,460],{},[7083,56854,55830],{"encoding":7085},[35,56856,56858],{"className":56857,"ariaHidden":7091},[7090],[35,56859,56861,56864,56867],{"className":56860},[7095],[35,56862],{"className":56863,"style":7303},[7099],[35,56865,13081],{"className":56866,"style":7310},[7104,7145],[35,56868,30606],{"className":56869},[7104],"-pulse moves the state to the equator, where the ground-state and excited-state measurement probabilities are equal.",[12,56872,56873,56874,56902],{},"Detuning tilts the rotation axis toward the ",[35,56875,56877,56890],{"className":56876},[7057],[35,56878,56880],{"className":56879},[7061],[7063,56881,56882],{"xmlns":7065},[7067,56883,56884,56888],{},[7070,56885,56886],{},[7076,56887,300],{},[7083,56889,300],{"encoding":7085},[35,56891,56893],{"className":56892,"ariaHidden":7091},[7090],[35,56894,56896,56899],{"className":56895},[7095],[35,56897],{"className":56898,"style":7173},[7099],[35,56900,300],{"className":56901,"style":25095},[7104,7145],"-axis. The state vector then precesses around the tilted axis and generally fails to reach the excited-state pole. This geometric picture explains the amplitude coefficient",[35,56904,56906],{"className":56905},[7318],[35,56907,56909,56939],{"className":56908},[7057],[35,56910,56912],{"className":56911},[7061],[7063,56913,56914],{"xmlns":7065,"display":7327},[7067,56915,56916,56936],{},[7070,56917,56918,56934],{},[14009,56919,56920,56926],{},[7073,56921,56922,56924],{},[7076,56923,47141],{"mathvariant":7282},[7227,56925,460],{},[8304,56927,56928,56930,56932],{},[7076,56929,47141],{"mathvariant":7282},[7076,56931,24883],{},[7227,56933,460],{},[7076,56935,5122],{"mathvariant":7282},[7083,56937,56938],{"encoding":7085},"\\frac{\\Omega^2}{\\Omega_R^2}.",[35,56940,56942],{"className":56941,"ariaHidden":7091},[7090],[35,56943,56945,56948,57084],{"className":56944},[7095],[35,56946],{"className":56947,"style":51312},[7099],[35,56949,56951,56954,57081],{"className":56950},[7104],[35,56952],{"className":56953},[7472,14311],[35,56955,56957],{"className":56956},[14009],[35,56958,56960,57073],{"className":56959},[7117,7731],[35,56961,56963,57070],{"className":56962},[7121],[35,56964,56966,57025,57033],{"className":56965,"style":51331},[7125],[35,56967,56968,56971],{"style":16234},[35,56969],{"className":56970,"style":8839},[7133],[35,56972,56974],{"className":56973},[7104],[35,56975,56977,56980],{"className":56976},[7104],[35,56978,47141],{"className":56979},[7104],[35,56981,56983],{"className":56982},[7113],[35,56984,56986,57017],{"className":56985},[7117,7731],[35,56987,56989,57014],{"className":56988},[7121],[35,56990,56992,57003],{"className":56991,"style":51358},[7125],[35,56993,56994,56997],{"style":51361},[35,56995],{"className":56996,"style":7134},[7133],[35,56998,57000],{"className":56999},[7138,7139,7140,7141],[35,57001,24883],{"className":57002,"style":24908},[7104,7145,7141],[35,57004,57005,57008],{"style":51373},[35,57006],{"className":57007,"style":7134},[7133],[35,57009,57011],{"className":57010},[7138,7139,7140,7141],[35,57012,460],{"className":57013},[7104,7141],[35,57015,7756],{"className":57016},[7755],[35,57018,57020],{"className":57019},[7121],[35,57021,57023],{"className":57022,"style":51392},[7125],[35,57024],{},[35,57026,57027,57030],{"style":14395},[35,57028],{"className":57029,"style":8839},[7133],[35,57031],{"className":57032,"style":14403},[14402],[35,57034,57035,57038],{"style":14406},[35,57036],{"className":57037,"style":8839},[7133],[35,57039,57041],{"className":57040},[7104],[35,57042,57044,57047],{"className":57043},[7104],[35,57045,47141],{"className":57046},[7104],[35,57048,57050],{"className":57049},[7113],[35,57051,57053],{"className":57052},[7117],[35,57054,57056],{"className":57055},[7121],[35,57057,57059],{"className":57058,"style":17739},[7125],[35,57060,57061,57064],{"style":7129},[35,57062],{"className":57063,"style":7134},[7133],[35,57065,57067],{"className":57066},[7138,7139,7140,7141],[35,57068,460],{"className":57069},[7104,7141],[35,57071,7756],{"className":57072},[7755],[35,57074,57076],{"className":57075},[7121],[35,57077,57079],{"className":57078,"style":51449},[7125],[35,57080],{},[35,57082],{"className":57083},[7314,14311],[35,57085,5122],{"className":57086},[7104],[12,57088,57089,57090,57129],{},"As ",[35,57091,57093,57111],{"className":57092},[7057],[35,57094,57096],{"className":57095},[7061],[7063,57097,57098],{"xmlns":7065},[7067,57099,57100,57108],{},[7070,57101,57102,57104,57106],{},[7223,57103,7283],{"stretchy":7289},[7076,57105,47224],{"mathvariant":7282},[7223,57107,7283],{"stretchy":7289},[7083,57109,57110],{"encoding":7085},"\\lvert\\Delta\\rvert",[35,57112,57114],{"className":57113,"ariaHidden":7091},[7090],[35,57115,57117,57120,57123,57126],{"className":57116},[7095],[35,57118],{"className":57119,"style":7303},[7099],[35,57121,7283],{"className":57122},[7472],[35,57124,47224],{"className":57125},[7104],[35,57127,7283],{"className":57128},[7314]," grows, the rotation axis becomes more longitudinal, the oscillation frequency increases, and the maximum excited-state population decreases.",[5019,57131,57133],{"id":57132},"fourier-analysis-dynamics","Fourier Analysis Dynamics",[12,57135,57136,57137,57165,57166,57264],{},"The spectral analysis computes a real Fast Fourier transform along the pulse-duration axis. For each sampled drive frequency ",[35,57138,57140,57153],{"className":57139},[7057],[35,57141,57143],{"className":57142},[7061],[7063,57144,57145],{"xmlns":7065},[7067,57146,57147,57151],{},[7070,57148,57149],{},[7076,57150,1675],{},[7083,57152,1675],{"encoding":7085},[35,57154,57156],{"className":57155,"ariaHidden":7091},[7090],[35,57157,57159,57162],{"className":57158},[7095],[35,57160],{"className":57161,"style":11562},[7099],[35,57163,1675],{"className":57164,"style":31824},[7104,7145],", the trace ",[35,57167,57169,57197],{"className":57168},[7057],[35,57170,57172],{"className":57171},[7061],[7063,57173,57174],{"xmlns":7065},[7067,57175,57176,57194],{},[7070,57177,57178,57184,57186,57188,57190,57192],{},[7699,57179,57180,57182],{},[7076,57181,7705],{},[7076,57183,7817],{},[7223,57185,275],{"stretchy":7289},[7076,57187,46183],{},[7223,57189,2877],{"separator":7091},[7076,57191,1675],{},[7223,57193,1694],{"stretchy":7289},[7083,57195,57196],{"encoding":7085},"P_e(\\tau,f)",[35,57198,57200],{"className":57199,"ariaHidden":7091},[7090],[35,57201,57203,57206,57246,57249,57252,57255,57258,57261],{"className":57202},[7095],[35,57204],{"className":57205,"style":7303},[7099],[35,57207,57209,57212],{"className":57208},[7104],[35,57210,7705],{"className":57211,"style":7751},[7104,7145],[35,57213,57215],{"className":57214},[7113],[35,57216,57218,57238],{"className":57217},[7117,7731],[35,57219,57221,57235],{"className":57220},[7121],[35,57222,57224],{"className":57223,"style":9035},[7125],[35,57225,57226,57229],{"style":12163},[35,57227],{"className":57228,"style":7134},[7133],[35,57230,57232],{"className":57231},[7138,7139,7140,7141],[35,57233,7817],{"className":57234},[7104,7145,7141],[35,57236,7756],{"className":57237},[7755],[35,57239,57241],{"className":57240},[7121],[35,57242,57244],{"className":57243,"style":7763},[7125],[35,57245],{},[35,57247,275],{"className":57248},[7472],[35,57250,46183],{"className":57251,"style":46255},[7104,7145],[35,57253,2877],{"className":57254},[7492],[35,57256],{"className":57257,"style":7496},[7248],[35,57259,1675],{"className":57260,"style":31824},[7104,7145],[35,57262,1694],{"className":57263},[7314]," is treated as a time-domain signal. The spectral amplitude is",[35,57266,57268],{"className":57267},[7318],[35,57269,57271,57334],{"className":57270},[7057],[35,57272,57274],{"className":57273},[7061],[7063,57275,57276],{"xmlns":7065,"display":7327},[7067,57277,57278,57331],{},[7070,57279,57280,57282,57284,57286,57288,57290,57292,57294,57329],{},[7076,57281,10384],{},[7223,57283,275],{"stretchy":7289},[7076,57285,47454],{},[7223,57287,2877],{"separator":7091},[7076,57289,1675],{},[7223,57291,1694],{"stretchy":7289},[7223,57293,242],{},[7070,57295,57296,57298,57305,57327],{},[7223,57297,7283],{"fence":7091},[7699,57299,57300,57303],{},[7076,57301,57302],{"mathvariant":7078},"F",[7076,57304,46183],{},[7070,57306,57307,57309,57315,57317,57319,57321,57323,57325],{},[7223,57308,1681],{"fence":7091},[7699,57310,57311,57313],{},[7076,57312,7705],{},[7076,57314,7817],{},[7223,57316,275],{"stretchy":7289},[7076,57318,46183],{},[7223,57320,2877],{"separator":7091},[7076,57322,1675],{},[7223,57324,1694],{"stretchy":7289},[7223,57326,1697],{"fence":7091},[7223,57328,7283],{"fence":7091},[7076,57330,5122],{"mathvariant":7282},[7083,57332,57333],{"encoding":7085},"S(\\nu,f)=\\left|\\mathcal{F}_{\\tau}\\left\\{P_e(\\tau,f)\\right\\}\\right|.",[35,57335,57337,57373],{"className":57336,"ariaHidden":7091},[7090],[35,57338,57340,57343,57346,57349,57352,57355,57358,57361,57364,57367,57370],{"className":57339},[7095],[35,57341],{"className":57342,"style":7303},[7099],[35,57344,10384],{"className":57345,"style":10419},[7104,7145],[35,57347,275],{"className":57348},[7472],[35,57350,47454],{"className":57351,"style":47475},[7104,7145],[35,57353,2877],{"className":57354},[7492],[35,57356],{"className":57357,"style":7496},[7248],[35,57359,1675],{"className":57360,"style":31824},[7104,7145],[35,57362,1694],{"className":57363},[7314],[35,57365],{"className":57366,"style":7450},[7248],[35,57368,242],{"className":57369},[7454],[35,57371],{"className":57372,"style":7450},[7248],[35,57374,57376,57379,57503,57506],{"className":57375},[7095],[35,57377],{"className":57378,"style":7303},[7099],[35,57380,57382,57385,57430,57433,57500],{"className":57381},[7468],[35,57383,7283],{"className":57384,"style":7474},[7472,7473],[35,57386,57388,57392],{"className":57387},[7104],[35,57389,57302],{"className":57390,"style":57391},[7104,7108],"margin-right:0.0993em;",[35,57393,57395],{"className":57394},[7113],[35,57396,57398,57422],{"className":57397},[7117,7731],[35,57399,57401,57419],{"className":57400},[7121],[35,57402,57404],{"className":57403,"style":9035},[7125],[35,57405,57407,57410],{"style":57406},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[35,57408],{"className":57409,"style":7134},[7133],[35,57411,57413],{"className":57412},[7138,7139,7140,7141],[35,57414,57416],{"className":57415},[7104,7141],[35,57417,46183],{"className":57418,"style":46255},[7104,7145,7141],[35,57420,7756],{"className":57421},[7755],[35,57423,57425],{"className":57424},[7121],[35,57426,57428],{"className":57427,"style":7763},[7125],[35,57429],{},[35,57431],{"className":57432,"style":7496},[7248],[35,57434,57436,57439,57479,57482,57485,57488,57491,57494,57497],{"className":57435},[7468],[35,57437,1681],{"className":57438,"style":7474},[7472,7473],[35,57440,57442,57445],{"className":57441},[7104],[35,57443,7705],{"className":57444,"style":7751},[7104,7145],[35,57446,57448],{"className":57447},[7113],[35,57449,57451,57471],{"className":57450},[7117,7731],[35,57452,57454,57468],{"className":57453},[7121],[35,57455,57457],{"className":57456,"style":9035},[7125],[35,57458,57459,57462],{"style":12163},[35,57460],{"className":57461,"style":7134},[7133],[35,57463,57465],{"className":57464},[7138,7139,7140,7141],[35,57466,7817],{"className":57467},[7104,7145,7141],[35,57469,7756],{"className":57470},[7755],[35,57472,57474],{"className":57473},[7121],[35,57475,57477],{"className":57476,"style":7763},[7125],[35,57478],{},[35,57480,275],{"className":57481},[7472],[35,57483,46183],{"className":57484,"style":46255},[7104,7145],[35,57486,2877],{"className":57487},[7492],[35,57489],{"className":57490,"style":7496},[7248],[35,57492,1675],{"className":57493,"style":31824},[7104,7145],[35,57495,1694],{"className":57496},[7314],[35,57498,1697],{"className":57499,"style":7474},[7314,7473],[35,57501,7283],{"className":57502,"style":7474},[7314,7473],[35,57504],{"className":57505,"style":7496},[7248],[35,57507,5122],{"className":57508},[7104],[12,57510,57511],{},"This probability expression contains a squared sinusoid. Using the identity",[35,57513,57515],{"className":57514},[7318],[35,57516,57518,57594],{"className":57517},[7057],[35,57519,57521],{"className":57520},[7061],[7063,57522,57523],{"xmlns":7065,"display":7327},[7067,57524,57525,57591],{},[7070,57526,57527,57537,57557,57559,57565],{},[7073,57528,57529,57535],{},[7070,57530,57531,57533],{},[7076,57532,51200],{},[7223,57534,41175],{},[7227,57536,460],{},[7070,57538,57539,57541,57555],{},[7223,57540,275],{"fence":7091},[14009,57542,57543,57553],{},[7070,57544,57545,57551],{},[7699,57546,57547,57549],{},[7076,57548,47141],{"mathvariant":7282},[7076,57550,24883],{},[7076,57552,46183],{},[7227,57554,460],{},[7223,57556,1694],{"fence":7091},[7223,57558,242],{},[14009,57560,57561,57563],{},[7227,57562,500],{},[7227,57564,460],{},[7070,57566,57567,57569,57571,57573,57575,57577,57579,57585,57587,57589],{},[7223,57568,1254],{"fence":7091},[7227,57570,500],{},[7223,57572,7225],{},[7076,57574,42416],{},[7223,57576,41175],{},[7223,57578,275],{"stretchy":7289},[7699,57580,57581,57583],{},[7076,57582,47141],{"mathvariant":7282},[7076,57584,24883],{},[7076,57586,46183],{},[7223,57588,1694],{"stretchy":7289},[7223,57590,16788],{"fence":7091},[7083,57592,57593],{"encoding":7085},"\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right)=\\frac{1}{2}\\left[1-\\cos(\\Omega_R\\tau)\\right]",[35,57595,57597,57761],{"className":57596,"ariaHidden":7091},[7090],[35,57598,57600,57603,57632,57635,57752,57755,57758],{"className":57599},[7095],[35,57601],{"className":57602,"style":42152},[7099],[35,57604,57606,57609],{"className":57605},[12096],[35,57607,51200],{"className":57608},[12096],[35,57610,57612],{"className":57611},[7113],[35,57613,57615],{"className":57614},[7117],[35,57616,57618],{"className":57617},[7121],[35,57619,57621],{"className":57620,"style":51476},[7125],[35,57622,57623,57626],{"style":51479},[35,57624],{"className":57625,"style":7134},[7133],[35,57627,57629],{"className":57628},[7138,7139,7140,7141],[35,57630,460],{"className":57631},[7104,7141],[35,57633],{"className":57634,"style":7496},[7248],[35,57636,57638,57644,57746],{"className":57637},[7468],[35,57639,57641],{"className":57640,"style":7474},[7472,7473],[35,57642,275],{"className":57643},[7478,7140],[35,57645,57647,57650,57743],{"className":57646},[7104],[35,57648],{"className":57649},[7472,14311],[35,57651,57653],{"className":57652},[14009],[35,57654,57656,57735],{"className":57655},[7117,7731],[35,57657,57659,57732],{"className":57658},[7121],[35,57660,57662,57673,57681],{"className":57661,"style":51519},[7125],[35,57663,57664,57667],{"style":16234},[35,57665],{"className":57666,"style":8839},[7133],[35,57668,57670],{"className":57669},[7104],[35,57671,460],{"className":57672},[7104],[35,57674,57675,57678],{"style":14395},[35,57676],{"className":57677,"style":8839},[7133],[35,57679],{"className":57680,"style":14403},[14402],[35,57682,57683,57686],{"style":14406},[35,57684],{"className":57685,"style":8839},[7133],[35,57687,57689,57729],{"className":57688},[7104],[35,57690,57692,57695],{"className":57691},[7104],[35,57693,47141],{"className":57694},[7104],[35,57696,57698],{"className":57697},[7113],[35,57699,57701,57721],{"className":57700},[7117,7731],[35,57702,57704,57718],{"className":57703},[7121],[35,57705,57707],{"className":57706,"style":7738},[7125],[35,57708,57709,57712],{"style":9926},[35,57710],{"className":57711,"style":7134},[7133],[35,57713,57715],{"className":57714},[7138,7139,7140,7141],[35,57716,24883],{"className":57717,"style":24908},[7104,7145,7141],[35,57719,7756],{"className":57720},[7755],[35,57722,57724],{"className":57723},[7121],[35,57725,57727],{"className":57726,"style":7763},[7125],[35,57728],{},[35,57730,46183],{"className":57731,"style":46255},[7104,7145],[35,57733,7756],{"className":57734},[7755],[35,57736,57738],{"className":57737},[7121],[35,57739,57741],{"className":57740,"style":16272},[7125],[35,57742],{},[35,57744],{"className":57745},[7314,14311],[35,57747,57749],{"className":57748,"style":7474},[7314,7473],[35,57750,1694],{"className":57751},[7478,7140],[35,57753],{"className":57754,"style":7450},[7248],[35,57756,242],{"className":57757},[7454],[35,57759],{"className":57760,"style":7450},[7248],[35,57762,57764,57767,57829,57832],{"className":57763},[7095],[35,57765],{"className":57766,"style":53281},[7099],[35,57768,57770,57773,57826],{"className":57769},[7104],[35,57771],{"className":57772},[7472,14311],[35,57774,57776],{"className":57775},[14009],[35,57777,57779,57818],{"className":57778},[7117,7731],[35,57780,57782,57815],{"className":57781},[7121],[35,57783,57785,57796,57804],{"className":57784,"style":14324},[7125],[35,57786,57787,57790],{"style":16234},[35,57788],{"className":57789,"style":8839},[7133],[35,57791,57793],{"className":57792},[7104],[35,57794,460],{"className":57795},[7104],[35,57797,57798,57801],{"style":14395},[35,57799],{"className":57800,"style":8839},[7133],[35,57802],{"className":57803,"style":14403},[14402],[35,57805,57806,57809],{"style":14406},[35,57807],{"className":57808,"style":8839},[7133],[35,57810,57812],{"className":57811},[7104],[35,57813,500],{"className":57814},[7104],[35,57816,7756],{"className":57817},[7755],[35,57819,57821],{"className":57820},[7121],[35,57822,57824],{"className":57823,"style":16272},[7125],[35,57825],{},[35,57827],{"className":57828},[7314,14311],[35,57830],{"className":57831,"style":7496},[7248],[35,57833,57835,57838,57841,57844,57847,57850,57853,57856,57896,57899,57902],{"className":57834},[7468],[35,57836,1254],{"className":57837,"style":7474},[7472,7473],[35,57839,500],{"className":57840},[7104],[35,57842],{"className":57843,"style":7249},[7248],[35,57845,7225],{"className":57846},[7253],[35,57848],{"className":57849,"style":7249},[7248],[35,57851,42416],{"className":57852},[12096],[35,57854,275],{"className":57855},[7472],[35,57857,57859,57862],{"className":57858},[7104],[35,57860,47141],{"className":57861},[7104],[35,57863,57865],{"className":57864},[7113],[35,57866,57868,57888],{"className":57867},[7117,7731],[35,57869,57871,57885],{"className":57870},[7121],[35,57872,57874],{"className":57873,"style":7738},[7125],[35,57875,57876,57879],{"style":9926},[35,57877],{"className":57878,"style":7134},[7133],[35,57880,57882],{"className":57881},[7138,7139,7140,7141],[35,57883,24883],{"className":57884,"style":24908},[7104,7145,7141],[35,57886,7756],{"className":57887},[7755],[35,57889,57891],{"className":57890},[7121],[35,57892,57894],{"className":57893,"style":7763},[7125],[35,57895],{},[35,57897,46183],{"className":57898,"style":46255},[7104,7145],[35,57900,1694],{"className":57901},[7314],[35,57903,16788],{"className":57904,"style":7474},[7314,7473],[12,57906,57907],{},"shows that the dominant oscillatory spectral component appears at the generalized frequency",[35,57909,57911],{"className":57910},[7318],[35,57912,57914,57950],{"className":57913},[7057],[35,57915,57917],{"className":57916},[7061],[7063,57918,57919],{"xmlns":7065,"display":7327},[7067,57920,57921,57947],{},[7070,57922,57923,57929,57931,57945],{},[7699,57924,57925,57927],{},[7076,57926,47454],{},[7076,57928,24883],{},[7223,57930,242],{},[14009,57932,57933,57939],{},[7699,57934,57935,57937],{},[7076,57936,47141],{"mathvariant":7282},[7076,57938,24883],{},[7070,57940,57941,57943],{},[7227,57942,460],{},[7076,57944,13081],{},[7076,57946,5122],{"mathvariant":7282},[7083,57948,57949],{"encoding":7085},"\\nu_R=\\frac{\\Omega_R}{2\\pi}.",[35,57951,57953,58008],{"className":57952,"ariaHidden":7091},[7090],[35,57954,57956,57959,57999,58002,58005],{"className":57955},[7095],[35,57957],{"className":57958,"style":29689},[7099],[35,57960,57962,57965],{"className":57961},[7104],[35,57963,47454],{"className":57964,"style":47475},[7104,7145],[35,57966,57968],{"className":57967},[7113],[35,57969,57971,57991],{"className":57970},[7117,7731],[35,57972,57974,57988],{"className":57973},[7121],[35,57975,57977],{"className":57976,"style":7738},[7125],[35,57978,57979,57982],{"style":47490},[35,57980],{"className":57981,"style":7134},[7133],[35,57983,57985],{"className":57984},[7138,7139,7140,7141],[35,57986,24883],{"className":57987,"style":24908},[7104,7145,7141],[35,57989,7756],{"className":57990},[7755],[35,57992,57994],{"className":57993},[7121],[35,57995,57997],{"className":57996,"style":7763},[7125],[35,57998],{},[35,58000],{"className":58001,"style":7450},[7248],[35,58003,242],{"className":58004},[7454],[35,58006],{"className":58007,"style":7450},[7248],[35,58009,58011,58014,58116],{"className":58010},[7095],[35,58012],{"className":58013,"style":52912},[7099],[35,58015,58017,58020,58113],{"className":58016},[7104],[35,58018],{"className":58019},[7472,14311],[35,58021,58023],{"className":58022},[14009],[35,58024,58026,58105],{"className":58025},[7117,7731],[35,58027,58029,58102],{"className":58028},[7121],[35,58030,58032,58046,58054],{"className":58031,"style":51519},[7125],[35,58033,58034,58037],{"style":16234},[35,58035],{"className":58036,"style":8839},[7133],[35,58038,58040,58043],{"className":58039},[7104],[35,58041,460],{"className":58042},[7104],[35,58044,13081],{"className":58045,"style":7310},[7104,7145],[35,58047,58048,58051],{"style":14395},[35,58049],{"className":58050,"style":8839},[7133],[35,58052],{"className":58053,"style":14403},[14402],[35,58055,58056,58059],{"style":14406},[35,58057],{"className":58058,"style":8839},[7133],[35,58060,58062],{"className":58061},[7104],[35,58063,58065,58068],{"className":58064},[7104],[35,58066,47141],{"className":58067},[7104],[35,58069,58071],{"className":58070},[7113],[35,58072,58074,58094],{"className":58073},[7117,7731],[35,58075,58077,58091],{"className":58076},[7121],[35,58078,58080],{"className":58079,"style":7738},[7125],[35,58081,58082,58085],{"style":9926},[35,58083],{"className":58084,"style":7134},[7133],[35,58086,58088],{"className":58087},[7138,7139,7140,7141],[35,58089,24883],{"className":58090,"style":24908},[7104,7145,7141],[35,58092,7756],{"className":58093},[7755],[35,58095,58097],{"className":58096},[7121],[35,58098,58100],{"className":58099,"style":7763},[7125],[35,58101],{},[35,58103,7756],{"className":58104},[7755],[35,58106,58108],{"className":58107},[7121],[35,58109,58111],{"className":58110,"style":16272},[7125],[35,58112],{},[35,58114],{"className":58115},[7314,14311],[35,58117,5122],{"className":58118},[7104],[12,58120,58121],{},"The computational workflow subtracts the mean signal along the pulse-duration axis, applies a Hann window to reduce spectral leakage, uses zero-padding to smooth the displayed Fourier grid, and overlays the exact generalized Rabi ridge on the Fourier heatmap. Zero-padding improves visual interpolation of the plotted spectrum, although the physical frequency resolution remains set by the sampled temporal window.",[5019,58123,58125],{"id":58124},"computational-methodology","Computational Methodology",[12,58127,58128,58129,58157,58158,58186,58187,15316,58215,58243,58244,58378,58379,58598],{},"The notebook uses a deterministic closed-form workflow, so it does not require numerical time stepping of the Schrödinger equation. It defines a drive-frequency grid ",[35,58130,58132,58145],{"className":58131},[7057],[35,58133,58135],{"className":58134},[7061],[7063,58136,58137],{"xmlns":7065},[7067,58138,58139,58143],{},[7070,58140,58141],{},[7076,58142,1675],{},[7083,58144,1675],{"encoding":7085},[35,58146,58148],{"className":58147,"ariaHidden":7091},[7090],[35,58149,58151,58154],{"className":58150},[7095],[35,58152],{"className":58153,"style":11562},[7099],[35,58155,1675],{"className":58156,"style":31824},[7104,7145]," in GHz and a pulse-duration grid ",[35,58159,58161,58174],{"className":58160},[7057],[35,58162,58164],{"className":58163},[7061],[7063,58165,58166],{"xmlns":7065},[7067,58167,58168,58172],{},[7070,58169,58170],{},[7076,58171,46183],{},[7083,58173,46309],{"encoding":7085},[35,58175,58177],{"className":58176,"ariaHidden":7091},[7090],[35,58178,58180,58183],{"className":58179},[7095],[35,58181],{"className":58182,"style":7173},[7099],[35,58184,46183],{"className":58185,"style":46255},[7104,7145]," in ns, then converts both to SI units. From a dense two-dimensional mesh over ",[35,58188,58190,58203],{"className":58189},[7057],[35,58191,58193],{"className":58192},[7061],[7063,58194,58195],{"xmlns":7065},[7067,58196,58197,58201],{},[7070,58198,58199],{},[7076,58200,1675],{},[7083,58202,1675],{"encoding":7085},[35,58204,58206],{"className":58205,"ariaHidden":7091},[7090],[35,58207,58209,58212],{"className":58208},[7095],[35,58210],{"className":58211,"style":11562},[7099],[35,58213,1675],{"className":58214,"style":31824},[7104,7145],[35,58216,58218,58231],{"className":58217},[7057],[35,58219,58221],{"className":58220},[7061],[7063,58222,58223],{"xmlns":7065},[7067,58224,58225,58229],{},[7070,58226,58227],{},[7076,58228,46183],{},[7083,58230,46309],{"encoding":7085},[35,58232,58234],{"className":58233,"ariaHidden":7091},[7090],[35,58235,58237,58240],{"className":58236},[7095],[35,58238],{"className":58239,"style":7173},[7099],[35,58241,46183],{"className":58242,"style":46255},[7104,7145],", the workflow computes the detuning ",[35,58245,58247,58281],{"className":58246},[7057],[35,58248,58250],{"className":58249},[7061],[7063,58251,58252],{"xmlns":7065},[7067,58253,58254,58278],{},[7070,58255,58256,58258,58260,58262,58264,58266,58268,58270,58276],{},[7076,58257,47224],{"mathvariant":7282},[7223,58259,242],{},[7227,58261,460],{},[7076,58263,13081],{},[7223,58265,275],{"stretchy":7289},[7076,58267,1675],{},[7223,58269,7225],{},[7699,58271,58272,58274],{},[7076,58273,1675],{},[7227,58275,5312],{},[7223,58277,1694],{"stretchy":7289},[7083,58279,58280],{"encoding":7085},"\\Delta=2\\pi(f-f_0)",[35,58282,58284,58302,58329],{"className":58283,"ariaHidden":7091},[7090],[35,58285,58287,58290,58293,58296,58299],{"className":58286},[7095],[35,58288],{"className":58289,"style":7100},[7099],[35,58291,47224],{"className":58292},[7104],[35,58294],{"className":58295,"style":7450},[7248],[35,58297,242],{"className":58298},[7454],[35,58300],{"className":58301,"style":7450},[7248],[35,58303,58305,58308,58311,58314,58317,58320,58323,58326],{"className":58304},[7095],[35,58306],{"className":58307,"style":7303},[7099],[35,58309,460],{"className":58310},[7104],[35,58312,13081],{"className":58313,"style":7310},[7104,7145],[35,58315,275],{"className":58316},[7472],[35,58318,1675],{"className":58319,"style":31824},[7104,7145],[35,58321],{"className":58322,"style":7249},[7248],[35,58324,7225],{"className":58325},[7253],[35,58327],{"className":58328,"style":7249},[7248],[35,58330,58332,58335,58375],{"className":58331},[7095],[35,58333],{"className":58334,"style":7303},[7099],[35,58336,58338,58341],{"className":58337},[7104],[35,58339,1675],{"className":58340,"style":31824},[7104,7145],[35,58342,58344],{"className":58343},[7113],[35,58345,58347,58367],{"className":58346},[7117,7731],[35,58348,58350,58364],{"className":58349},[7121],[35,58351,58353],{"className":58352,"style":18794},[7125],[35,58354,58355,58358],{"style":46746},[35,58356],{"className":58357,"style":7134},[7133],[35,58359,58361],{"className":58360},[7138,7139,7140,7141],[35,58362,5312],{"className":58363},[7104,7141],[35,58365,7756],{"className":58366},[7755],[35,58368,58370],{"className":58369},[7121],[35,58371,58373],{"className":58372,"style":7763},[7125],[35,58374],{},[35,58376,1694],{"className":58377},[7314],", evaluates the generalized Rabi angular frequency ",[35,58380,58382,58419],{"className":58381},[7057],[35,58383,58385],{"className":58384},[7061],[7063,58386,58387],{"xmlns":7065},[7067,58388,58389,58417],{},[7070,58390,58391,58397,58399],{},[7699,58392,58393,58395],{},[7076,58394,47141],{"mathvariant":7282},[7076,58396,24883],{},[7223,58398,242],{},[14014,58400,58401],{},[7070,58402,58403,58409,58411],{},[7073,58404,58405,58407],{},[7076,58406,47141],{"mathvariant":7282},[7227,58408,460],{},[7223,58410,445],{},[7073,58412,58413,58415],{},[7076,58414,47224],{"mathvariant":7282},[7227,58416,460],{},[7083,58418,51662],{"encoding":7085},[35,58420,58422,58477],{"className":58421,"ariaHidden":7091},[7090],[35,58423,58425,58428,58468,58471,58474],{"className":58424},[7095],[35,58426],{"className":58427,"style":7718},[7099],[35,58429,58431,58434],{"className":58430},[7104],[35,58432,47141],{"className":58433},[7104],[35,58435,58437],{"className":58436},[7113],[35,58438,58440,58460],{"className":58439},[7117,7731],[35,58441,58443,58457],{"className":58442},[7121],[35,58444,58446],{"className":58445,"style":7738},[7125],[35,58447,58448,58451],{"style":9926},[35,58449],{"className":58450,"style":7134},[7133],[35,58452,58454],{"className":58453},[7138,7139,7140,7141],[35,58455,24883],{"className":58456,"style":24908},[7104,7145,7141],[35,58458,7756],{"className":58459},[7755],[35,58461,58463],{"className":58462},[7121],[35,58464,58466],{"className":58465,"style":7763},[7125],[35,58467],{},[35,58469],{"className":58470,"style":7450},[7248],[35,58472,242],{"className":58473},[7454],[35,58475],{"className":58476,"style":7450},[7248],[35,58478,58480,58484],{"className":58479},[7095],[35,58481],{"className":58482,"style":58483},[7099],"height:1.04em;vertical-align:-0.1266em;",[35,58485,58487],{"className":58486},[7104,490],[35,58488,58490,58589],{"className":58489},[7117,7731],[35,58491,58493,58586],{"className":58492},[7121],[35,58494,58497,58573],{"className":58495,"style":58496},[7125],"height:0.9134em;",[35,58498,58500,58503],{"className":58499,"style":8835},[14350],[35,58501],{"className":58502,"style":8839},[7133],[35,58504,58506,58535,58538,58541,58544],{"className":58505,"style":14357},[7104],[35,58507,58509,58512],{"className":58508},[7104],[35,58510,47141],{"className":58511},[7104],[35,58513,58515],{"className":58514},[7113],[35,58516,58518],{"className":58517},[7117],[35,58519,58521],{"className":58520},[7121],[35,58522,58524],{"className":58523,"style":33894},[7125],[35,58525,58526,58529],{"style":33897},[35,58527],{"className":58528,"style":7134},[7133],[35,58530,58532],{"className":58531},[7138,7139,7140,7141],[35,58533,460],{"className":58534},[7104,7141],[35,58536],{"className":58537,"style":7249},[7248],[35,58539,445],{"className":58540},[7253],[35,58542],{"className":58543,"style":7249},[7248],[35,58545,58547,58550],{"className":58546},[7104],[35,58548,47224],{"className":58549},[7104],[35,58551,58553],{"className":58552},[7113],[35,58554,58556],{"className":58555},[7117],[35,58557,58559],{"className":58558},[7121],[35,58560,58562],{"className":58561,"style":33894},[7125],[35,58563,58564,58567],{"style":33897},[35,58565],{"className":58566,"style":7134},[7133],[35,58568,58570],{"className":58569},[7138,7139,7140,7141],[35,58571,460],{"className":58572},[7104,7141],[35,58574,58576,58579],{"style":58575},"top:-2.8734em;",[35,58577],{"className":58578,"style":8839},[7133],[35,58580,58582],{"className":58581,"style":14371},[14370],[10808,58583,58584],{"xmlns":10810,"width":14374,"height":14375,"viewBox":14376,"preserveAspectRatio":14377},[10817,58585],{"d":14380},[35,58587,7756],{"className":58588},[7755],[35,58590,58592],{"className":58591},[7121],[35,58593,58596],{"className":58594,"style":58595},[7125],"height:0.1266em;",[35,58597],{},", and calculates the analytic excited-state probability",[35,58600,58602],{"className":58601},[7318],[35,58603,58605,58683],{"className":58604},[7057],[35,58606,58608],{"className":58607},[7061],[7063,58609,58610],{"xmlns":7065,"display":7327},[7067,58611,58612,58680],{},[7070,58613,58614,58620,58622,58624,58626,58628,58630,58632,58648,58658,58678],{},[7699,58615,58616,58618],{},[7076,58617,7705],{},[7076,58619,7817],{},[7223,58621,275],{"stretchy":7289},[7076,58623,1675],{},[7223,58625,2877],{"separator":7091},[7076,58627,46183],{},[7223,58629,1694],{"stretchy":7289},[7223,58631,242],{},[14009,58633,58634,58640],{},[7073,58635,58636,58638],{},[7076,58637,47141],{"mathvariant":7282},[7227,58639,460],{},[8304,58641,58642,58644,58646],{},[7076,58643,47141],{"mathvariant":7282},[7076,58645,24883],{},[7227,58647,460],{},[7073,58649,58650,58656],{},[7070,58651,58652,58654],{},[7076,58653,51200],{},[7223,58655,41175],{},[7227,58657,460],{},[7070,58659,58660,58662,58676],{},[7223,58661,275],{"fence":7091},[14009,58663,58664,58674],{},[7070,58665,58666,58672],{},[7699,58667,58668,58670],{},[7076,58669,47141],{"mathvariant":7282},[7076,58671,24883],{},[7076,58673,46183],{},[7227,58675,460],{},[7223,58677,1694],{"fence":7091},[7076,58679,5122],{"mathvariant":7282},[7083,58681,58682],{"encoding":7085},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right).",[35,58684,58686,58759],{"className":58685,"ariaHidden":7091},[7090],[35,58687,58689,58692,58732,58735,58738,58741,58744,58747,58750,58753,58756],{"className":58688},[7095],[35,58690],{"className":58691,"style":7303},[7099],[35,58693,58695,58698],{"className":58694},[7104],[35,58696,7705],{"className":58697,"style":7751},[7104,7145],[35,58699,58701],{"className":58700},[7113],[35,58702,58704,58724],{"className":58703},[7117,7731],[35,58705,58707,58721],{"className":58706},[7121],[35,58708,58710],{"className":58709,"style":9035},[7125],[35,58711,58712,58715],{"style":12163},[35,58713],{"className":58714,"style":7134},[7133],[35,58716,58718],{"className":58717},[7138,7139,7140,7141],[35,58719,7817],{"className":58720},[7104,7145,7141],[35,58722,7756],{"className":58723},[7755],[35,58725,58727],{"className":58726},[7121],[35,58728,58730],{"className":58729,"style":7763},[7125],[35,58731],{},[35,58733,275],{"className":58734},[7472],[35,58736,1675],{"className":58737,"style":31824},[7104,7145],[35,58739,2877],{"className":58740},[7492],[35,58742],{"className":58743,"style":7496},[7248],[35,58745,46183],{"className":58746,"style":46255},[7104,7145],[35,58748,1694],{"className":58749},[7314],[35,58751],{"className":58752,"style":7450},[7248],[35,58754,242],{"className":58755},[7454],[35,58757],{"className":58758,"style":7450},[7248],[35,58760,58762,58765,58901,58904,58933,58936,59053,59056],{"className":58761},[7095],[35,58763],{"className":58764,"style":51312},[7099],[35,58766,58768,58771,58898],{"className":58767},[7104],[35,58769],{"className":58770},[7472,14311],[35,58772,58774],{"className":58773},[14009],[35,58775,58777,58890],{"className":58776},[7117,7731],[35,58778,58780,58887],{"className":58779},[7121],[35,58781,58783,58842,58850],{"className":58782,"style":51331},[7125],[35,58784,58785,58788],{"style":16234},[35,58786],{"className":58787,"style":8839},[7133],[35,58789,58791],{"className":58790},[7104],[35,58792,58794,58797],{"className":58793},[7104],[35,58795,47141],{"className":58796},[7104],[35,58798,58800],{"className":58799},[7113],[35,58801,58803,58834],{"className":58802},[7117,7731],[35,58804,58806,58831],{"className":58805},[7121],[35,58807,58809,58820],{"className":58808,"style":51358},[7125],[35,58810,58811,58814],{"style":51361},[35,58812],{"className":58813,"style":7134},[7133],[35,58815,58817],{"className":58816},[7138,7139,7140,7141],[35,58818,24883],{"className":58819,"style":24908},[7104,7145,7141],[35,58821,58822,58825],{"style":51373},[35,58823],{"className":58824,"style":7134},[7133],[35,58826,58828],{"className":58827},[7138,7139,7140,7141],[35,58829,460],{"className":58830},[7104,7141],[35,58832,7756],{"className":58833},[7755],[35,58835,58837],{"className":58836},[7121],[35,58838,58840],{"className":58839,"style":51392},[7125],[35,58841],{},[35,58843,58844,58847],{"style":14395},[35,58845],{"className":58846,"style":8839},[7133],[35,58848],{"className":58849,"style":14403},[14402],[35,58851,58852,58855],{"style":14406},[35,58853],{"className":58854,"style":8839},[7133],[35,58856,58858],{"className":58857},[7104],[35,58859,58861,58864],{"className":58860},[7104],[35,58862,47141],{"className":58863},[7104],[35,58865,58867],{"className":58866},[7113],[35,58868,58870],{"className":58869},[7117],[35,58871,58873],{"className":58872},[7121],[35,58874,58876],{"className":58875,"style":17739},[7125],[35,58877,58878,58881],{"style":7129},[35,58879],{"className":58880,"style":7134},[7133],[35,58882,58884],{"className":58883},[7138,7139,7140,7141],[35,58885,460],{"className":58886},[7104,7141],[35,58888,7756],{"className":58889},[7755],[35,58891,58893],{"className":58892},[7121],[35,58894,58896],{"className":58895,"style":51449},[7125],[35,58897],{},[35,58899],{"className":58900},[7314,14311],[35,58902],{"className":58903,"style":7496},[7248],[35,58905,58907,58910],{"className":58906},[12096],[35,58908,51200],{"className":58909},[12096],[35,58911,58913],{"className":58912},[7113],[35,58914,58916],{"className":58915},[7117],[35,58917,58919],{"className":58918},[7121],[35,58920,58922],{"className":58921,"style":51476},[7125],[35,58923,58924,58927],{"style":51479},[35,58925],{"className":58926,"style":7134},[7133],[35,58928,58930],{"className":58929},[7138,7139,7140,7141],[35,58931,460],{"className":58932},[7104,7141],[35,58934],{"className":58935,"style":7496},[7248],[35,58937,58939,58945,59047],{"className":58938},[7468],[35,58940,58942],{"className":58941,"style":7474},[7472,7473],[35,58943,275],{"className":58944},[7478,7140],[35,58946,58948,58951,59044],{"className":58947},[7104],[35,58949],{"className":58950},[7472,14311],[35,58952,58954],{"className":58953},[14009],[35,58955,58957,59036],{"className":58956},[7117,7731],[35,58958,58960,59033],{"className":58959},[7121],[35,58961,58963,58974,58982],{"className":58962,"style":51519},[7125],[35,58964,58965,58968],{"style":16234},[35,58966],{"className":58967,"style":8839},[7133],[35,58969,58971],{"className":58970},[7104],[35,58972,460],{"className":58973},[7104],[35,58975,58976,58979],{"style":14395},[35,58977],{"className":58978,"style":8839},[7133],[35,58980],{"className":58981,"style":14403},[14402],[35,58983,58984,58987],{"style":14406},[35,58985],{"className":58986,"style":8839},[7133],[35,58988,58990,59030],{"className":58989},[7104],[35,58991,58993,58996],{"className":58992},[7104],[35,58994,47141],{"className":58995},[7104],[35,58997,58999],{"className":58998},[7113],[35,59000,59002,59022],{"className":59001},[7117,7731],[35,59003,59005,59019],{"className":59004},[7121],[35,59006,59008],{"className":59007,"style":7738},[7125],[35,59009,59010,59013],{"style":9926},[35,59011],{"className":59012,"style":7134},[7133],[35,59014,59016],{"className":59015},[7138,7139,7140,7141],[35,59017,24883],{"className":59018,"style":24908},[7104,7145,7141],[35,59020,7756],{"className":59021},[7755],[35,59023,59025],{"className":59024},[7121],[35,59026,59028],{"className":59027,"style":7763},[7125],[35,59029],{},[35,59031,46183],{"className":59032,"style":46255},[7104,7145],[35,59034,7756],{"className":59035},[7755],[35,59037,59039],{"className":59038},[7121],[35,59040,59042],{"className":59041,"style":16272},[7125],[35,59043],{},[35,59045],{"className":59046},[7314,14311],[35,59048,59050],{"className":59049,"style":7474},[7314,7473],[35,59051,1694],{"className":59052},[7478,7140],[35,59054],{"className":59055,"style":7496},[7248],[35,59057,5122],{"className":59058},[7104],[12,59060,59061,59062,59177],{},"The routine optionally applies the damping envelope ",[35,59063,59065,59099],{"className":59064},[7057],[35,59066,59068],{"className":59067},[7061],[7063,59069,59070],{"xmlns":7065},[7067,59071,59072,59096],{},[7070,59073,59074,59076,59078,59080,59082,59084,59086,59094],{},[7076,59075,28563],{},[7223,59077,41175],{},[7223,59079,275],{"stretchy":7289},[7223,59081,7225],{},[7076,59083,46183],{},[7076,59085,398],{"mathvariant":7282},[8304,59087,59088,59090,59092],{},[7076,59089,41530],{},[7227,59091,460],{},[7223,59093,25282],{},[7223,59095,1694],{"stretchy":7289},[7083,59097,59098],{"encoding":7085},"\\exp(-\\tau\u002FT_2^\\ast)",[35,59100,59102],{"className":59101,"ariaHidden":7091},[7090],[35,59103,59105,59108,59111,59114,59117,59120,59123,59174],{"className":59104},[7095],[35,59106],{"className":59107,"style":7303},[7099],[35,59109,28563],{"className":59110},[12096],[35,59112,275],{"className":59113},[7472],[35,59115,7225],{"className":59116},[7104],[35,59118,46183],{"className":59119,"style":46255},[7104,7145],[35,59121,398],{"className":59122},[7104],[35,59124,59126,59129],{"className":59125},[7104],[35,59127,41530],{"className":59128,"style":7751},[7104,7145],[35,59130,59132],{"className":59131},[7113],[35,59133,59135,59166],{"className":59134},[7117,7731],[35,59136,59138,59163],{"className":59137},[7121],[35,59139,59141,59152],{"className":59140,"style":41067},[7125],[35,59142,59143,59146],{"style":47751},[35,59144],{"className":59145,"style":7134},[7133],[35,59147,59149],{"className":59148},[7138,7139,7140,7141],[35,59150,460],{"className":59151},[7104,7141],[35,59153,59154,59157],{"style":7129},[35,59155],{"className":59156,"style":7134},[7133],[35,59158,59160],{"className":59159},[7138,7139,7140,7141],[35,59161,25282],{"className":59162},[7253,7141],[35,59164,7756],{"className":59165},[7755],[35,59167,59169],{"className":59168},[7121],[35,59170,59172],{"className":59171,"style":47781},[7125],[35,59173],{},[35,59175,1694],{"className":59176},[7314]," and clips the result to the physical interval",[35,59179,59181],{"className":59180},[7318],[35,59182,59184,59210],{"className":59183},[7057],[35,59185,59187],{"className":59186},[7061],[7063,59188,59189],{"xmlns":7065,"display":7327},[7067,59190,59191,59207],{},[7070,59192,59193,59195,59197,59203,59205],{},[7227,59194,5312],{},[7223,59196,31994],{},[7699,59198,59199,59201],{},[7076,59200,7705],{},[7076,59202,7817],{},[7223,59204,31994],{},[7227,59206,48822],{},[7083,59208,59209],{"encoding":7085},"0\\le P_e\\le 1.",[35,59211,59213,59232,59287],{"className":59212,"ariaHidden":7091},[7090],[35,59214,59216,59220,59223,59226,59229],{"className":59215},[7095],[35,59217],{"className":59218,"style":59219},[7099],"height:0.7804em;vertical-align:-0.136em;",[35,59221,5312],{"className":59222},[7104],[35,59224],{"className":59225,"style":7450},[7248],[35,59227,31994],{"className":59228},[7454],[35,59230],{"className":59231,"style":7450},[7248],[35,59233,59235,59238,59278,59281,59284],{"className":59234},[7095],[35,59236],{"className":59237,"style":7718},[7099],[35,59239,59241,59244],{"className":59240},[7104],[35,59242,7705],{"className":59243,"style":7751},[7104,7145],[35,59245,59247],{"className":59246},[7113],[35,59248,59250,59270],{"className":59249},[7117,7731],[35,59251,59253,59267],{"className":59252},[7121],[35,59254,59256],{"className":59255,"style":9035},[7125],[35,59257,59258,59261],{"style":12163},[35,59259],{"className":59260,"style":7134},[7133],[35,59262,59264],{"className":59263},[7138,7139,7140,7141],[35,59265,7817],{"className":59266},[7104,7145,7141],[35,59268,7756],{"className":59269},[7755],[35,59271,59273],{"className":59272},[7121],[35,59274,59276],{"className":59275,"style":7763},[7125],[35,59277],{},[35,59279],{"className":59280,"style":7450},[7248],[35,59282,31994],{"className":59283},[7454],[35,59285],{"className":59286,"style":7450},[7248],[35,59288,59290,59293],{"className":59289},[7095],[35,59291],{"className":59292,"style":7263},[7099],[35,59294,48822],{"className":59295},[7104],[12,59297,59298],{},"Finally, the output is rendered as heatmaps, three-dimensional surfaces, one-dimensional slices, and Fourier-domain spectra. Standard scientific Python libraries provide the numerical arrays, visualization routines, and optional interactive rendering tools.",[5019,59300,59302],{"id":59301},"graphical-interpretation","Graphical Interpretation",[12,59304,59305,59306,59404],{},"The ",[35,59307,59309,59337],{"className":59308},[7057],[35,59310,59312],{"className":59311},[7061],[7063,59313,59314],{"xmlns":7065},[7067,59315,59316,59334],{},[7070,59317,59318,59324,59326,59328,59330,59332],{},[7699,59319,59320,59322],{},[7076,59321,7705],{},[7076,59323,7817],{},[7223,59325,275],{"stretchy":7289},[7076,59327,1675],{},[7223,59329,2877],{"separator":7091},[7076,59331,46183],{},[7223,59333,1694],{"stretchy":7289},[7083,59335,59336],{"encoding":7085},"P_e(f,\\tau)",[35,59338,59340],{"className":59339,"ariaHidden":7091},[7090],[35,59341,59343,59346,59386,59389,59392,59395,59398,59401],{"className":59342},[7095],[35,59344],{"className":59345,"style":7303},[7099],[35,59347,59349,59352],{"className":59348},[7104],[35,59350,7705],{"className":59351,"style":7751},[7104,7145],[35,59353,59355],{"className":59354},[7113],[35,59356,59358,59378],{"className":59357},[7117,7731],[35,59359,59361,59375],{"className":59360},[7121],[35,59362,59364],{"className":59363,"style":9035},[7125],[35,59365,59366,59369],{"style":12163},[35,59367],{"className":59368,"style":7134},[7133],[35,59370,59372],{"className":59371},[7138,7139,7140,7141],[35,59373,7817],{"className":59374},[7104,7145,7141],[35,59376,7756],{"className":59377},[7755],[35,59379,59381],{"className":59380},[7121],[35,59382,59384],{"className":59383,"style":7763},[7125],[35,59385],{},[35,59387,275],{"className":59388},[7472],[35,59390,1675],{"className":59391,"style":31824},[7104,7145],[35,59393,2877],{"className":59394},[7492],[35,59396],{"className":59397,"style":7496},[7248],[35,59399,46183],{"className":59400,"style":46255},[7104,7145],[35,59402,1694],{"className":59403},[7314]," heatmap shows the predicted probability of detecting the qubit in the excited state after applying a control pulse. Its horizontal axis gives the applied drive frequency. The vertical axis gives the pulse duration. Brighter regions indicate higher excited-state probability.",[12,59406,59407],{},"The resonant interaction region appears near",[35,59409,59411],{"className":59410},[7318],[35,59412,59414,59437],{"className":59413},[7057],[35,59415,59417],{"className":59416},[7061],[7063,59418,59419],{"xmlns":7065,"display":7327},[7067,59420,59421,59435],{},[7070,59422,59423,59425,59427,59433],{},[7076,59424,1675],{},[7223,59426,242],{},[7699,59428,59429,59431],{},[7076,59430,1675],{},[7227,59432,5312],{},[7076,59434,5122],{"mathvariant":7282},[7083,59436,48518],{"encoding":7085},[35,59438,59440,59458],{"className":59439,"ariaHidden":7091},[7090],[35,59441,59443,59446,59449,59452,59455],{"className":59442},[7095],[35,59444],{"className":59445,"style":11562},[7099],[35,59447,1675],{"className":59448,"style":31824},[7104,7145],[35,59450],{"className":59451,"style":7450},[7248],[35,59453,242],{"className":59454},[7454],[35,59456],{"className":59457,"style":7450},[7248],[35,59459,59461,59464,59504],{"className":59460},[7095],[35,59462],{"className":59463,"style":11562},[7099],[35,59465,59467,59470],{"className":59466},[7104],[35,59468,1675],{"className":59469,"style":31824},[7104,7145],[35,59471,59473],{"className":59472},[7113],[35,59474,59476,59496],{"className":59475},[7117,7731],[35,59477,59479,59493],{"className":59478},[7121],[35,59480,59482],{"className":59481,"style":18794},[7125],[35,59483,59484,59487],{"style":46746},[35,59485],{"className":59486,"style":7134},[7133],[35,59488,59490],{"className":59489},[7138,7139,7140,7141],[35,59491,5312],{"className":59492},[7104,7141],[35,59494,7756],{"className":59495},[7755],[35,59497,59499],{"className":59498},[7121],[35,59500,59502],{"className":59501,"style":7763},[7125],[35,59503],{},[35,59505,5122],{"className":59506},[7104],[12,59508,59509,59510,59589],{},"At resonance, contrast is maximal and the pulse duration alone sets the rotation angle. With the default drive strength ",[35,59511,59513,59547],{"className":59512},[7057],[35,59514,59516],{"className":59515},[7061],[7063,59517,59518],{"xmlns":7065},[7067,59519,59520,59544],{},[7070,59521,59522,59524,59526,59528,59530,59532,59534,59536],{},[7076,59523,47141],{"mathvariant":7282},[7076,59525,398],{"mathvariant":7282},[7227,59527,460],{},[7076,59529,13081],{},[7223,59531,242],{},[7227,59533,52886],{},[10694,59535,10696],{},[7070,59537,59538,59540,59542],{},[7076,59539,52893],{"mathvariant":7282},[7076,59541,7669],{"mathvariant":7282},[7076,59543,300],{"mathvariant":7282},[7083,59545,59546],{"encoding":7085},"\\Omega\u002F2\\pi=20~\\mathrm{MHz}",[35,59548,59550,59571],{"className":59549,"ariaHidden":7091},[7090],[35,59551,59553,59556,59559,59562,59565,59568],{"className":59552},[7095],[35,59554],{"className":59555,"style":7303},[7099],[35,59557,47199],{"className":59558},[7104],[35,59560,13081],{"className":59561,"style":7310},[7104,7145],[35,59563],{"className":59564,"style":7450},[7248],[35,59566,242],{"className":59567},[7454],[35,59569],{"className":59570,"style":7450},[7248],[35,59572,59574,59577,59580,59583],{"className":59573},[7095],[35,59575],{"className":59576,"style":7100},[7099],[35,59578,52886],{"className":59579},[7104],[35,59581,10696],{"className":59582},[7248,48368],[35,59584,59586],{"className":59585},[7104],[35,59587,53006],{"className":59588},[7104,48375],", the ideal resonant maxima appear near",[35,59591,59593],{"className":59592},[7318],[35,59594,59596,59669],{"className":59595},[7057],[35,59597,59599],{"className":59598},[7061],[7063,59600,59601],{"xmlns":7065,"display":7327},[7067,59602,59603,59666],{},[7070,59604,59605,59607,59609,59611,59613,59619,59621,59623,59626,59628,59634,59636,59638,59641,59643,59649,59651,59653,59656,59658,59664],{},[7076,59606,46183],{},[7223,59608,242],{},[7227,59610,2680],{},[10694,59612,10696],{},[7070,59614,59615,59617],{},[7076,59616,291],{"mathvariant":7282},[7076,59618,19046],{"mathvariant":7282},[7223,59620,2877],{"separator":7091},[10694,59622,10696],{},[7227,59624,59625],{},"75",[10694,59627,10696],{},[7070,59629,59630,59632],{},[7076,59631,291],{"mathvariant":7282},[7076,59633,19046],{"mathvariant":7282},[7223,59635,2877],{"separator":7091},[10694,59637,10696],{},[7227,59639,59640],{},"125",[10694,59642,10696],{},[7070,59644,59645,59647],{},[7076,59646,291],{"mathvariant":7282},[7076,59648,19046],{"mathvariant":7282},[7223,59650,2877],{"separator":7091},[10694,59652,10696],{},[7227,59654,59655],{},"175",[10694,59657,10696],{},[7070,59659,59660,59662],{},[7076,59661,291],{"mathvariant":7282},[7076,59663,19046],{"mathvariant":7282},[7223,59665,2877],{"separator":7091},[7083,59667,59668],{"encoding":7085},"\\tau=25~\\mathrm{ns},\\ 75~\\mathrm{ns},\\ 125~\\mathrm{ns},\\ 175~\\mathrm{ns},",[35,59670,59672,59690],{"className":59671,"ariaHidden":7091},[7090],[35,59673,59675,59678,59681,59684,59687],{"className":59674},[7095],[35,59676],{"className":59677,"style":7173},[7099],[35,59679,46183],{"className":59680,"style":46255},[7104,7145],[35,59682],{"className":59683,"style":7450},[7248],[35,59685,242],{"className":59686},[7454],[35,59688],{"className":59689,"style":7450},[7248],[35,59691,59693,59697,59700,59703,59709,59712,59715,59718,59721,59724,59730,59733,59736,59739,59742,59745,59751,59754,59757,59760,59763,59766,59772],{"className":59692},[7095],[35,59694],{"className":59695,"style":59696},[7099],"height:0.8389em;vertical-align:-0.1944em;",[35,59698,2680],{"className":59699},[7104],[35,59701,10696],{"className":59702},[7248,48368],[35,59704,59706],{"className":59705},[7104],[35,59707,53156],{"className":59708},[7104,48375],[35,59710,2877],{"className":59711},[7492],[35,59713,10696],{"className":59714},[7248],[35,59716],{"className":59717,"style":7496},[7248],[35,59719,59625],{"className":59720},[7104],[35,59722,10696],{"className":59723},[7248,48368],[35,59725,59727],{"className":59726},[7104],[35,59728,53156],{"className":59729},[7104,48375],[35,59731,2877],{"className":59732},[7492],[35,59734,10696],{"className":59735},[7248],[35,59737],{"className":59738,"style":7496},[7248],[35,59740,59640],{"className":59741},[7104],[35,59743,10696],{"className":59744},[7248,48368],[35,59746,59748],{"className":59747},[7104],[35,59749,53156],{"className":59750},[7104,48375],[35,59752,2877],{"className":59753},[7492],[35,59755,10696],{"className":59756},[7248],[35,59758],{"className":59759,"style":7496},[7248],[35,59761,59655],{"className":59762},[7104],[35,59764,10696],{"className":59765},[7248,48368],[35,59767,59769],{"className":59768},[7104],[35,59770,53156],{"className":59771},[7104,48375],[35,59773,2877],{"className":59774},[7492],[12,59776,59777],{},"before the optional phenomenological damping factor is applied.",[12,59779,59780,59781,59850,59851,59977],{},"Off resonance, the fringes show lower contrast and faster temporal oscillations. This behavior follows from the generalized Rabi frequency ",[35,59782,59784,59801],{"className":59783},[7057],[35,59785,59787],{"className":59786},[7061],[7063,59788,59789],{"xmlns":7065},[7067,59790,59791,59799],{},[7070,59792,59793],{},[7699,59794,59795,59797],{},[7076,59796,47141],{"mathvariant":7282},[7076,59798,24883],{},[7083,59800,47381],{"encoding":7085},[35,59802,59804],{"className":59803,"ariaHidden":7091},[7090],[35,59805,59807,59810],{"className":59806},[7095],[35,59808],{"className":59809,"style":7718},[7099],[35,59811,59813,59816],{"className":59812},[7104],[35,59814,47141],{"className":59815},[7104],[35,59817,59819],{"className":59818},[7113],[35,59820,59822,59842],{"className":59821},[7117,7731],[35,59823,59825,59839],{"className":59824},[7121],[35,59826,59828],{"className":59827,"style":7738},[7125],[35,59829,59830,59833],{"style":9926},[35,59831],{"className":59832,"style":7134},[7133],[35,59834,59836],{"className":59835},[7138,7139,7140,7141],[35,59837,24883],{"className":59838,"style":24908},[7104,7145,7141],[35,59840,7756],{"className":59841},[7755],[35,59843,59845],{"className":59844},[7121],[35,59846,59848],{"className":59847,"style":7763},[7125],[35,59849],{},", which increases with detuning, and from the amplitude coefficient ",[35,59852,59854,59882],{"className":59853},[7057],[35,59855,59857],{"className":59856},[7061],[7063,59858,59859],{"xmlns":7065},[7067,59860,59861,59879],{},[7070,59862,59863,59869,59871],{},[7073,59864,59865,59867],{},[7076,59866,47141],{"mathvariant":7282},[7227,59868,460],{},[7076,59870,398],{"mathvariant":7282},[8304,59872,59873,59875,59877],{},[7076,59874,47141],{"mathvariant":7282},[7076,59876,24883],{},[7227,59878,460],{},[7083,59880,59881],{"encoding":7085},"\\Omega^2\u002F\\Omega_R^2",[35,59883,59885],{"className":59884,"ariaHidden":7091},[7090],[35,59886,59888,59892,59921,59924],{"className":59887},[7095],[35,59889],{"className":59890,"style":59891},[7099],"height:1.0894em;vertical-align:-0.2753em;",[35,59893,59895,59898],{"className":59894},[7104],[35,59896,47141],{"className":59897},[7104],[35,59899,59901],{"className":59900},[7113],[35,59902,59904],{"className":59903},[7117],[35,59905,59907],{"className":59906},[7121],[35,59908,59910],{"className":59909,"style":17739},[7125],[35,59911,59912,59915],{"style":7129},[35,59913],{"className":59914,"style":7134},[7133],[35,59916,59918],{"className":59917},[7138,7139,7140,7141],[35,59919,460],{"className":59920},[7104,7141],[35,59922,398],{"className":59923},[7104],[35,59925,59927,59930],{"className":59926},[7104],[35,59928,47141],{"className":59929},[7104],[35,59931,59933],{"className":59932},[7113],[35,59934,59936,59968],{"className":59935},[7117,7731],[35,59937,59939,59965],{"className":59938},[7121],[35,59940,59942,59954],{"className":59941,"style":17739},[7125],[35,59943,59945,59948],{"style":59944},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[35,59946],{"className":59947,"style":7134},[7133],[35,59949,59951],{"className":59950},[7138,7139,7140,7141],[35,59952,24883],{"className":59953,"style":24908},[7104,7145,7141],[35,59955,59956,59959],{"style":7129},[35,59957],{"className":59958,"style":7134},[7133],[35,59960,59962],{"className":59961},[7138,7139,7140,7141],[35,59963,460],{"className":59964},[7104,7141],[35,59966,7756],{"className":59967},[7755],[35,59969,59971],{"className":59970},[7121],[35,59972,59975],{"className":59973,"style":59974},[7125],"height:0.2753em;",[35,59976],{},", which decreases with detuning.",[12,59979,59980],{},"The three-dimensional surface plot represents the same probability data as a height field. This view emphasizes the central resonant ridge and the suppressed off-resonant oscillations. Cross-section plots isolate one-dimensional behavior. A fixed-frequency slice shows how probability evolves with pulse duration. At a fixed duration, a complementary slice shows how probability changes across the drive-frequency sweep.",[12,59982,59983,59984,60081],{},"The FFT heatmap characterizes the oscillation-frequency content of ",[35,59985,59987,60014],{"className":59986},[7057],[35,59988,59990],{"className":59989},[7061],[7063,59991,59992],{"xmlns":7065},[7067,59993,59994,60012],{},[7070,59995,59996,60002,60004,60006,60008,60010],{},[7699,59997,59998,60000],{},[7076,59999,7705],{},[7076,60001,7817],{},[7223,60003,275],{"stretchy":7289},[7076,60005,46183],{},[7223,60007,2877],{"separator":7091},[7076,60009,1675],{},[7223,60011,1694],{"stretchy":7289},[7083,60013,57196],{"encoding":7085},[35,60015,60017],{"className":60016,"ariaHidden":7091},[7090],[35,60018,60020,60023,60063,60066,60069,60072,60075,60078],{"className":60019},[7095],[35,60021],{"className":60022,"style":7303},[7099],[35,60024,60026,60029],{"className":60025},[7104],[35,60027,7705],{"className":60028,"style":7751},[7104,7145],[35,60030,60032],{"className":60031},[7113],[35,60033,60035,60055],{"className":60034},[7117,7731],[35,60036,60038,60052],{"className":60037},[7121],[35,60039,60041],{"className":60040,"style":9035},[7125],[35,60042,60043,60046],{"style":12163},[35,60044],{"className":60045,"style":7134},[7133],[35,60047,60049],{"className":60048},[7138,7139,7140,7141],[35,60050,7817],{"className":60051},[7104,7145,7141],[35,60053,7756],{"className":60054},[7755],[35,60056,60058],{"className":60057},[7121],[35,60059,60061],{"className":60060,"style":7763},[7125],[35,60062],{},[35,60064,275],{"className":60065},[7472],[35,60067,46183],{"className":60068,"style":46255},[7104,7145],[35,60070,2877],{"className":60071},[7492],[35,60073],{"className":60074,"style":7496},[7248],[35,60076,1675],{"className":60077,"style":31824},[7104,7145],[35,60079,1694],{"className":60080},[7314],". An overlaid theoretical curve follows",[35,60083,60085],{"className":60084},[7318],[35,60086,60088,60160],{"className":60087},[7057],[35,60089,60091],{"className":60090},[7061],[7063,60092,60093],{"xmlns":7065,"display":7327},[7067,60094,60095,60157],{},[7070,60096,60097,60103,60105,60107,60109,60111,60155],{},[7699,60098,60099,60101],{},[7076,60100,47454],{},[7076,60102,24883],{},[7223,60104,275],{"stretchy":7289},[7076,60106,1675],{},[7223,60108,1694],{"stretchy":7289},[7223,60110,242],{},[14014,60112,60113],{},[7070,60114,60115,60135,60137,60139,60141,60143,60149],{},[7073,60116,60117,60133],{},[7070,60118,60119,60121,60131],{},[7223,60120,275],{"fence":7091},[14009,60122,60123,60125],{},[7076,60124,47141],{"mathvariant":7282},[7070,60126,60127,60129],{},[7227,60128,460],{},[7076,60130,13081],{},[7223,60132,1694],{"fence":7091},[7227,60134,460],{},[7223,60136,445],{},[7223,60138,275],{"stretchy":7289},[7076,60140,1675],{},[7223,60142,7225],{},[7699,60144,60145,60147],{},[7076,60146,1675],{},[7227,60148,5312],{},[7073,60150,60151,60153],{},[7223,60152,1694],{"stretchy":7289},[7227,60154,460],{},[7076,60156,5122],{"mathvariant":7282},[7083,60158,60159],{"encoding":7085},"\\nu_R(f)=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[35,60161,60163,60227],{"className":60162,"ariaHidden":7091},[7090],[35,60164,60166,60169,60209,60212,60215,60218,60221,60224],{"className":60165},[7095],[35,60167],{"className":60168,"style":7303},[7099],[35,60170,60172,60175],{"className":60171},[7104],[35,60173,47454],{"className":60174,"style":47475},[7104,7145],[35,60176,60178],{"className":60177},[7113],[35,60179,60181,60201],{"className":60180},[7117,7731],[35,60182,60184,60198],{"className":60183},[7121],[35,60185,60187],{"className":60186,"style":7738},[7125],[35,60188,60189,60192],{"style":47490},[35,60190],{"className":60191,"style":7134},[7133],[35,60193,60195],{"className":60194},[7138,7139,7140,7141],[35,60196,24883],{"className":60197,"style":24908},[7104,7145,7141],[35,60199,7756],{"className":60200},[7755],[35,60202,60204],{"className":60203},[7121],[35,60205,60207],{"className":60206,"style":7763},[7125],[35,60208],{},[35,60210,275],{"className":60211},[7472],[35,60213,1675],{"className":60214,"style":31824},[7104,7145],[35,60216,1694],{"className":60217},[7314],[35,60219],{"className":60220,"style":7450},[7248],[35,60222,242],{"className":60223},[7454],[35,60225],{"className":60226,"style":7450},[7248],[35,60228,60230,60233,60476],{"className":60229},[7095],[35,60231],{"className":60232,"style":54502},[7099],[35,60234,60236],{"className":60235},[7104,490],[35,60237,60239,60468],{"className":60238},[7117,7731],[35,60240,60242,60465],{"className":60241},[7121],[35,60243,60245,60453],{"className":60244,"style":54515},[7125],[35,60246,60248,60251],{"className":60247,"style":54519},[14350],[35,60249],{"className":60250,"style":54523},[7133],[35,60252,60254,60360,60363,60366,60369,60372,60375,60378,60381,60384,60424],{"className":60253,"style":51752},[7104],[35,60255,60257,60337],{"className":60256},[7468],[35,60258,60260,60266,60331],{"className":60259},[7468],[35,60261,60263],{"className":60262,"style":7474},[7472,7473],[35,60264,275],{"className":60265},[7478,7140],[35,60267,60269,60272,60328],{"className":60268},[7104],[35,60270],{"className":60271},[7472,14311],[35,60273,60275],{"className":60274},[14009],[35,60276,60278,60320],{"className":60277},[7117,7731],[35,60279,60281,60317],{"className":60280},[7121],[35,60282,60284,60298,60306],{"className":60283,"style":51519},[7125],[35,60285,60286,60289],{"style":16234},[35,60287],{"className":60288,"style":8839},[7133],[35,60290,60292,60295],{"className":60291},[7104],[35,60293,460],{"className":60294},[7104],[35,60296,13081],{"className":60297,"style":7310},[7104,7145],[35,60299,60300,60303],{"style":14395},[35,60301],{"className":60302,"style":8839},[7133],[35,60304],{"className":60305,"style":14403},[14402],[35,60307,60308,60311],{"style":14406},[35,60309],{"className":60310,"style":8839},[7133],[35,60312,60314],{"className":60313},[7104],[35,60315,47141],{"className":60316},[7104],[35,60318,7756],{"className":60319},[7755],[35,60321,60323],{"className":60322},[7121],[35,60324,60326],{"className":60325,"style":16272},[7125],[35,60327],{},[35,60329],{"className":60330},[7314,14311],[35,60332,60334],{"className":60333,"style":7474},[7314,7473],[35,60335,1694],{"className":60336},[7478,7140],[35,60338,60340],{"className":60339},[7113],[35,60341,60343],{"className":60342},[7117],[35,60344,60346],{"className":60345},[7121],[35,60347,60349],{"className":60348,"style":54622},[7125],[35,60350,60351,60354],{"style":54625},[35,60352],{"className":60353,"style":7134},[7133],[35,60355,60357],{"className":60356},[7138,7139,7140,7141],[35,60358,460],{"className":60359},[7104,7141],[35,60361],{"className":60362,"style":7249},[7248],[35,60364,445],{"className":60365},[7253],[35,60367],{"className":60368,"style":7249},[7248],[35,60370,275],{"className":60371},[7472],[35,60373,1675],{"className":60374,"style":31824},[7104,7145],[35,60376],{"className":60377,"style":7249},[7248],[35,60379,7225],{"className":60380},[7253],[35,60382],{"className":60383,"style":7249},[7248],[35,60385,60387,60390],{"className":60386},[7104],[35,60388,1675],{"className":60389,"style":31824},[7104,7145],[35,60391,60393],{"className":60392},[7113],[35,60394,60396,60416],{"className":60395},[7117,7731],[35,60397,60399,60413],{"className":60398},[7121],[35,60400,60402],{"className":60401,"style":18794},[7125],[35,60403,60404,60407],{"style":46746},[35,60405],{"className":60406,"style":7134},[7133],[35,60408,60410],{"className":60409},[7138,7139,7140,7141],[35,60411,5312],{"className":60412},[7104,7141],[35,60414,7756],{"className":60415},[7755],[35,60417,60419],{"className":60418},[7121],[35,60420,60422],{"className":60421,"style":7763},[7125],[35,60423],{},[35,60425,60427,60430],{"className":60426},[7314],[35,60428,1694],{"className":60429},[7314],[35,60431,60433],{"className":60432},[7113],[35,60434,60436],{"className":60435},[7117],[35,60437,60439],{"className":60438},[7121],[35,60440,60442],{"className":60441,"style":33894},[7125],[35,60443,60444,60447],{"style":33897},[35,60445],{"className":60446,"style":7134},[7133],[35,60448,60450],{"className":60449},[7138,7139,7140,7141],[35,60451,460],{"className":60452},[7104,7141],[35,60454,60455,60458],{"style":54730},[35,60456],{"className":60457,"style":54523},[7133],[35,60459,60461],{"className":60460,"style":54737},[14370],[10808,60462,60463],{"xmlns":10810,"width":14374,"height":54740,"viewBox":54741,"preserveAspectRatio":14377},[10817,60464],{"d":54744},[35,60466,7756],{"className":60467},[7755],[35,60469,60471],{"className":60470},[7121],[35,60472,60474],{"className":60473,"style":54754},[7125],[35,60475],{},[35,60477,5122],{"className":60478},[7104],[5019,60480,60482],{"id":60481},"primary-modeling-assumptions","Primary Modeling Assumptions",[12,60484,60485],{},"This notebook is an idealized mathematical physics visualization designed for conceptual clarity and calibration intuition.",[46331,60487,60488,60498],{},[46334,60489,60490],{},[46337,60491,60492,60495],{},[46340,60493,60494],{},"Assumption",[46340,60496,60497],{},"Implication",[46347,60499,60500,60584,60592,60599,60607,60644,60734,60891,60939,60947,60955,60963],{},[46337,60501,60502,60505],{},[46352,60503,60504],{},"Two-level truncation",[46352,60506,60507,60508,15316,60546,5122],{},"The simulated model includes only ",[35,60509,60511,60528],{"className":60510},[7057],[35,60512,60514],{"className":60513},[7061],[7063,60515,60516],{"xmlns":7065},[7067,60517,60518,60526],{},[7070,60519,60520,60522,60524],{},[7223,60521,7283],{"stretchy":7289},[7076,60523,42397],{},[7223,60525,7290],{"stretchy":7289},[7083,60527,46374],{"encoding":7085},[35,60529,60531],{"className":60530,"ariaHidden":7091},[7090],[35,60532,60534,60537,60540,60543],{"className":60533},[7095],[35,60535],{"className":60536,"style":7303},[7099],[35,60538,7283],{"className":60539},[7472],[35,60541,42397],{"className":60542,"style":7310},[7104,7145],[35,60544,7290],{"className":60545},[7314],[35,60547,60549,60566],{"className":60548},[7057],[35,60550,60552],{"className":60551},[7061],[7063,60553,60554],{"xmlns":7065},[7067,60555,60556,60564],{},[7070,60557,60558,60560,60562],{},[7223,60559,7283],{"stretchy":7289},[7076,60561,7817],{},[7223,60563,7290],{"stretchy":7289},[7083,60565,46420],{"encoding":7085},[35,60567,60569],{"className":60568,"ariaHidden":7091},[7090],[35,60570,60572,60575,60578,60581],{"className":60571},[7095],[35,60573],{"className":60574,"style":7303},[7099],[35,60576,7283],{"className":60577},[7472],[35,60579,7817],{"className":60580},[7104,7145],[35,60582,7290],{"className":60583},[7314],[46337,60585,60586,60589],{},[46352,60587,60588],{},"Classical drive",[46352,60590,60591],{},"The microwave field is prescribed externally instead of quantized as a field mode.",[46337,60593,60594,60596],{},[46352,60595,47794],{},[46352,60597,60598],{},"Fast counter-rotating terms are omitted.",[46337,60600,60601,60604],{},[46352,60602,60603],{},"Square pulse",[46352,60605,60606],{},"The drive is assumed to turn on and off instantaneously.",[46337,60608,60609,60612],{},[46352,60610,60611],{},"Uniform drive amplitude",[46352,60613,60614,60615,60643],{},"The parameter ",[35,60616,60618,60631],{"className":60617},[7057],[35,60619,60621],{"className":60620},[7061],[7063,60622,60623],{"xmlns":7065},[7067,60624,60625,60629],{},[7070,60626,60627],{},[7076,60628,47141],{"mathvariant":7282},[7083,60630,47144],{"encoding":7085},[35,60632,60634],{"className":60633,"ariaHidden":7091},[7090],[35,60635,60637,60640],{"className":60636},[7095],[35,60638],{"className":60639,"style":7100},[7099],[35,60641,47141],{"className":60642},[7104]," remains constant across the frequency sweep.",[46337,60645,60646,60649],{},[46352,60647,60648],{},"Optional damping envelope",[46352,60650,60614,60651,60733],{},[35,60652,60654,60673],{"className":60653},[7057],[35,60655,60657],{"className":60656},[7061],[7063,60658,60659],{"xmlns":7065},[7067,60660,60661,60671],{},[7070,60662,60663],{},[8304,60664,60665,60667,60669],{},[7076,60666,41530],{},[7227,60668,460],{},[7223,60670,25282],{},[7083,60672,47720],{"encoding":7085},[35,60674,60676],{"className":60675,"ariaHidden":7091},[7090],[35,60677,60679,60682],{"className":60678},[7095],[35,60680],{"className":60681,"style":47730},[7099],[35,60683,60685,60688],{"className":60684},[7104],[35,60686,41530],{"className":60687,"style":7751},[7104,7145],[35,60689,60691],{"className":60690},[7113],[35,60692,60694,60725],{"className":60693},[7117,7731],[35,60695,60697,60722],{"className":60696},[7121],[35,60698,60700,60711],{"className":60699,"style":41067},[7125],[35,60701,60702,60705],{"style":47751},[35,60703],{"className":60704,"style":7134},[7133],[35,60706,60708],{"className":60707},[7138,7139,7140,7141],[35,60709,460],{"className":60710},[7104,7141],[35,60712,60713,60716],{"style":7129},[35,60714],{"className":60715,"style":7134},[7133],[35,60717,60719],{"className":60718},[7138,7139,7140,7141],[35,60720,25282],{"className":60721},[7253,7141],[35,60723,7756],{"className":60724},[7755],[35,60726,60728],{"className":60727},[7121],[35,60729,60731],{"className":60730,"style":47781},[7125],[35,60732],{}," enters as a simple exponential contrast envelope.",[46337,60735,60736,60810],{},[46352,60737,60738,60739,60809],{},"No explicit ",[35,60740,60742,60760],{"className":60741},[7057],[35,60743,60745],{"className":60744},[7061],[7063,60746,60747],{"xmlns":7065},[7067,60748,60749,60757],{},[7070,60750,60751],{},[7699,60752,60753,60755],{},[7076,60754,41530],{},[7227,60756,500],{},[7083,60758,60759],{"encoding":7085},"T_1",[35,60761,60763],{"className":60762,"ariaHidden":7091},[7090],[35,60764,60766,60769],{"className":60765},[7095],[35,60767],{"className":60768,"style":7718},[7099],[35,60770,60772,60775],{"className":60771},[7104],[35,60773,41530],{"className":60774,"style":7751},[7104,7145],[35,60776,60778],{"className":60777},[7113],[35,60779,60781,60801],{"className":60780},[7117,7731],[35,60782,60784,60798],{"className":60783},[7121],[35,60785,60787],{"className":60786,"style":18794},[7125],[35,60788,60789,60792],{"style":12163},[35,60790],{"className":60791,"style":7134},[7133],[35,60793,60795],{"className":60794},[7138,7139,7140,7141],[35,60796,500],{"className":60797},[7104,7141],[35,60799,7756],{"className":60800},[7755],[35,60802,60804],{"className":60803},[7121],[35,60805,60807],{"className":60806,"style":7763},[7125],[35,60808],{}," relaxation",[46352,60811,60812,60813,60851,60852,60890],{},"Energy decay from ",[35,60814,60816,60833],{"className":60815},[7057],[35,60817,60819],{"className":60818},[7061],[7063,60820,60821],{"xmlns":7065},[7067,60822,60823,60831],{},[7070,60824,60825,60827,60829],{},[7223,60826,7283],{"stretchy":7289},[7076,60828,7817],{},[7223,60830,7290],{"stretchy":7289},[7083,60832,46420],{"encoding":7085},[35,60834,60836],{"className":60835,"ariaHidden":7091},[7090],[35,60837,60839,60842,60845,60848],{"className":60838},[7095],[35,60840],{"className":60841,"style":7303},[7099],[35,60843,7283],{"className":60844},[7472],[35,60846,7817],{"className":60847},[7104,7145],[35,60849,7290],{"className":60850},[7314]," to ",[35,60853,60855,60872],{"className":60854},[7057],[35,60856,60858],{"className":60857},[7061],[7063,60859,60860],{"xmlns":7065},[7067,60861,60862,60870],{},[7070,60863,60864,60866,60868],{},[7223,60865,7283],{"stretchy":7289},[7076,60867,42397],{},[7223,60869,7290],{"stretchy":7289},[7083,60871,46374],{"encoding":7085},[35,60873,60875],{"className":60874,"ariaHidden":7091},[7090],[35,60876,60878,60881,60884,60887],{"className":60877},[7095],[35,60879],{"className":60880,"style":7303},[7099],[35,60882,7283],{"className":60883},[7472],[35,60885,42397],{"className":60886,"style":7310},[7104,7145],[35,60888,7290],{"className":60889},[7314]," is omitted.",[46337,60892,60893,60896],{},[46352,60894,60895],{},"No higher-state leakage",[46352,60897,60898,60899,60938],{},"Higher levels, including a transmon ",[35,60900,60902,60920],{"className":60901},[7057],[35,60903,60905],{"className":60904},[7061],[7063,60906,60907],{"xmlns":7065},[7067,60908,60909,60917],{},[7070,60910,60911,60913,60915],{},[7223,60912,7283],{"stretchy":7289},[7076,60914,1675],{},[7223,60916,7290],{"stretchy":7289},[7083,60918,60919],{"encoding":7085},"\\lvert f\\rangle",[35,60921,60923],{"className":60922,"ariaHidden":7091},[7090],[35,60924,60926,60929,60932,60935],{"className":60925},[7095],[35,60927],{"className":60928,"style":7303},[7099],[35,60930,7283],{"className":60931},[7472],[35,60933,1675],{"className":60934,"style":31824},[7104,7145],[35,60936,7290],{"className":60937},[7314]," state, are excluded.",[46337,60940,60941,60944],{},[46352,60942,60943],{},"No AC Stark shift",[46352,60945,60946],{},"Drive-induced transition-frequency shifts are excluded.",[46337,60948,60949,60952],{},[46352,60950,60951],{},"No Bloch-Siegert shift",[46352,60953,60954],{},"Counter-rotating corrections are excluded.",[46337,60956,60957,60960],{},[46352,60958,60959],{},"No experimental readout model",[46352,60961,60962],{},"Assignment errors and finite signal-to-noise effects are omitted.",[46337,60964,60965,60968],{},[46352,60966,60967],{},"No pulse-envelope shaping",[46352,60969,60970],{},"Gaussian, DRAG, cosine, and hardware-specific envelopes are omitted.",[12,60972,60973],{},"A hardware-accurate superconducting-qubit simulation requires additional structure, including anharmonic multilevel dynamics, calibrated microwave pulse envelopes, drive phase control, amplitude-dependent frequency shifts, finite thermal populations, leakage channels, and measurement infidelity modeled with density matrix dynamics.",[5019,60975,60977],{"id":60976},"notebook-control-knobs","Notebook Control Knobs",[12,60979,60980],{},"The first code section centralizes the main physical and numerical parameters for direct adjustment.",[46331,60982,60983,60996],{},[46334,60984,60985],{},[46337,60986,60987,60990,60994],{},[46340,60988,60989],{},"Parameter",[46340,60991,60993],{"align":60992},"right","Default",[46340,60995,46345],{},[46347,60997,60998,61082,61135,61150,61165,61180,61194,61209,61223],{},[46337,60999,61000,61005,61009],{},[46352,61001,61002],{},[32,61003,61004],{},"F0_GHZ",[46352,61006,61007],{"align":60992},[32,61008,48281],{},[46352,61010,61011,61012,61081],{},"Qubit transition frequency ",[35,61013,61015,61032],{"className":61014},[7057],[35,61016,61018],{"className":61017},[7061],[7063,61019,61020],{"xmlns":7065},[7067,61021,61022,61030],{},[7070,61023,61024],{},[7699,61025,61026,61028],{},[7076,61027,1675],{},[7227,61029,5312],{},[7083,61031,46716],{"encoding":7085},[35,61033,61035],{"className":61034,"ariaHidden":7091},[7090],[35,61036,61038,61041],{"className":61037},[7095],[35,61039],{"className":61040,"style":11562},[7099],[35,61042,61044,61047],{"className":61043},[7104],[35,61045,1675],{"className":61046,"style":31824},[7104,7145],[35,61048,61050],{"className":61049},[7113],[35,61051,61053,61073],{"className":61052},[7117,7731],[35,61054,61056,61070],{"className":61055},[7121],[35,61057,61059],{"className":61058,"style":18794},[7125],[35,61060,61061,61064],{"style":46746},[35,61062],{"className":61063,"style":7134},[7133],[35,61065,61067],{"className":61066},[7138,7139,7140,7141],[35,61068,5312],{"className":61069},[7104,7141],[35,61071,7756],{"className":61072},[7755],[35,61074,61076],{"className":61075},[7121],[35,61077,61079],{"className":61078,"style":7763},[7125],[35,61080],{}," in GHz",[46337,61083,61084,61089,61094],{},[46352,61085,61086],{},[32,61087,61088],{},"OMEGA_ONRESONANCE_MHZ",[46352,61090,61091],{"align":60992},[32,61092,61093],{},"20.0",[46352,61095,61096,61097,61134],{},"On-resonance Rabi rate ",[35,61098,61100,61119],{"className":61099},[7057],[35,61101,61103],{"className":61102},[7061],[7063,61104,61105],{"xmlns":7065},[7067,61106,61107,61117],{},[7070,61108,61109,61111,61113,61115],{},[7076,61110,47141],{"mathvariant":7282},[7076,61112,398],{"mathvariant":7282},[7227,61114,460],{},[7076,61116,13081],{},[7083,61118,47186],{"encoding":7085},[35,61120,61122],{"className":61121,"ariaHidden":7091},[7090],[35,61123,61125,61128,61131],{"className":61124},[7095],[35,61126],{"className":61127,"style":7303},[7099],[35,61129,47199],{"className":61130},[7104],[35,61132,13081],{"className":61133,"style":7310},[7104,7145]," in MHz",[46337,61136,61137,61142,61147],{},[46352,61138,61139],{},[32,61140,61141],{},"FREQ_MIN_GHZ",[46352,61143,61144],{"align":60992},[32,61145,61146],{},"4.90",[46352,61148,61149],{},"Minimum swept drive frequency in GHz",[46337,61151,61152,61157,61162],{},[46352,61153,61154],{},[32,61155,61156],{},"FREQ_MAX_GHZ",[46352,61158,61159],{"align":60992},[32,61160,61161],{},"5.10",[46352,61163,61164],{},"Maximum swept drive frequency in GHz",[46337,61166,61167,61172,61177],{},[46352,61168,61169],{},[32,61170,61171],{},"N_FREQ",[46352,61173,61174],{"align":60992},[32,61175,61176],{},"401",[46352,61178,61179],{},"Number of drive-frequency samples",[46337,61181,61182,61187,61191],{},[46352,61183,61184],{},[32,61185,61186],{},"DUR_MIN_NS",[46352,61188,61189],{"align":60992},[32,61190,376],{},[46352,61192,61193],{},"Minimum pulse duration in ns",[46337,61195,61196,61201,61206],{},[46352,61197,61198],{},[32,61199,61200],{},"DUR_MAX_NS",[46352,61202,61203],{"align":60992},[32,61204,61205],{},"200.0",[46352,61207,61208],{},"Maximum pulse duration in ns",[46337,61210,61211,61216,61220],{},[46352,61212,61213],{},[32,61214,61215],{},"N_DUR",[46352,61217,61218],{"align":60992},[32,61219,61176],{},[46352,61221,61222],{},"Number of pulse-duration samples",[46337,61224,61225,61230,61235],{},[46352,61226,61227],{},[32,61228,61229],{},"T2STAR_NS",[46352,61231,61232],{"align":60992},[32,61233,61234],{},"500.0",[46352,61236,61237,61238,61241],{},"Optional damping time in ns, with ",[32,61239,61240],{},"None"," disabling damping",[12,61243,61244],{},"Additional cross-section plotting cells expose localized slice controls.",[46331,61246,61247,61257],{},[46334,61248,61249],{},[46337,61250,61251,61253,61255],{},[46340,61252,60989],{},[46340,61254,60993],{"align":60992},[46340,61256,46345],{},[46347,61258,61259,61358],{},[46337,61260,61261,61266,61270],{},[46352,61262,61263],{},[32,61264,61265],{},"F_CROSS_GHZ",[46352,61267,61268],{"align":60992},[32,61269,48281],{},[46352,61271,61272,61273],{},"Drive-frequency slice used to plot ",[35,61274,61276,61300],{"className":61275},[7057],[35,61277,61279],{"className":61278},[7061],[7063,61280,61281],{"xmlns":7065},[7067,61282,61283,61297],{},[7070,61284,61285,61291,61293,61295],{},[7699,61286,61287,61289],{},[7076,61288,7705],{},[7076,61290,7817],{},[7223,61292,275],{"stretchy":7289},[7076,61294,46183],{},[7223,61296,1694],{"stretchy":7289},[7083,61298,61299],{"encoding":7085},"P_e(\\tau)",[35,61301,61303],{"className":61302,"ariaHidden":7091},[7090],[35,61304,61306,61309,61349,61352,61355],{"className":61305},[7095],[35,61307],{"className":61308,"style":7303},[7099],[35,61310,61312,61315],{"className":61311},[7104],[35,61313,7705],{"className":61314,"style":7751},[7104,7145],[35,61316,61318],{"className":61317},[7113],[35,61319,61321,61341],{"className":61320},[7117,7731],[35,61322,61324,61338],{"className":61323},[7121],[35,61325,61327],{"className":61326,"style":9035},[7125],[35,61328,61329,61332],{"style":12163},[35,61330],{"className":61331,"style":7134},[7133],[35,61333,61335],{"className":61334},[7138,7139,7140,7141],[35,61336,7817],{"className":61337},[7104,7145,7141],[35,61339,7756],{"className":61340},[7755],[35,61342,61344],{"className":61343},[7121],[35,61345,61347],{"className":61346,"style":7763},[7125],[35,61348],{},[35,61350,275],{"className":61351},[7472],[35,61353,46183],{"className":61354,"style":46255},[7104,7145],[35,61356,1694],{"className":61357},[7314],[46337,61359,61360,61365,61370],{},[46352,61361,61362],{},[32,61363,61364],{},"T_CROSS_NS",[46352,61366,61367],{"align":60992},[32,61368,61369],{},"100.0",[46352,61371,61372,61373],{},"Pulse-duration slice used to plot ",[35,61374,61376,61400],{"className":61375},[7057],[35,61377,61379],{"className":61378},[7061],[7063,61380,61381],{"xmlns":7065},[7067,61382,61383,61397],{},[7070,61384,61385,61391,61393,61395],{},[7699,61386,61387,61389],{},[7076,61388,7705],{},[7076,61390,7817],{},[7223,61392,275],{"stretchy":7289},[7076,61394,1675],{},[7223,61396,1694],{"stretchy":7289},[7083,61398,61399],{"encoding":7085},"P_e(f)",[35,61401,61403],{"className":61402,"ariaHidden":7091},[7090],[35,61404,61406,61409,61449,61452,61455],{"className":61405},[7095],[35,61407],{"className":61408,"style":7303},[7099],[35,61410,61412,61415],{"className":61411},[7104],[35,61413,7705],{"className":61414,"style":7751},[7104,7145],[35,61416,61418],{"className":61417},[7113],[35,61419,61421,61441],{"className":61420},[7117,7731],[35,61422,61424,61438],{"className":61423},[7121],[35,61425,61427],{"className":61426,"style":9035},[7125],[35,61428,61429,61432],{"style":12163},[35,61430],{"className":61431,"style":7134},[7133],[35,61433,61435],{"className":61434},[7138,7139,7140,7141],[35,61436,7817],{"className":61437},[7104,7145,7141],[35,61439,7756],{"className":61440},[7755],[35,61442,61444],{"className":61443},[7121],[35,61445,61447],{"className":61446,"style":7763},[7125],[35,61448],{},[35,61450,275],{"className":61451},[7472],[35,61453,1675],{"className":61454,"style":31824},[7104,7145],[35,61456,1694],{"className":61457},[7314],[12,61459,61460],{},"Separate FFT processing controls support spectral analysis.",[46331,61462,61463,61473],{},[46334,61464,61465],{},[46337,61466,61467,61469,61471],{},[46340,61468,60989],{},[46340,61470,60993],{"align":60992},[46340,61472,46345],{},[46347,61474,61475,61489,61504,61518,61533,61548],{},[46337,61476,61477,61482,61486],{},[46352,61478,61479],{},[32,61480,61481],{},"FFT_PAD",[46352,61483,61484],{"align":60992},[32,61485,40225],{},[46352,61487,61488],{},"Zero-padding factor used in the Fourier transform",[46337,61490,61491,61496,61501],{},[46352,61492,61493],{},[32,61494,61495],{},"WINDOW",[46352,61497,61498],{"align":60992},[32,61499,61500],{},"\"hann\"",[46352,61502,61503],{},"Window function applied along the pulse-duration axis",[46337,61505,61506,61511,61515],{},[46352,61507,61508],{},[32,61509,61510],{},"DETREND_MEAN",[46352,61512,61513],{"align":60992},[32,61514,828],{},[46352,61516,61517],{},"Flag that subtracts the mean before the FFT",[46337,61519,61520,61525,61530],{},[46352,61521,61522],{},[32,61523,61524],{},"AMP_MODE",[46352,61526,61527],{"align":60992},[32,61528,61529],{},"\"magnitude\"",[46352,61531,61532],{},"Display mode for magnitude, power, or dB-scaled amplitude",[46337,61534,61535,61540,61545],{},[46352,61536,61537],{},[32,61538,61539],{},"NORM_MODE",[46352,61541,61542],{"align":60992},[32,61543,61544],{},"\"global\"",[46352,61546,61547],{},"Normalization mode applied globally or per drive-frequency column",[46337,61549,61550,61555,61560],{},[46352,61551,61552],{},[32,61553,61554],{},"FREQ_UNITS",[46352,61556,61557],{"align":60992},[32,61558,61559],{},"\"MHz\"",[46352,61561,61562],{},"Frequency units used on the Fourier-frequency axis",[12,61564,61565],{},"Three-dimensional visualization cells add rendering controls for upsampling, colormaps, face-count limits, and viewing angles. These settings change the appearance of the rendered figures while preserving the analytic probability model.",[25,61567,61569],{"className":27,"code":61568,"language":29,"meta":30,"style":30},"# @title Install dependencies\nimport sys\n%pip -q install pyvista pythreejs trame\n",[32,61570,61571,61576,61583],{"__ignoreMap":30},[35,61572,61573],{"class":37,"line":38},[35,61574,61575],{"class":41},"# @title Install dependencies\n",[35,61577,61578,61580],{"class":37,"line":45},[35,61579,110],{"class":109},[35,61581,61582],{"class":113}," sys\n",[35,61584,61585,61587,61590,61592],{"class":37,"line":51},[35,61586,13287],{"class":241},[35,61588,61589],{"class":113},"pip ",[35,61591,564],{"class":241},[35,61593,61594],{"class":113},"q install pyvista pythreejs trame\n",[25,61596,61598],{"className":27,"code":61597,"language":29,"meta":30,"style":30},"# @title Controls & imports\nimport math\nfrom typing import Optional, Tuple\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n\ntry:\n    import pyvista as pv  # optional; only for 3D interactive\n    PV_AVAILABLE = True\nexcept Exception:\n    PV_AVAILABLE = False\n\nplt.rcParams['figure.dpi'] = 250\nplt.rcParams['axes.grid'] = False\n\n# ---------- Control knobs ----------\nF0_GHZ: float = 5.000        # Qubit transition freq f0 [GHz]\nOMEGA_ONRESONANCE_MHZ: float = 20.0  # Ω\u002F2π on-resonance [MHz]\nFREQ_MIN_GHZ: float = 4.90   # Sweep min f [GHz]\nFREQ_MAX_GHZ: float = 5.10   # Sweep max f [GHz]\nN_FREQ: int = 401            # # of frequency points\nDUR_MIN_NS: float = 0.0      # Sweep min duration τ [ns]\nDUR_MAX_NS: float = 200.0    # Sweep max duration τ [ns]\nN_DUR: int = 401             # # of duration points\nT2STAR_NS: Optional[float] = 500.0  # None to disable exp(-τ\u002FT2*)\n\nprint('Controls loaded.')\n",[32,61599,61600,61605,61611,61622,61632,61642,61652,61656,61663,61679,61689,61697,61706,61710,61723,61736,61740,61745,61761,61777,61793,61809,61825,61840,61856,61871,61890,61894],{"__ignoreMap":30},[35,61601,61602],{"class":37,"line":38},[35,61603,61604],{"class":41},"# @title Controls & imports\n",[35,61606,61607,61609],{"class":37,"line":45},[35,61608,110],{"class":109},[35,61610,122],{"class":113},[35,61612,61613,61615,61617,61619],{"class":37,"line":51},[35,61614,128],{"class":109},[35,61616,144],{"class":113},[35,61618,110],{"class":109},[35,61620,61621],{"class":113}," Optional, Tuple\n",[35,61623,61624,61626,61628,61630],{"class":37,"line":57},[35,61625,110],{"class":109},[35,61627,162],{"class":113},[35,61629,165],{"class":109},[35,61631,168],{"class":113},[35,61633,61634,61636,61638,61640],{"class":37,"line":63},[35,61635,110],{"class":109},[35,61637,189],{"class":113},[35,61639,165],{"class":109},[35,61641,194],{"class":113},[35,61643,61644,61646,61648,61650],{"class":37,"line":69},[35,61645,110],{"class":109},[35,61647,176],{"class":113},[35,61649,165],{"class":109},[35,61651,181],{"class":113},[35,61653,61654],{"class":37,"line":75},[35,61655,103],{"emptyLinePlaceholder":102},[35,61657,61658,61661],{"class":37,"line":81},[35,61659,61660],{"class":109},"try",[35,61662,364],{"class":113},[35,61664,61665,61668,61671,61673,61676],{"class":37,"line":87},[35,61666,61667],{"class":109},"    import",[35,61669,61670],{"class":113}," pyvista ",[35,61672,165],{"class":109},[35,61674,61675],{"class":113}," pv  ",[35,61677,61678],{"class":41},"# optional; only for 3D interactive\n",[35,61680,61681,61684,61686],{"class":37,"line":93},[35,61682,61683],{"class":245},"    PV_AVAILABLE",[35,61685,308],{"class":241},[35,61687,61688],{"class":245}," True\n",[35,61690,61691,61694],{"class":37,"line":99},[35,61692,61693],{"class":109},"except",[35,61695,61696],{"class":113}," Exception:\n",[35,61698,61699,61701,61703],{"class":37,"line":106},[35,61700,61683],{"class":245},[35,61702,308],{"class":241},[35,61704,61705],{"class":245}," False\n",[35,61707,61708],{"class":37,"line":117},[35,61709,103],{"emptyLinePlaceholder":102},[35,61711,61712,61714,61716,61718,61720],{"class":37,"line":125},[35,61713,231],{"class":113},[35,61715,235],{"class":234},[35,61717,238],{"class":113},[35,61719,242],{"class":241},[35,61721,61722],{"class":245}," 250\n",[35,61724,61725,61727,61730,61732,61734],{"class":37,"line":139},[35,61726,231],{"class":113},[35,61728,61729],{"class":234},"'axes.grid'",[35,61731,238],{"class":113},[35,61733,242],{"class":241},[35,61735,61705],{"class":245},[35,61737,61738],{"class":37,"line":152},[35,61739,103],{"emptyLinePlaceholder":102},[35,61741,61742],{"class":37,"line":157},[35,61743,61744],{"class":41},"# ---------- Control knobs ----------\n",[35,61746,61747,61749,61751,61753,61755,61758],{"class":37,"line":171},[35,61748,61004],{"class":245},[35,61750,282],{"class":113},[35,61752,305],{"class":241},[35,61754,308],{"class":241},[35,61756,61757],{"class":245}," 5.000",[35,61759,61760],{"class":41},"        # Qubit transition freq f0 [GHz]\n",[35,61762,61763,61765,61767,61769,61771,61774],{"class":37,"line":184},[35,61764,61088],{"class":245},[35,61766,282],{"class":113},[35,61768,305],{"class":241},[35,61770,308],{"class":241},[35,61772,61773],{"class":245}," 20.0",[35,61775,61776],{"class":41},"  # Ω\u002F2π on-resonance [MHz]\n",[35,61778,61779,61781,61783,61785,61787,61790],{"class":37,"line":197},[35,61780,61141],{"class":245},[35,61782,282],{"class":113},[35,61784,305],{"class":241},[35,61786,308],{"class":241},[35,61788,61789],{"class":245}," 4.90",[35,61791,61792],{"class":41},"   # Sweep min f [GHz]\n",[35,61794,61795,61797,61799,61801,61803,61806],{"class":37,"line":210},[35,61796,61156],{"class":245},[35,61798,282],{"class":113},[35,61800,305],{"class":241},[35,61802,308],{"class":241},[35,61804,61805],{"class":245}," 5.10",[35,61807,61808],{"class":41},"   # Sweep max f [GHz]\n",[35,61810,61811,61813,61815,61817,61819,61822],{"class":37,"line":223},[35,61812,61171],{"class":245},[35,61814,282],{"class":113},[35,61816,285],{"class":241},[35,61818,308],{"class":241},[35,61820,61821],{"class":245}," 401",[35,61823,61824],{"class":41},"            # # of frequency points\n",[35,61826,61827,61829,61831,61833,61835,61837],{"class":37,"line":228},[35,61828,61186],{"class":245},[35,61830,282],{"class":113},[35,61832,305],{"class":241},[35,61834,308],{"class":241},[35,61836,724],{"class":245},[35,61838,61839],{"class":41},"      # Sweep min duration τ [ns]\n",[35,61841,61842,61844,61846,61848,61850,61853],{"class":37,"line":249},[35,61843,61200],{"class":245},[35,61845,282],{"class":113},[35,61847,305],{"class":241},[35,61849,308],{"class":241},[35,61851,61852],{"class":245}," 200.0",[35,61854,61855],{"class":41},"    # Sweep max duration τ [ns]\n",[35,61857,61858,61860,61862,61864,61866,61868],{"class":37,"line":254},[35,61859,61215],{"class":245},[35,61861,282],{"class":113},[35,61863,285],{"class":241},[35,61865,308],{"class":241},[35,61867,61821],{"class":245},[35,61869,61870],{"class":41},"             # # of duration points\n",[35,61872,61873,61875,61878,61880,61882,61884,61887],{"class":37,"line":260},[35,61874,61229],{"class":245},[35,61876,61877],{"class":113},": Optional[",[35,61879,305],{"class":241},[35,61881,238],{"class":113},[35,61883,242],{"class":241},[35,61885,61886],{"class":245}," 500.0",[35,61888,61889],{"class":41},"  # None to disable exp(-τ\u002FT2*)\n",[35,61891,61892],{"class":37,"line":265},[35,61893,103],{"emptyLinePlaceholder":102},[35,61895,61896,61898,61900,61903],{"class":37,"line":326},[35,61897,1861],{"class":241},[35,61899,275],{"class":113},[35,61901,61902],{"class":234},"'Controls loaded.'",[35,61904,384],{"class":113},[25,61906,61909],{"className":61907,"code":61908,"language":4991,"meta":30},[4989],"Controls loaded.\n",[32,61910,61908],{"__ignoreMap":30},[25,61912,61914],{"className":27,"code":61913,"language":29,"meta":30,"style":30},"def rabi_probability(\n    f_ghz: np.ndarray,\n    tau_ns: np.ndarray,\n    f0_ghz: float,\n    omega_onres_mhz: float,\n    t2star_ns: Optional[float] = None,\n) -> np.ndarray:\n    \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n\n    Args:\n        f_ghz: 1D array of drive frequencies [GHz].\n        tau_ns: 1D array of pulse durations [ns].\n        f0_ghz: Qubit frequency on resonance [GHz].\n        omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n        t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n\n    Returns:\n        2D array P_e with shape (len(tau_ns), len(f_ghz)).\n    \"\"\"\n    two_pi = 2.0 * math.pi\n    f = f_ghz * 1e9\n    f0 = f0_ghz * 1e9\n    tau = tau_ns * 1e-9\n    omega = two_pi * omega_onres_mhz * 1e6  # rad\u002Fs\n\n    F, TAU = np.meshgrid(f, tau, indexing='xy')\n    Delta = two_pi * (F - f0)               # rad\u002Fs\n    Omega_R = np.sqrt(omega**2 + Delta**2)\n\n    with np.errstate(divide='ignore', invalid='ignore'):\n        amp = (omega \u002F Omega_R) ** 2\n        phase = 0.5 * Omega_R * TAU\n        Pe = amp * np.sin(phase) ** 2\n\n    if t2star_ns is not None and t2star_ns > 0.0:\n        decay = np.exp(-TAU \u002F (t2star_ns * 1e-9))\n        Pe = Pe * decay\n\n    return np.clip(Pe, 0.0, 1.0)\n",[32,61915,61916,61925,61932,61939,61950,61961,61979,61984,61989,61993,61997,62002,62007,62012,62017,62022,62026,62030,62035,62039,62054,62069,62083,62098,62121,62125,62153,62175,62204,62208,62239,62259,62279,62302,62306,62330,62360,62374,62378],{"__ignoreMap":30},[35,61917,61918,61920,61923],{"class":37,"line":38},[35,61919,268],{"class":109},[35,61921,61922],{"class":271}," rabi_probability",[35,61924,903],{"class":113},[35,61926,61927,61930],{"class":37,"line":45},[35,61928,61929],{"class":278},"    f_ghz",[35,61931,6047],{"class":113},[35,61933,61934,61937],{"class":37,"line":51},[35,61935,61936],{"class":278},"    tau_ns",[35,61938,6047],{"class":113},[35,61940,61941,61944,61946,61948],{"class":37,"line":57},[35,61942,61943],{"class":278},"    f0_ghz",[35,61945,282],{"class":113},[35,61947,305],{"class":241},[35,61949,912],{"class":113},[35,61951,61952,61955,61957,61959],{"class":37,"line":63},[35,61953,61954],{"class":278},"    omega_onres_mhz",[35,61956,282],{"class":113},[35,61958,305],{"class":241},[35,61960,912],{"class":113},[35,61962,61963,61966,61968,61970,61972,61974,61977],{"class":37,"line":69},[35,61964,61965],{"class":278},"    t2star_ns",[35,61967,61877],{"class":113},[35,61969,305],{"class":241},[35,61971,238],{"class":113},[35,61973,242],{"class":241},[35,61975,61976],{"class":245}," None",[35,61978,912],{"class":113},[35,61980,61981],{"class":37,"line":75},[35,61982,61983],{"class":113},") -> np.ndarray:\n",[35,61985,61986],{"class":37,"line":81},[35,61987,61988],{"class":234},"    \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n",[35,61990,61991],{"class":37,"line":87},[35,61992,103],{"emptyLinePlaceholder":102},[35,61994,61995],{"class":37,"line":93},[35,61996,6131],{"class":234},[35,61998,61999],{"class":37,"line":99},[35,62000,62001],{"class":234},"        f_ghz: 1D array of drive frequencies [GHz].\n",[35,62003,62004],{"class":37,"line":106},[35,62005,62006],{"class":234},"        tau_ns: 1D array of pulse durations [ns].\n",[35,62008,62009],{"class":37,"line":117},[35,62010,62011],{"class":234},"        f0_ghz: Qubit frequency on resonance [GHz].\n",[35,62013,62014],{"class":37,"line":125},[35,62015,62016],{"class":234},"        omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n",[35,62018,62019],{"class":37,"line":139},[35,62020,62021],{"class":234},"        t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n",[35,62023,62024],{"class":37,"line":152},[35,62025,103],{"emptyLinePlaceholder":102},[35,62027,62028],{"class":37,"line":157},[35,62029,6190],{"class":234},[35,62031,62032],{"class":37,"line":171},[35,62033,62034],{"class":234},"        2D array P_e with shape (len(tau_ns), len(f_ghz)).\n",[35,62036,62037],{"class":37,"line":184},[35,62038,329],{"class":234},[35,62040,62041,62044,62046,62049,62051],{"class":37,"line":197},[35,62042,62043],{"class":113},"    two_pi ",[35,62045,242],{"class":241},[35,62047,62048],{"class":245}," 2.0",[35,62050,463],{"class":241},[35,62052,62053],{"class":113}," math.pi\n",[35,62055,62056,62059,62061,62064,62066],{"class":37,"line":210},[35,62057,62058],{"class":113},"    f ",[35,62060,242],{"class":241},[35,62062,62063],{"class":113}," f_ghz ",[35,62065,484],{"class":241},[35,62067,62068],{"class":245}," 1e9\n",[35,62070,62071,62074,62076,62079,62081],{"class":37,"line":223},[35,62072,62073],{"class":113},"    f0 ",[35,62075,242],{"class":241},[35,62077,62078],{"class":113}," f0_ghz ",[35,62080,484],{"class":241},[35,62082,62068],{"class":245},[35,62084,62085,62088,62090,62093,62095],{"class":37,"line":228},[35,62086,62087],{"class":113},"    tau ",[35,62089,242],{"class":241},[35,62091,62092],{"class":113}," tau_ns ",[35,62094,484],{"class":241},[35,62096,62097],{"class":245}," 1e-9\n",[35,62099,62100,62103,62105,62108,62110,62113,62115,62118],{"class":37,"line":249},[35,62101,62102],{"class":113},"    omega ",[35,62104,242],{"class":241},[35,62106,62107],{"class":113}," two_pi ",[35,62109,484],{"class":241},[35,62111,62112],{"class":113}," omega_onres_mhz ",[35,62114,484],{"class":241},[35,62116,62117],{"class":245}," 1e6",[35,62119,62120],{"class":41},"  # rad\u002Fs\n",[35,62122,62123],{"class":37,"line":254},[35,62124,103],{"emptyLinePlaceholder":102},[35,62126,62127,62130,62133,62135,62137,62140,62143,62146,62148,62151],{"class":37,"line":260},[35,62128,62129],{"class":113},"    F, ",[35,62131,62132],{"class":245},"TAU",[35,62134,308],{"class":241},[35,62136,2412],{"class":113},[35,62138,62139],{"class":271},"meshgrid",[35,62141,62142],{"class":113},"(f, tau, ",[35,62144,62145],{"class":822},"indexing",[35,62147,242],{"class":241},[35,62149,62150],{"class":234},"'xy'",[35,62152,384],{"class":113},[35,62154,62155,62158,62160,62162,62164,62167,62169,62172],{"class":37,"line":265},[35,62156,62157],{"class":113},"    Delta ",[35,62159,242],{"class":241},[35,62161,62107],{"class":113},[35,62163,484],{"class":241},[35,62165,62166],{"class":113}," (F ",[35,62168,564],{"class":241},[35,62170,62171],{"class":113}," f0)               ",[35,62173,62174],{"class":41},"# rad\u002Fs\n",[35,62176,62177,62180,62182,62184,62186,62189,62191,62193,62195,62198,62200,62202],{"class":37,"line":326},[35,62178,62179],{"class":113},"    Omega_R ",[35,62181,242],{"class":241},[35,62183,2412],{"class":113},[35,62185,490],{"class":271},[35,62187,62188],{"class":113},"(omega",[35,62190,421],{"class":241},[35,62192,460],{"class":245},[35,62194,415],{"class":241},[35,62196,62197],{"class":113}," Delta",[35,62199,421],{"class":241},[35,62201,460],{"class":245},[35,62203,384],{"class":113},[35,62205,62206],{"class":37,"line":332},[35,62207,103],{"emptyLinePlaceholder":102},[35,62209,62210,62213,62215,62218,62220,62223,62225,62228,62230,62233,62235,62237],{"class":37,"line":338},[35,62211,62212],{"class":109},"    with",[35,62214,2412],{"class":113},[35,62216,62217],{"class":271},"errstate",[35,62219,275],{"class":113},[35,62221,62222],{"class":822},"divide",[35,62224,242],{"class":241},[35,62226,62227],{"class":234},"'ignore'",[35,62229,288],{"class":113},[35,62231,62232],{"class":822},"invalid",[35,62234,242],{"class":241},[35,62236,62227],{"class":234},[35,62238,6479],{"class":113},[35,62240,62241,62244,62246,62249,62251,62254,62256],{"class":37,"line":344},[35,62242,62243],{"class":113},"        amp ",[35,62245,242],{"class":241},[35,62247,62248],{"class":113}," (omega ",[35,62250,398],{"class":241},[35,62252,62253],{"class":113}," Omega_R) ",[35,62255,421],{"class":241},[35,62257,62258],{"class":245}," 2\n",[35,62260,62261,62264,62266,62269,62271,62274,62276],{"class":37,"line":349},[35,62262,62263],{"class":113},"        phase ",[35,62265,242],{"class":241},[35,62267,62268],{"class":245}," 0.5",[35,62270,463],{"class":241},[35,62272,62273],{"class":113}," Omega_R ",[35,62275,484],{"class":241},[35,62277,62278],{"class":245}," TAU\n",[35,62280,62281,62284,62286,62289,62291,62293,62295,62298,62300],{"class":37,"line":367},[35,62282,62283],{"class":113},"        Pe ",[35,62285,242],{"class":241},[35,62287,62288],{"class":113}," amp ",[35,62290,484],{"class":241},[35,62292,2412],{"class":113},[35,62294,51200],{"class":271},[35,62296,62297],{"class":113},"(phase) ",[35,62299,421],{"class":241},[35,62301,62258],{"class":245},[35,62303,62304],{"class":37,"line":387},[35,62305,103],{"emptyLinePlaceholder":102},[35,62307,62308,62310,62313,62315,62317,62319,62322,62324,62326,62328],{"class":37,"line":404},[35,62309,352],{"class":109},[35,62311,62312],{"class":113}," t2star_ns ",[35,62314,41492],{"class":109},[35,62316,26490],{"class":109},[35,62318,61976],{"class":245},[35,62320,62321],{"class":109}," and",[35,62323,62312],{"class":113},[35,62325,1497],{"class":241},[35,62327,724],{"class":245},[35,62329,364],{"class":113},[35,62331,62332,62335,62337,62339,62341,62343,62345,62347,62350,62353,62355,62358],{"class":37,"line":434},[35,62333,62334],{"class":113},"        decay ",[35,62336,242],{"class":241},[35,62338,2412],{"class":113},[35,62340,28563],{"class":271},[35,62342,275],{"class":113},[35,62344,564],{"class":241},[35,62346,62132],{"class":245},[35,62348,62349],{"class":241}," \u002F",[35,62351,62352],{"class":113}," (t2star_ns ",[35,62354,484],{"class":241},[35,62356,62357],{"class":245}," 1e-9",[35,62359,2401],{"class":113},[35,62361,62362,62364,62366,62369,62371],{"class":37,"line":474},[35,62363,62283],{"class":113},[35,62365,242],{"class":241},[35,62367,62368],{"class":113}," Pe ",[35,62370,484],{"class":241},[35,62372,62373],{"class":113}," decay\n",[35,62375,62376],{"class":37,"line":546},[35,62377,103],{"emptyLinePlaceholder":102},[35,62379,62380,62382,62384,62387,62390,62392,62394,62396],{"class":37,"line":584},[35,62381,549],{"class":109},[35,62383,2412],{"class":113},[35,62385,62386],{"class":271},"clip",[35,62388,62389],{"class":113},"(Pe, ",[35,62391,376],{"class":245},[35,62393,288],{"class":113},[35,62395,381],{"class":245},[35,62397,384],{"class":113},[25,62399,62401],{"className":27,"code":62400,"language":29,"meta":30,"style":30},"freq_ghz = np.linspace(FREQ_MIN_GHZ, FREQ_MAX_GHZ, N_FREQ)\ndur_ns = np.linspace(DUR_MIN_NS, DUR_MAX_NS, N_DUR)\nPe = rabi_probability(freq_ghz, dur_ns, F0_GHZ, OMEGA_ONRESONANCE_MHZ, T2STAR_NS)\nprint('Grid shape (τ × f):', Pe.shape)\n",[32,62402,62403,62429,62454,62478],{"__ignoreMap":30},[35,62404,62405,62408,62410,62412,62415,62417,62419,62421,62423,62425,62427],{"class":37,"line":38},[35,62406,62407],{"class":113},"freq_ghz ",[35,62409,242],{"class":241},[35,62411,2412],{"class":113},[35,62413,62414],{"class":271},"linspace",[35,62416,275],{"class":113},[35,62418,61141],{"class":245},[35,62420,288],{"class":113},[35,62422,61156],{"class":245},[35,62424,288],{"class":113},[35,62426,61171],{"class":245},[35,62428,384],{"class":113},[35,62430,62431,62434,62436,62438,62440,62442,62444,62446,62448,62450,62452],{"class":37,"line":45},[35,62432,62433],{"class":113},"dur_ns ",[35,62435,242],{"class":241},[35,62437,2412],{"class":113},[35,62439,62414],{"class":271},[35,62441,275],{"class":113},[35,62443,61186],{"class":245},[35,62445,288],{"class":113},[35,62447,61200],{"class":245},[35,62449,288],{"class":113},[35,62451,61215],{"class":245},[35,62453,384],{"class":113},[35,62455,62456,62459,62461,62463,62466,62468,62470,62472,62474,62476],{"class":37,"line":51},[35,62457,62458],{"class":113},"Pe ",[35,62460,242],{"class":241},[35,62462,61922],{"class":271},[35,62464,62465],{"class":113},"(freq_ghz, dur_ns, ",[35,62467,61004],{"class":245},[35,62469,288],{"class":113},[35,62471,61088],{"class":245},[35,62473,288],{"class":113},[35,62475,61229],{"class":245},[35,62477,384],{"class":113},[35,62479,62480,62482,62484,62487],{"class":37,"line":57},[35,62481,1861],{"class":241},[35,62483,275],{"class":113},[35,62485,62486],{"class":234},"'Grid shape (τ × f):'",[35,62488,62489],{"class":113},", Pe.shape)\n",[25,62491,62494],{"className":62492,"code":62493,"language":4991,"meta":30},[4989],"Grid shape (τ × f): (401, 401)\n",[32,62495,62493],{"__ignoreMap":30},[25,62497,62499],{"className":27,"code":62498,"language":29,"meta":30,"style":30},"controls_table = pd.DataFrame(\n    [\n        ['f0 (GHz)', F0_GHZ],\n        ['Ω\u002F2π on-res (MHz)', OMEGA_ONRESONANCE_MHZ],\n        ['f sweep min (GHz)', FREQ_MIN_GHZ],\n        ['f sweep max (GHz)', FREQ_MAX_GHZ],\n        ['N_f', N_FREQ],\n        ['τ sweep min (ns)', DUR_MIN_NS],\n        ['τ sweep max (ns)', DUR_MAX_NS],\n        ['N_τ', N_DUR],\n        ['T2* (ns; None=off)', T2STAR_NS],\n    ],\n    columns=['Parameter', 'Value']\n)\ncontrols_table\n",[32,62500,62501,62514,62519,62534,62547,62560,62573,62586,62599,62612,62625,62638,62643,62662,62666],{"__ignoreMap":30},[35,62502,62503,62506,62508,62510,62512],{"class":37,"line":38},[35,62504,62505],{"class":113},"controls_table ",[35,62507,242],{"class":241},[35,62509,1961],{"class":113},[35,62511,1964],{"class":271},[35,62513,903],{"class":113},[35,62515,62516],{"class":37,"line":45},[35,62517,62518],{"class":113},"    [\n",[35,62520,62521,62524,62527,62529,62531],{"class":37,"line":51},[35,62522,62523],{"class":113},"        [",[35,62525,62526],{"class":234},"'f0 (GHz)'",[35,62528,288],{"class":113},[35,62530,61004],{"class":245},[35,62532,62533],{"class":113},"],\n",[35,62535,62536,62538,62541,62543,62545],{"class":37,"line":57},[35,62537,62523],{"class":113},[35,62539,62540],{"class":234},"'Ω\u002F2π on-res (MHz)'",[35,62542,288],{"class":113},[35,62544,61088],{"class":245},[35,62546,62533],{"class":113},[35,62548,62549,62551,62554,62556,62558],{"class":37,"line":63},[35,62550,62523],{"class":113},[35,62552,62553],{"class":234},"'f sweep min (GHz)'",[35,62555,288],{"class":113},[35,62557,61141],{"class":245},[35,62559,62533],{"class":113},[35,62561,62562,62564,62567,62569,62571],{"class":37,"line":69},[35,62563,62523],{"class":113},[35,62565,62566],{"class":234},"'f sweep max (GHz)'",[35,62568,288],{"class":113},[35,62570,61156],{"class":245},[35,62572,62533],{"class":113},[35,62574,62575,62577,62580,62582,62584],{"class":37,"line":75},[35,62576,62523],{"class":113},[35,62578,62579],{"class":234},"'N_f'",[35,62581,288],{"class":113},[35,62583,61171],{"class":245},[35,62585,62533],{"class":113},[35,62587,62588,62590,62593,62595,62597],{"class":37,"line":81},[35,62589,62523],{"class":113},[35,62591,62592],{"class":234},"'τ sweep min (ns)'",[35,62594,288],{"class":113},[35,62596,61186],{"class":245},[35,62598,62533],{"class":113},[35,62600,62601,62603,62606,62608,62610],{"class":37,"line":87},[35,62602,62523],{"class":113},[35,62604,62605],{"class":234},"'τ sweep max (ns)'",[35,62607,288],{"class":113},[35,62609,61200],{"class":245},[35,62611,62533],{"class":113},[35,62613,62614,62616,62619,62621,62623],{"class":37,"line":93},[35,62615,62523],{"class":113},[35,62617,62618],{"class":234},"'N_τ'",[35,62620,288],{"class":113},[35,62622,61215],{"class":245},[35,62624,62533],{"class":113},[35,62626,62627,62629,62632,62634,62636],{"class":37,"line":99},[35,62628,62523],{"class":113},[35,62630,62631],{"class":234},"'T2* (ns; None=off)'",[35,62633,288],{"class":113},[35,62635,61229],{"class":245},[35,62637,62533],{"class":113},[35,62639,62640],{"class":37,"line":106},[35,62641,62642],{"class":113},"    ],\n",[35,62644,62645,62648,62650,62652,62655,62657,62660],{"class":37,"line":117},[35,62646,62647],{"class":822},"    columns",[35,62649,242],{"class":241},[35,62651,1254],{"class":113},[35,62653,62654],{"class":234},"'Parameter'",[35,62656,288],{"class":113},[35,62658,62659],{"class":234},"'Value'",[35,62661,1074],{"class":113},[35,62663,62664],{"class":37,"line":125},[35,62665,384],{"class":113},[35,62667,62668],{"class":37,"line":139},[35,62669,62670],{"class":113},"controls_table\n",[25,62672,62675],{"className":62673,"code":62674,"language":4991,"meta":30},[4989],"Parameter  Value\n0            f0 (GHz)    5.0\n1   Ω\u002F2π on-res (MHz)   20.0\n2   f sweep min (GHz)    4.9\n3   f sweep max (GHz)    5.1\n4                 N_f  401.0\n5    τ sweep min (ns)    0.0\n6    τ sweep max (ns)  200.0\n7                 N_τ  401.0\n8  T2* (ns; None=off)  500.0\n",[32,62676,62674],{"__ignoreMap":30},[25,62678,62680],{"className":27,"code":62679,"language":29,"meta":30,"style":30},"fig, ax = plt.subplots(figsize=(7, 5))\nc = ax.pcolormesh(freq_ghz, dur_ns, Pe, shading='auto')\ncb = fig.colorbar(c, ax=ax, label=r'$P_e$')\nax.set_xlabel('Drive frequency f (GHz)')\nax.set_ylabel('Pulse duration τ (ns)')\nax.set_title('Rabi excited-state probability $P_e(f,\\\\,\\\\tau)$')\nfig.tight_layout()\nplt.show()\n",[32,62681,62682,62712,62738,62773,62788,62802,62826,62835],{"__ignoreMap":30},[35,62683,62684,62687,62689,62692,62695,62697,62699,62701,62703,62706,62708,62710],{"class":37,"line":38},[35,62685,62686],{"class":113},"fig, ax ",[35,62688,242],{"class":241},[35,62690,62691],{"class":113}," plt.",[35,62693,62694],{"class":271},"subplots",[35,62696,275],{"class":113},[35,62698,2386],{"class":822},[35,62700,242],{"class":241},[35,62702,275],{"class":113},[35,62704,62705],{"class":245},"7",[35,62707,288],{"class":113},[35,62709,2629],{"class":245},[35,62711,2401],{"class":113},[35,62713,62714,62717,62719,62722,62725,62728,62731,62733,62736],{"class":37,"line":45},[35,62715,62716],{"class":113},"c ",[35,62718,242],{"class":241},[35,62720,62721],{"class":113}," ax.",[35,62723,62724],{"class":271},"pcolormesh",[35,62726,62727],{"class":113},"(freq_ghz, dur_ns, Pe, ",[35,62729,62730],{"class":822},"shading",[35,62732,242],{"class":241},[35,62734,62735],{"class":234},"'auto'",[35,62737,384],{"class":113},[35,62739,62740,62743,62745,62748,62751,62754,62757,62759,62762,62764,62766,62768,62771],{"class":37,"line":51},[35,62741,62742],{"class":113},"cb ",[35,62744,242],{"class":241},[35,62746,62747],{"class":113}," fig.",[35,62749,62750],{"class":271},"colorbar",[35,62752,62753],{"class":113},"(c, ",[35,62755,62756],{"class":822},"ax",[35,62758,242],{"class":241},[35,62760,62761],{"class":113},"ax, ",[35,62763,2966],{"class":822},[35,62765,242],{"class":241},[35,62767,6778],{"class":109},[35,62769,62770],{"class":3409},"'$P_e$'",[35,62772,384],{"class":113},[35,62774,62775,62778,62781,62783,62786],{"class":37,"line":57},[35,62776,62777],{"class":113},"ax.",[35,62779,62780],{"class":271},"set_xlabel",[35,62782,275],{"class":113},[35,62784,62785],{"class":234},"'Drive frequency f (GHz)'",[35,62787,384],{"class":113},[35,62789,62790,62792,62795,62797,62800],{"class":37,"line":63},[35,62791,62777],{"class":113},[35,62793,62794],{"class":271},"set_ylabel",[35,62796,275],{"class":113},[35,62798,62799],{"class":234},"'Pulse duration τ (ns)'",[35,62801,384],{"class":113},[35,62803,62804,62806,62809,62811,62814,62817,62819,62821,62824],{"class":37,"line":69},[35,62805,62777],{"class":113},[35,62807,62808],{"class":271},"set_title",[35,62810,275],{"class":113},[35,62812,62813],{"class":234},"'Rabi excited-state probability $P_e(f,",[35,62815,62816],{"class":241},"\\\\",[35,62818,2877],{"class":234},[35,62820,62816],{"class":241},[35,62822,62823],{"class":234},"tau)$'",[35,62825,384],{"class":113},[35,62827,62828,62831,62833],{"class":37,"line":75},[35,62829,62830],{"class":113},"fig.",[35,62832,2740],{"class":271},[35,62834,2101],{"class":113},[35,62836,62837,62839,62841],{"class":37,"line":81},[35,62838,2378],{"class":113},[35,62840,2792],{"class":271},[35,62842,2101],{"class":113},[4995,62844],{"alt":4997,"src":62845},"\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.webp",[25,62847,62849],{"className":27,"code":62848,"language":29,"meta":30,"style":30},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n\nimport importlib.util\nimport subprocess\nimport sys\n\n# -------------------------------------------------------------------------\n# Control knobs\n# -------------------------------------------------------------------------\nINSTALL_BLOCH_DEPENDENCIES = True\n\nBLOCH_REQUIRED_PACKAGES = {\n    \"numpy\": \"numpy\",\n    \"matplotlib\": \"matplotlib\",\n    \"scipy\": \"scipy\",\n    \"qutip\": \"qutip\",\n}\n\nBLOCH_F0_GHZ = globals().get(\"F0_GHZ\", 5.000)\nBLOCH_OMEGA_ONRESONANCE_MHZ = globals().get(\n    \"OMEGA_ONRESONANCE_MHZ\",\n    20.0,\n)\n\nBLOCH_RESONANT_DETUNING_MHZ = 0.0\nBLOCH_DETUNED_DETUNING_MHZ = 40.0\n\n# Use None to automatically plot one resonant pi pulse.\nBLOCH_DURATION_NS = None\n\nBLOCH_NUM_TIME_POINTS = 300\nBLOCH_FIGSIZE = (8.8, 7.8)\nBLOCH_DPI = 250\nBLOCH_VIEW = [-60, 25]\n\nBLOCH_SHOW_CONTROL_AXIS_ARROWS = True\nBLOCH_SHOW_ARROW_TIP_LABELS = True\nBLOCH_SHOW_ENDPOINT_MARKERS = True\nBLOCH_SHOW_LEGEND = True\nBLOCH_PRINT_SUMMARY = True\n\nCOLOR_RESONANT_TRAJECTORY = \"tab:blue\"\nCOLOR_DETUNED_TRAJECTORY = \"tab:orange\"\nCOLOR_RESONANT_AXIS = \"tab:green\"\nCOLOR_DETUNED_AXIS = \"tab:red\"\nCOLOR_INITIAL_STATE = \"black\"\n\nTRAJECTORY_LINEWIDTH = 2.4\nCONTROL_AXIS_LINEWIDTH = 2.2\nENDPOINT_MARKER_SIZE = 45\nARROW_LENGTH_RATIO = 0.12\nARROW_LABEL_SCALE = 1.16\n\nLEGEND_LOCATION = \"upper left\"\nLEGEND_BBOX_TO_ANCHOR = (1.02, 1.02)\n\n\ndef package_is_available(package_name: str) -> bool:\n    \"\"\"Return True when a package can be imported.\"\"\"\n    return importlib.util.find_spec(package_name) is not None\n\n\ndef install_missing_packages(\n    required_packages: dict[str, str],\n) -> None:\n    \"\"\"Install missing notebook packages using uv pip with a pip fallback.\"\"\"\n    if not INSTALL_BLOCH_DEPENDENCIES:\n        return\n\n    missing_package_names = [\n        pip_name\n        for import_name, pip_name in required_packages.items()\n        if not package_is_available(import_name)\n    ]\n\n    if not missing_package_names:\n        return\n\n    try:\n        subprocess.run(\n            [sys.executable, \"-m\", \"pip\", \"install\", \"-q\", \"uv\"],\n            check=True,\n        )\n        subprocess.run(\n            [\n                sys.executable,\n                \"-m\",\n                \"uv\",\n                \"pip\",\n                \"install\",\n                \"--system\",\n                \"-q\",\n                *missing_package_names,\n            ],\n            check=True,\n        )\n    except (subprocess.CalledProcessError, FileNotFoundError):\n        subprocess.run(\n            [\n                sys.executable,\n                \"-m\",\n                \"pip\",\n                \"install\",\n                \"-q\",\n                *missing_package_names,\n            ],\n            check=True,\n        )\n\n\ninstall_missing_packages(BLOCH_REQUIRED_PACKAGES)\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport qutip as qt\nfrom matplotlib.lines import Line2D\n\ntry:\n    from IPython import get_ipython\n\n    ipython = get_ipython()\n    if ipython is not None:\n        ipython.run_line_magic(\"matplotlib\", \"inline\")\nexcept ImportError:\n    pass\n\nplt.rcParams.update(\n    {\n        \"figure.dpi\": BLOCH_DPI,\n    }\n)\n\n\ndef get_pi_pulse_duration_ns(omega_onresonance_mhz: float) -> float:\n    \"\"\"Return the resonant pi-pulse duration in nanoseconds.\"\"\"\n    return 1.0e3 \u002F (2.0 * omega_onresonance_mhz)\n\n\ndef solve_rwa_bloch_trajectory(\n    detuning_mhz: float,\n    duration_ns: float,\n    num_time_points: int,\n) -> dict[str, np.ndarray | float]:\n    \"\"\"Solve the RWA two-level trajectory and return Bloch observables.\"\"\"\n    omega_rad_s = 2.0 * np.pi * BLOCH_OMEGA_ONRESONANCE_MHZ * 1.0e6\n    delta_rad_s = 2.0 * np.pi * detuning_mhz * 1.0e6\n    omega_generalized_rad_s = np.hypot(omega_rad_s, delta_rad_s)\n\n    time_s = np.linspace(0.0, duration_ns * 1.0e-9, num_time_points)\n\n    ground_state = qt.basis(2, 0)\n    excited_state = qt.basis(2, 1)\n    excited_projector = excited_state * excited_state.dag()\n\n    hamiltonian = 0.5 * (\n        delta_rad_s * qt.sigmaz()\n        + omega_rad_s * qt.sigmax()\n    )\n\n    result = qt.sesolve(\n        hamiltonian,\n        ground_state,\n        time_s,\n    )\n\n    x_expectation = np.real(np.asarray(qt.expect(qt.sigmax(), result.states)))\n    y_expectation = np.real(np.asarray(qt.expect(qt.sigmay(), result.states)))\n    z_expectation = np.real(np.asarray(qt.expect(qt.sigmaz(), result.states)))\n\n    excited_probability = np.real(\n        np.asarray(qt.expect(excited_projector, result.states))\n    )\n\n    bloch_vectors = np.vstack(\n        [\n            x_expectation,\n            y_expectation,\n            z_expectation,\n        ]\n    )\n\n    control_axis = np.array(\n        [\n            omega_rad_s \u002F omega_generalized_rad_s,\n            0.0,\n            delta_rad_s \u002F omega_generalized_rad_s,\n        ]\n    )\n\n    return {\n        \"time_ns\": time_s * 1.0e9,\n        \"vectors\": bloch_vectors,\n        \"excited_probability\": excited_probability,\n        \"control_axis\": control_axis,\n        \"generalized_rabi_mhz\": (\n            omega_generalized_rad_s \u002F (2.0 * np.pi * 1.0e6)\n        ),\n    }\n\n\ndef render_empty_bloch_sphere(\n    bloch_sphere: qt.Bloch,\n    fallback_figure: plt.Figure,\n):\n    \"\"\"Render an empty QuTiP Bloch sphere and return its Matplotlib axis.\"\"\"\n    if hasattr(bloch_sphere, \"render\"):\n        bloch_sphere.render()\n    elif hasattr(bloch_sphere, \"make_sphere\"):\n        bloch_sphere.make_sphere()\n    else:\n        bloch_sphere.show()\n\n    axis = getattr(bloch_sphere, \"axes\", None)\n\n    if axis is None and fallback_figure.axes:\n        axis = fallback_figure.axes[-1]\n\n    if axis is None:\n        raise RuntimeError(\"Could not access the Matplotlib Bloch axis.\")\n\n    return axis\n\n\ndef draw_bloch_trajectory(\n    axis,\n    vectors: np.ndarray,\n    color: str,\n    label: str,\n) -> None:\n    \"\"\"Draw a Bloch-vector trajectory on an existing 3D axis.\"\"\"\n    axis.plot(\n        vectors[0],\n        vectors[1],\n        vectors[2],\n        color=color,\n        linewidth=TRAJECTORY_LINEWIDTH,\n        label=label,\n    )\n\n\ndef draw_endpoint_marker(\n    axis,\n    vector: np.ndarray,\n    color: str,\n    label: str,\n) -> None:\n    \"\"\"Draw a marker at the final point of a Bloch trajectory.\"\"\"\n    axis.scatter(\n        vector[0],\n        vector[1],\n        vector[2],\n        color=color,\n        s=ENDPOINT_MARKER_SIZE,\n        depthshade=True,\n        label=label,\n    )\n\n\ndef draw_control_axis_arrow(\n    axis,\n    vector: np.ndarray,\n    color: str,\n    label: str,\n) -> None:\n    \"\"\"Draw a normalized effective-control-axis arrow.\"\"\"\n    axis.quiver(\n        0.0,\n        0.0,\n        0.0,\n        vector[0],\n        vector[1],\n        vector[2],\n        color=color,\n        linewidth=CONTROL_AXIS_LINEWIDTH,\n        arrow_length_ratio=ARROW_LENGTH_RATIO,\n        normalize=False,\n        label=label,\n    )\n\n\ndef add_arrow_tip_label(\n    axis,\n    vector: np.ndarray,\n    text: str,\n    color: str,\n) -> None:\n    \"\"\"Place a text label slightly beyond the arrow tip.\"\"\"\n    label_position = ARROW_LABEL_SCALE * vector\n\n    axis.text(\n        label_position[0],\n        label_position[1],\n        label_position[2],\n        text,\n        color=color,\n        fontsize=10,\n        ha=\"center\",\n        va=\"center\",\n    )\n\n\ndef add_initial_state_label(axis) -> None:\n    \"\"\"Label the initialized ground-state pole.\"\"\"\n    axis.scatter(\n        0.0,\n        0.0,\n        1.0,\n        color=COLOR_INITIAL_STATE,\n        s=ENDPOINT_MARKER_SIZE,\n        depthshade=True,\n    )\n    axis.text(\n        0.0,\n        0.0,\n        1.16,\n        r\"initial $|g\\rangle$\",\n        color=COLOR_INITIAL_STATE,\n        fontsize=9,\n        ha=\"center\",\n        va=\"center\",\n    )\n\n\nif BLOCH_DURATION_NS is None:\n    bloch_duration_ns = get_pi_pulse_duration_ns(\n        BLOCH_OMEGA_ONRESONANCE_MHZ\n    )\nelse:\n    bloch_duration_ns = BLOCH_DURATION_NS\n\nresonant_trajectory = solve_rwa_bloch_trajectory(\n    detuning_mhz=BLOCH_RESONANT_DETUNING_MHZ,\n    duration_ns=bloch_duration_ns,\n    num_time_points=BLOCH_NUM_TIME_POINTS,\n)\n\ndetuned_trajectory = solve_rwa_bloch_trajectory(\n    detuning_mhz=BLOCH_DETUNED_DETUNING_MHZ,\n    duration_ns=bloch_duration_ns,\n    num_time_points=BLOCH_NUM_TIME_POINTS,\n)\n\nresonant_drive_ghz = (\n    BLOCH_F0_GHZ\n    + BLOCH_RESONANT_DETUNING_MHZ * 1.0e-3\n)\ndetuned_drive_ghz = (\n    BLOCH_F0_GHZ\n    + BLOCH_DETUNED_DETUNING_MHZ * 1.0e-3\n)\n\nfigure = plt.figure(figsize=BLOCH_FIGSIZE)\n\nbloch = qt.Bloch(fig=figure)\nbloch.view = BLOCH_VIEW\nbloch.zlabel = [r\"$|g\\rangle$\", r\"$|e\\rangle$\"]\nbloch.title = (\n    \"Bloch-sphere Rabi trajectories\\n\"\n    rf\"resonant $\\Delta\u002F2\\pi={BLOCH_RESONANT_DETUNING_MHZ:.1f}$ MHz, \"\n    rf\"detuned $\\Delta\u002F2\\pi={BLOCH_DETUNED_DETUNING_MHZ:.1f}$ MHz\"\n)\n\nbloch_axis = render_empty_bloch_sphere(\n    bloch_sphere=bloch,\n    fallback_figure=figure,\n)\n\ndraw_bloch_trajectory(\n    axis=bloch_axis,\n    vectors=resonant_trajectory[\"vectors\"],\n    color=COLOR_RESONANT_TRAJECTORY,\n    label=\"Resonant state trajectory\",\n)\ndraw_bloch_trajectory(\n    axis=bloch_axis,\n    vectors=detuned_trajectory[\"vectors\"],\n    color=COLOR_DETUNED_TRAJECTORY,\n    label=\"Detuned state trajectory\",\n)\n\nif BLOCH_SHOW_ENDPOINT_MARKERS:\n    draw_endpoint_marker(\n        axis=bloch_axis,\n        vector=resonant_trajectory[\"vectors\"][:, -1],\n        color=COLOR_RESONANT_TRAJECTORY,\n        label=\"Resonant final state\",\n    )\n    draw_endpoint_marker(\n        axis=bloch_axis,\n        vector=detuned_trajectory[\"vectors\"][:, -1],\n        color=COLOR_DETUNED_TRAJECTORY,\n        label=\"Detuned final state\",\n    )\n    add_initial_state_label(bloch_axis)\n\nif BLOCH_SHOW_CONTROL_AXIS_ARROWS:\n    draw_control_axis_arrow(\n        axis=bloch_axis,\n        vector=resonant_trajectory[\"control_axis\"],\n        color=COLOR_RESONANT_AXIS,\n        label=r\"Resonant control axis $\\hat{n}_{\\mathrm{res}}$\",\n    )\n    draw_control_axis_arrow(\n        axis=bloch_axis,\n        vector=detuned_trajectory[\"control_axis\"],\n        color=COLOR_DETUNED_AXIS,\n        label=r\"Detuned control axis $\\hat{n}_{\\mathrm{det}}$\",\n    )\n\nif BLOCH_SHOW_ARROW_TIP_LABELS and BLOCH_SHOW_CONTROL_AXIS_ARROWS:\n    add_arrow_tip_label(\n        axis=bloch_axis,\n        vector=resonant_trajectory[\"control_axis\"],\n        text=r\"$\\hat{n}_{\\mathrm{res}}$\",\n        color=COLOR_RESONANT_AXIS,\n    )\n    add_arrow_tip_label(\n        axis=bloch_axis,\n        vector=detuned_trajectory[\"control_axis\"],\n        text=r\"$\\hat{n}_{\\mathrm{det}}$\",\n        color=COLOR_DETUNED_AXIS,\n    )\n\nif BLOCH_SHOW_LEGEND:\n    legend_handles = [\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_RESONANT_TRAJECTORY,\n            linewidth=TRAJECTORY_LINEWIDTH,\n            label=\"Resonant state trajectory\",\n        ),\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_DETUNED_TRAJECTORY,\n            linewidth=TRAJECTORY_LINEWIDTH,\n            label=\"Detuned state trajectory\",\n        ),\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_RESONANT_AXIS,\n            linewidth=CONTROL_AXIS_LINEWIDTH,\n            label=r\"Resonant control axis $\\hat{n}_{\\mathrm{res}}$\",\n        ),\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_DETUNED_AXIS,\n            linewidth=CONTROL_AXIS_LINEWIDTH,\n            label=r\"Detuned control axis $\\hat{n}_{\\mathrm{det}}$\",\n        ),\n    ]\n\n    bloch_axis.legend(\n        handles=legend_handles,\n        loc=LEGEND_LOCATION,\n        bbox_to_anchor=LEGEND_BBOX_TO_ANCHOR,\n        frameon=True,\n        borderaxespad=0.0,\n    )\n\nplt.show()\n\nif BLOCH_PRINT_SUMMARY:\n    resonant_axis = resonant_trajectory[\"control_axis\"]\n    detuned_axis = detuned_trajectory[\"control_axis\"]\n\n    print(\"Bloch-sphere trajectory summary\")\n    print(f\"Natural transition frequency f0: {BLOCH_F0_GHZ:.6f} GHz\")\n    print(f\"Resonant drive frequency: {resonant_drive_ghz:.6f} GHz\")\n    print(f\"Detuned drive frequency: {detuned_drive_ghz:.6f} GHz\")\n    print(f\"On-resonance Rabi rate: {BLOCH_OMEGA_ONRESONANCE_MHZ:.3f} MHz\")\n    print(f\"Plotted duration: {bloch_duration_ns:.3f} ns\")\n    print(\n        \"Resonant control axis n_res: \"\n        f\"({resonant_axis[0]:.6f}, \"\n        f\"{resonant_axis[1]:.6f}, \"\n        f\"{resonant_axis[2]:.6f})\"\n    )\n    print(\n        \"Detuned control axis n_det: \"\n        f\"({detuned_axis[0]:.6f}, \"\n        f\"{detuned_axis[1]:.6f}, \"\n        f\"{detuned_axis[2]:.6f})\"\n    )\n    print(\n        \"Resonant final excited-state probability: \"\n        f\"{resonant_trajectory['excited_probability'][-1]:.6f}\"\n    )\n    print(\n        \"Detuned final excited-state probability: \"\n        f\"{detuned_trajectory['excited_probability'][-1]:.6f}\"\n    )\n    print(\n        \"Detuned generalized Rabi frequency: \"\n        f\"{detuned_trajectory['generalized_rabi_mhz']:.6f} MHz\"\n    )\n",[32,62850,62851,62856,62861,62865,62872,62879,62885,62889,62894,62899,62903,62912,62916,62925,62937,62949,62961,62973,62977,62981,63007,63022,63029,63036,63040,63044,63054,63064,63068,63073,63083,63087,63096,63115,63124,63144,63148,63157,63166,63175,63184,63193,63197,63207,63217,63227,63237,63247,63251,63261,63271,63281,63291,63301,63305,63315,63333,63337,63341,63363,63368,63387,63391,63395,63404,63420,63428,63433,63444,63449,63453,63462,63467,63483,63494,63498,63502,63511,63515,63519,63526,63535,63565,63576,63581,63589,63594,63599,63606,63613,63620,63627,63634,63641,63649,63654,63664,63668,63676,63684,63688,63692,63698,63704,63710,63716,63722,63726,63736,63740,63744,63748,63759,63763,63773,63783,63795,63807,63811,63817,63830,63834,63846,63861,63880,63887,63892,63896,63906,63911,63922,63926,63930,63934,63938,63960,63965,63984,63988,63992,64001,64012,64023,64034,64051,64056,64079,64101,64116,64120,64146,64150,64173,64194,64214,64218,64231,64245,64262,64266,64270,64284,64289,64294,64299,64303,64307,64338,64364,64389,64393,64406,64420,64424,64428,64441,64445,64450,64455,64460,64464,64468,64472,64485,64489,64499,64506,64515,64519,64523,64527,64533,64548,64556,64564,64572,64580,64602,64607,64611,64615,64619,64628,64636,64644,64648,64653,64668,64678,64692,64701,64707,64715,64719,64740,64744,64760,64776,64780,64792,64805,64809,64816,64820,64824,64833,64840,64847,64857,64867,64875,64880,64889,64898,64906,64914,64924,64935,64945,64949,64953,64957,64966,64972,64979,64989,64999,65007,65012,65021,65030,65038,65046,65054,65065,65076,65084,65088,65092,65096,65105,65111,65117,65127,65137,65145,65150,65159,65166,65172,65178,65186,65194,65202,65210,65220,65231,65242,65250,65254,65258,65262,65271,65277,65283,65294,65304,65312,65317,65332,65336,65344,65353,65361,65369,65374,65382,65393,65405,65416,65420,65424,65428,65446,65451,65459,65465,65471,65478,65488,65498,65508,65512,65520,65526,65532,65539,65559,65569,65580,65590,65600,65604,65608,65612,65625,65636,65641,65645,65652,65661,65665,65676,65686,65695,65705,65709,65713,65724,65734,65742,65752,65756,65760,65769,65774,65787,65791,65800,65804,65815,65819,65823,65844,65848,65870,65880,65917,65926,65935,65954,65971,65975,65979,65990,65999,66008,66012,66016,66023,66032,66046,66056,66067,66071,66077,66085,66098,66108,66119,66123,66127,66136,66143,66153,66174,66185,66197,66202,66209,66218,66237,66248,66260,66265,66274,66279,66289,66297,66306,66320,66331,66357,66362,66369,66378,66391,66402,66425,66430,66435,66449,66457,66466,66479,66501,66512,66517,66524,66533,66546,66567,66578,66583,66588,66598,66608,66616,66626,66635,66647,66659,66671,66676,66683,66692,66701,66712,66723,66734,66739,66746,66755,66764,66775,66786,66807,66812,66819,66828,66837,66848,66859,66880,66885,66890,66895,66905,66916,66928,66940,66952,66964,66969,66974,66983,66988,66998,67013,67028,67033,67045,67070,67095,67120,67145,67171,67178,67184,67209,67230,67252,67257,67264,67270,67292,67313,67334,67339,67346,67352,67380,67385,67392,67398,67425,67430,67437,67443,67466],{"__ignoreMap":30},[35,62852,62853],{"class":37,"line":38},[35,62854,62855],{"class":41},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n",[35,62857,62858],{"class":37,"line":45},[35,62859,62860],{"class":234},"\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n",[35,62862,62863],{"class":37,"line":51},[35,62864,103],{"emptyLinePlaceholder":102},[35,62866,62867,62869],{"class":37,"line":57},[35,62868,110],{"class":109},[35,62870,62871],{"class":113}," importlib.util\n",[35,62873,62874,62876],{"class":37,"line":63},[35,62875,110],{"class":109},[35,62877,62878],{"class":113}," subprocess\n",[35,62880,62881,62883],{"class":37,"line":69},[35,62882,110],{"class":109},[35,62884,61582],{"class":113},[35,62886,62887],{"class":37,"line":75},[35,62888,103],{"emptyLinePlaceholder":102},[35,62890,62891],{"class":37,"line":81},[35,62892,62893],{"class":41},"# -------------------------------------------------------------------------\n",[35,62895,62896],{"class":37,"line":87},[35,62897,62898],{"class":41},"# Control knobs\n",[35,62900,62901],{"class":37,"line":93},[35,62902,62893],{"class":41},[35,62904,62905,62908,62910],{"class":37,"line":99},[35,62906,62907],{"class":245},"INSTALL_BLOCH_DEPENDENCIES",[35,62909,308],{"class":241},[35,62911,61688],{"class":245},[35,62913,62914],{"class":37,"line":106},[35,62915,103],{"emptyLinePlaceholder":102},[35,62917,62918,62921,62923],{"class":37,"line":117},[35,62919,62920],{"class":245},"BLOCH_REQUIRED_PACKAGES",[35,62922,308],{"class":241},[35,62924,1104],{"class":113},[35,62926,62927,62930,62932,62935],{"class":37,"line":125},[35,62928,62929],{"class":234},"    \"numpy\"",[35,62931,282],{"class":113},[35,62933,62934],{"class":234},"\"numpy\"",[35,62936,912],{"class":113},[35,62938,62939,62942,62944,62947],{"class":37,"line":139},[35,62940,62941],{"class":234},"    \"matplotlib\"",[35,62943,282],{"class":113},[35,62945,62946],{"class":234},"\"matplotlib\"",[35,62948,912],{"class":113},[35,62950,62951,62954,62956,62959],{"class":37,"line":152},[35,62952,62953],{"class":234},"    \"scipy\"",[35,62955,282],{"class":113},[35,62957,62958],{"class":234},"\"scipy\"",[35,62960,912],{"class":113},[35,62962,62963,62966,62968,62971],{"class":37,"line":157},[35,62964,62965],{"class":234},"    \"qutip\"",[35,62967,282],{"class":113},[35,62969,62970],{"class":234},"\"qutip\"",[35,62972,912],{"class":113},[35,62974,62975],{"class":37,"line":171},[35,62976,1228],{"class":113},[35,62978,62979],{"class":37,"line":184},[35,62980,103],{"emptyLinePlaceholder":102},[35,62982,62983,62986,62988,62991,62993,62996,62998,63001,63003,63005],{"class":37,"line":197},[35,62984,62985],{"class":245},"BLOCH_F0_GHZ",[35,62987,308],{"class":241},[35,62989,62990],{"class":241}," globals",[35,62992,18496],{"class":113},[35,62994,62995],{"class":271},"get",[35,62997,275],{"class":113},[35,62999,63000],{"class":234},"\"F0_GHZ\"",[35,63002,288],{"class":113},[35,63004,48281],{"class":245},[35,63006,384],{"class":113},[35,63008,63009,63012,63014,63016,63018,63020],{"class":37,"line":210},[35,63010,63011],{"class":245},"BLOCH_OMEGA_ONRESONANCE_MHZ",[35,63013,308],{"class":241},[35,63015,62990],{"class":241},[35,63017,18496],{"class":113},[35,63019,62995],{"class":271},[35,63021,903],{"class":113},[35,63023,63024,63027],{"class":37,"line":223},[35,63025,63026],{"class":234},"    \"OMEGA_ONRESONANCE_MHZ\"",[35,63028,912],{"class":113},[35,63030,63031,63034],{"class":37,"line":228},[35,63032,63033],{"class":245},"    20.0",[35,63035,912],{"class":113},[35,63037,63038],{"class":37,"line":249},[35,63039,384],{"class":113},[35,63041,63042],{"class":37,"line":254},[35,63043,103],{"emptyLinePlaceholder":102},[35,63045,63046,63049,63051],{"class":37,"line":260},[35,63047,63048],{"class":245},"BLOCH_RESONANT_DETUNING_MHZ",[35,63050,308],{"class":241},[35,63052,63053],{"class":245}," 0.0\n",[35,63055,63056,63059,63061],{"class":37,"line":265},[35,63057,63058],{"class":245},"BLOCH_DETUNED_DETUNING_MHZ",[35,63060,308],{"class":241},[35,63062,63063],{"class":245}," 40.0\n",[35,63065,63066],{"class":37,"line":326},[35,63067,103],{"emptyLinePlaceholder":102},[35,63069,63070],{"class":37,"line":332},[35,63071,63072],{"class":41},"# Use None to automatically plot one resonant pi pulse.\n",[35,63074,63075,63078,63080],{"class":37,"line":338},[35,63076,63077],{"class":245},"BLOCH_DURATION_NS",[35,63079,308],{"class":241},[35,63081,63082],{"class":245}," None\n",[35,63084,63085],{"class":37,"line":344},[35,63086,103],{"emptyLinePlaceholder":102},[35,63088,63089,63092,63094],{"class":37,"line":349},[35,63090,63091],{"class":245},"BLOCH_NUM_TIME_POINTS",[35,63093,308],{"class":241},[35,63095,28122],{"class":245},[35,63097,63098,63101,63103,63105,63108,63110,63113],{"class":37,"line":367},[35,63099,63100],{"class":245},"BLOCH_FIGSIZE",[35,63102,308],{"class":241},[35,63104,373],{"class":113},[35,63106,63107],{"class":245},"8.8",[35,63109,288],{"class":113},[35,63111,63112],{"class":245},"7.8",[35,63114,384],{"class":113},[35,63116,63117,63120,63122],{"class":37,"line":387},[35,63118,63119],{"class":245},"BLOCH_DPI",[35,63121,308],{"class":241},[35,63123,61722],{"class":245},[35,63125,63126,63129,63131,63133,63135,63138,63140,63142],{"class":37,"line":404},[35,63127,63128],{"class":245},"BLOCH_VIEW",[35,63130,308],{"class":241},[35,63132,2871],{"class":113},[35,63134,564],{"class":241},[35,63136,63137],{"class":245},"60",[35,63139,288],{"class":113},[35,63141,2680],{"class":245},[35,63143,1074],{"class":113},[35,63145,63146],{"class":37,"line":434},[35,63147,103],{"emptyLinePlaceholder":102},[35,63149,63150,63153,63155],{"class":37,"line":474},[35,63151,63152],{"class":245},"BLOCH_SHOW_CONTROL_AXIS_ARROWS",[35,63154,308],{"class":241},[35,63156,61688],{"class":245},[35,63158,63159,63162,63164],{"class":37,"line":546},[35,63160,63161],{"class":245},"BLOCH_SHOW_ARROW_TIP_LABELS",[35,63163,308],{"class":241},[35,63165,61688],{"class":245},[35,63167,63168,63171,63173],{"class":37,"line":584},[35,63169,63170],{"class":245},"BLOCH_SHOW_ENDPOINT_MARKERS",[35,63172,308],{"class":241},[35,63174,61688],{"class":245},[35,63176,63177,63180,63182],{"class":37,"line":589},[35,63178,63179],{"class":245},"BLOCH_SHOW_LEGEND",[35,63181,308],{"class":241},[35,63183,61688],{"class":245},[35,63185,63186,63189,63191],{"class":37,"line":594},[35,63187,63188],{"class":245},"BLOCH_PRINT_SUMMARY",[35,63190,308],{"class":241},[35,63192,61688],{"class":245},[35,63194,63195],{"class":37,"line":643},[35,63196,103],{"emptyLinePlaceholder":102},[35,63198,63199,63202,63204],{"class":37,"line":648},[35,63200,63201],{"class":245},"COLOR_RESONANT_TRAJECTORY",[35,63203,308],{"class":241},[35,63205,63206],{"class":234}," \"tab:blue\"\n",[35,63208,63209,63212,63214],{"class":37,"line":654},[35,63210,63211],{"class":245},"COLOR_DETUNED_TRAJECTORY",[35,63213,308],{"class":241},[35,63215,63216],{"class":234}," \"tab:orange\"\n",[35,63218,63219,63222,63224],{"class":37,"line":660},[35,63220,63221],{"class":245},"COLOR_RESONANT_AXIS",[35,63223,308],{"class":241},[35,63225,63226],{"class":234}," \"tab:green\"\n",[35,63228,63229,63232,63234],{"class":37,"line":665},[35,63230,63231],{"class":245},"COLOR_DETUNED_AXIS",[35,63233,308],{"class":241},[35,63235,63236],{"class":234}," \"tab:red\"\n",[35,63238,63239,63242,63244],{"class":37,"line":692},[35,63240,63241],{"class":245},"COLOR_INITIAL_STATE",[35,63243,308],{"class":241},[35,63245,63246],{"class":234}," \"black\"\n",[35,63248,63249],{"class":37,"line":714},[35,63250,103],{"emptyLinePlaceholder":102},[35,63252,63253,63256,63258],{"class":37,"line":729},[35,63254,63255],{"class":245},"TRAJECTORY_LINEWIDTH",[35,63257,308],{"class":241},[35,63259,63260],{"class":245}," 2.4\n",[35,63262,63263,63266,63268],{"class":37,"line":737},[35,63264,63265],{"class":245},"CONTROL_AXIS_LINEWIDTH",[35,63267,308],{"class":241},[35,63269,63270],{"class":245}," 2.2\n",[35,63272,63273,63276,63278],{"class":37,"line":768},[35,63274,63275],{"class":245},"ENDPOINT_MARKER_SIZE",[35,63277,308],{"class":241},[35,63279,63280],{"class":245}," 45\n",[35,63282,63283,63286,63288],{"class":37,"line":793},[35,63284,63285],{"class":245},"ARROW_LENGTH_RATIO",[35,63287,308],{"class":241},[35,63289,63290],{"class":245}," 0.12\n",[35,63292,63293,63296,63298],{"class":37,"line":798},[35,63294,63295],{"class":245},"ARROW_LABEL_SCALE",[35,63297,308],{"class":241},[35,63299,63300],{"class":245}," 1.16\n",[35,63302,63303],{"class":37,"line":803},[35,63304,103],{"emptyLinePlaceholder":102},[35,63306,63307,63310,63312],{"class":37,"line":809},[35,63308,63309],{"class":245},"LEGEND_LOCATION",[35,63311,308],{"class":241},[35,63313,63314],{"class":234}," \"upper left\"\n",[35,63316,63317,63320,63322,63324,63327,63329,63331],{"class":37,"line":814},[35,63318,63319],{"class":245},"LEGEND_BBOX_TO_ANCHOR",[35,63321,308],{"class":241},[35,63323,373],{"class":113},[35,63325,63326],{"class":245},"1.02",[35,63328,288],{"class":113},[35,63330,63326],{"class":245},[35,63332,384],{"class":113},[35,63334,63335],{"class":37,"line":833},[35,63336,103],{"emptyLinePlaceholder":102},[35,63338,63339],{"class":37,"line":845},[35,63340,103],{"emptyLinePlaceholder":102},[35,63342,63343,63345,63348,63350,63353,63355,63357,63359,63361],{"class":37,"line":854},[35,63344,268],{"class":109},[35,63346,63347],{"class":271}," package_is_available",[35,63349,275],{"class":113},[35,63351,63352],{"class":278},"package_name",[35,63354,282],{"class":113},[35,63356,1831],{"class":241},[35,63358,636],{"class":113},[35,63360,6091],{"class":241},[35,63362,364],{"class":113},[35,63364,63365],{"class":37,"line":865},[35,63366,63367],{"class":234},"    \"\"\"Return True when a package can be imported.\"\"\"\n",[35,63369,63370,63372,63375,63378,63381,63383,63385],{"class":37,"line":873},[35,63371,549],{"class":109},[35,63373,63374],{"class":113}," importlib.util.",[35,63376,63377],{"class":271},"find_spec",[35,63379,63380],{"class":113},"(package_name) ",[35,63382,41492],{"class":109},[35,63384,26490],{"class":109},[35,63386,63082],{"class":245},[35,63388,63389],{"class":37,"line":878},[35,63390,103],{"emptyLinePlaceholder":102},[35,63392,63393],{"class":37,"line":883},[35,63394,103],{"emptyLinePlaceholder":102},[35,63396,63397,63399,63402],{"class":37,"line":897},[35,63398,268],{"class":109},[35,63400,63401],{"class":271}," install_missing_packages",[35,63403,903],{"class":113},[35,63405,63406,63409,63412,63414,63416,63418],{"class":37,"line":906},[35,63407,63408],{"class":278},"    required_packages",[35,63410,63411],{"class":113},": dict[",[35,63413,1831],{"class":241},[35,63415,288],{"class":113},[35,63417,1831],{"class":241},[35,63419,62533],{"class":113},[35,63421,63422,63424,63426],{"class":37,"line":915},[35,63423,636],{"class":113},[35,63425,61240],{"class":245},[35,63427,364],{"class":113},[35,63429,63430],{"class":37,"line":923},[35,63431,63432],{"class":234},"    \"\"\"Install missing notebook packages using uv pip with a pip fallback.\"\"\"\n",[35,63434,63435,63437,63439,63442],{"class":37,"line":929},[35,63436,352],{"class":109},[35,63438,26490],{"class":109},[35,63440,63441],{"class":245}," INSTALL_BLOCH_DEPENDENCIES",[35,63443,364],{"class":113},[35,63445,63446],{"class":37,"line":935},[35,63447,63448],{"class":109},"        return\n",[35,63450,63451],{"class":37,"line":942},[35,63452,103],{"emptyLinePlaceholder":102},[35,63454,63455,63458,63460],{"class":37,"line":950},[35,63456,63457],{"class":113},"    missing_package_names ",[35,63459,242],{"class":241},[35,63461,894],{"class":113},[35,63463,63464],{"class":37,"line":958},[35,63465,63466],{"class":113},"        pip_name\n",[35,63468,63469,63471,63474,63476,63479,63481],{"class":37,"line":964},[35,63470,2087],{"class":109},[35,63472,63473],{"class":113}," import_name, pip_name ",[35,63475,1663],{"class":109},[35,63477,63478],{"class":113}," 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x_expectation,\n",[35,64451,64452],{"class":37,"line":2151},[35,64453,64454],{"class":113},"            y_expectation,\n",[35,64456,64457],{"class":37,"line":2190},[35,64458,64459],{"class":113},"            z_expectation,\n",[35,64461,64462],{"class":37,"line":2223},[35,64463,27131],{"class":113},[35,64465,64466],{"class":37,"line":2228},[35,64467,6280],{"class":113},[35,64469,64470],{"class":37,"line":2239},[35,64471,103],{"emptyLinePlaceholder":102},[35,64473,64474,64477,64479,64481,64483],{"class":37,"line":2258},[35,64475,64476],{"class":113},"    control_axis ",[35,64478,242],{"class":241},[35,64480,2412],{"class":113},[35,64482,6245],{"class":271},[35,64484,903],{"class":113},[35,64486,64487],{"class":37,"line":2263},[35,64488,27086],{"class":113},[35,64490,64491,64494,64496],{"class":37,"line":2269},[35,64492,64493],{"class":113},"            omega_rad_s ",[35,64495,398],{"class":241},[35,64497,64498],{"class":113}," 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resonant_trajectory[",[35,67010,66317],{"class":234},[35,67012,1074],{"class":113},[35,67014,67016,67019,67021,67024,67026],{"class":37,"line":67015},469,[35,67017,67018],{"class":113},"    detuned_axis ",[35,67020,242],{"class":241},[35,67022,67023],{"class":113}," detuned_trajectory[",[35,67025,66317],{"class":234},[35,67027,1074],{"class":113},[35,67029,67031],{"class":37,"line":67030},470,[35,67032,103],{"emptyLinePlaceholder":102},[35,67034,67036,67038,67040,67043],{"class":37,"line":67035},471,[35,67037,6637],{"class":241},[35,67039,275],{"class":113},[35,67041,67042],{"class":234},"\"Bloch-sphere trajectory summary\"",[35,67044,384],{"class":113},[35,67046,67048,67050,67052,67054,67057,67060,67063,67065,67068],{"class":37,"line":67047},472,[35,67049,6637],{"class":241},[35,67051,275],{"class":113},[35,67053,1675],{"class":109},[35,67055,67056],{"class":234},"\"Natural transition frequency f0: ",[35,67058,67059],{"class":245},"{BLOCH_F0_GHZ",[35,67061,67062],{"class":109},":.6f",[35,67064,1697],{"class":245},[35,67066,67067],{"class":234}," GHz\"",[35,67069,384],{"class":113},[35,67071,67073,67075,67077,67079,67082,67084,67087,67089,67091,67093],{"class":37,"line":67072},473,[35,67074,6637],{"class":241},[35,67076,275],{"class":113},[35,67078,1675],{"class":109},[35,67080,67081],{"class":234},"\"Resonant drive frequency: ",[35,67083,1681],{"class":245},[35,67085,67086],{"class":113},"resonant_drive_ghz",[35,67088,67062],{"class":109},[35,67090,1697],{"class":245},[35,67092,67067],{"class":234},[35,67094,384],{"class":113},[35,67096,67098,67100,67102,67104,67107,67109,67112,67114,67116,67118],{"class":37,"line":67097},474,[35,67099,6637],{"class":241},[35,67101,275],{"class":113},[35,67103,1675],{"class":109},[35,67105,67106],{"class":234},"\"Detuned drive frequency: ",[35,67108,1681],{"class":245},[35,67110,67111],{"class":113},"detuned_drive_ghz",[35,67113,67062],{"class":109},[35,67115,1697],{"class":245},[35,67117,67067],{"class":234},[35,67119,384],{"class":113},[35,67121,67123,67125,67127,67129,67132,67135,67138,67140,67143],{"class":37,"line":67122},475,[35,67124,6637],{"class":241},[35,67126,275],{"class":113},[35,67128,1675],{"class":109},[35,67130,67131],{"class":234},"\"On-resonance Rabi rate: ",[35,67133,67134],{"class":245},"{BLOCH_OMEGA_ONRESONANCE_MHZ",[35,67136,67137],{"class":109},":.3f",[35,67139,1697],{"class":245},[35,67141,67142],{"class":234}," MHz\"",[35,67144,384],{"class":113},[35,67146,67148,67150,67152,67154,67157,67159,67162,67164,67166,67169],{"class":37,"line":67147},476,[35,67149,6637],{"class":241},[35,67151,275],{"class":113},[35,67153,1675],{"class":109},[35,67155,67156],{"class":234},"\"Plotted duration: ",[35,67158,1681],{"class":245},[35,67160,67161],{"class":113},"bloch_duration_ns",[35,67163,67137],{"class":109},[35,67165,1697],{"class":245},[35,67167,67168],{"class":234}," ns\"",[35,67170,384],{"class":113},[35,67172,67174,67176],{"class":37,"line":67173},477,[35,67175,6637],{"class":241},[35,67177,903],{"class":113},[35,67179,67181],{"class":37,"line":67180},478,[35,67182,67183],{"class":234},"        \"Resonant control axis n_res: \"\n",[35,67185,67187,67190,67193,67195,67198,67200,67202,67204,67206],{"class":37,"line":67186},479,[35,67188,67189],{"class":109},"        f",[35,67191,67192],{"class":234},"\"(",[35,67194,1681],{"class":245},[35,67196,67197],{"class":113},"resonant_axis[",[35,67199,5312],{"class":245},[35,67201,16788],{"class":113},[35,67203,67062],{"class":109},[35,67205,1697],{"class":245},[35,67207,67208],{"class":234},", \"\n",[35,67210,67212,67214,67216,67218,67220,67222,67224,67226,67228],{"class":37,"line":67211},480,[35,67213,67189],{"class":109},[35,67215,2986],{"class":234},[35,67217,1681],{"class":245},[35,67219,67197],{"class":113},[35,67221,500],{"class":245},[35,67223,16788],{"class":113},[35,67225,67062],{"class":109},[35,67227,1697],{"class":245},[35,67229,67208],{"class":234},[35,67231,67233,67235,67237,67239,67241,67243,67245,67247,67249],{"class":37,"line":67232},481,[35,67234,67189],{"class":109},[35,67236,2986],{"class":234},[35,67238,1681],{"class":245},[35,67240,67197],{"class":113},[35,67242,460],{"class":245},[35,67244,16788],{"class":113},[35,67246,67062],{"class":109},[35,67248,1697],{"class":245},[35,67250,67251],{"class":234},")\"\n",[35,67253,67255],{"class":37,"line":67254},482,[35,67256,6280],{"class":113},[35,67258,67260,67262],{"class":37,"line":67259},483,[35,67261,6637],{"class":241},[35,67263,903],{"class":113},[35,67265,67267],{"class":37,"line":67266},484,[35,67268,67269],{"class":234},"        \"Detuned control axis n_det: \"\n",[35,67271,67273,67275,67277,67279,67282,67284,67286,67288,67290],{"class":37,"line":67272},485,[35,67274,67189],{"class":109},[35,67276,67192],{"class":234},[35,67278,1681],{"class":245},[35,67280,67281],{"class":113},"detuned_axis[",[35,67283,5312],{"class":245},[35,67285,16788],{"class":113},[35,67287,67062],{"class":109},[35,67289,1697],{"class":245},[35,67291,67208],{"class":234},[35,67293,67295,67297,67299,67301,67303,67305,67307,67309,67311],{"class":37,"line":67294},486,[35,67296,67189],{"class":109},[35,67298,2986],{"class":234},[35,67300,1681],{"class":245},[35,67302,67281],{"class":113},[35,67304,500],{"class":245},[35,67306,16788],{"class":113},[35,67308,67062],{"class":109},[35,67310,1697],{"class":245},[35,67312,67208],{"class":234},[35,67314,67316,67318,67320,67322,67324,67326,67328,67330,67332],{"class":37,"line":67315},487,[35,67317,67189],{"class":109},[35,67319,2986],{"class":234},[35,67321,1681],{"class":245},[35,67323,67281],{"class":113},[35,67325,460],{"class":245},[35,67327,16788],{"class":113},[35,67329,67062],{"class":109},[35,67331,1697],{"class":245},[35,67333,67251],{"class":234},[35,67335,67337],{"class":37,"line":67336},488,[35,67338,6280],{"class":113},[35,67340,67342,67344],{"class":37,"line":67341},489,[35,67343,6637],{"class":241},[35,67345,903],{"class":113},[35,67347,67349],{"class":37,"line":67348},490,[35,67350,67351],{"class":234},"        \"Resonant final excited-state probability: \"\n",[35,67353,67355,67357,67359,67361,67363,67366,67368,67370,67372,67374,67376,67378],{"class":37,"line":67354},491,[35,67356,67189],{"class":109},[35,67358,2986],{"class":234},[35,67360,1681],{"class":245},[35,67362,66040],{"class":113},[35,67364,67365],{"class":234},"'excited_probability'",[35,67367,1997],{"class":113},[35,67369,564],{"class":241},[35,67371,500],{"class":245},[35,67373,16788],{"class":113},[35,67375,67062],{"class":109},[35,67377,1697],{"class":245},[35,67379,3368],{"class":234},[35,67381,67383],{"class":37,"line":67382},492,[35,67384,6280],{"class":113},[35,67386,67388,67390],{"class":37,"line":67387},493,[35,67389,6637],{"class":241},[35,67391,903],{"class":113},[35,67393,67395],{"class":37,"line":67394},494,[35,67396,67397],{"class":234},"        \"Detuned final excited-state probability: \"\n",[35,67399,67401,67403,67405,67407,67409,67411,67413,67415,67417,67419,67421,67423],{"class":37,"line":67400},495,[35,67402,67189],{"class":109},[35,67404,2986],{"class":234},[35,67406,1681],{"class":245},[35,67408,66093],{"class":113},[35,67410,67365],{"class":234},[35,67412,1997],{"class":113},[35,67414,564],{"class":241},[35,67416,500],{"class":245},[35,67418,16788],{"class":113},[35,67420,67062],{"class":109},[35,67422,1697],{"class":245},[35,67424,3368],{"class":234},[35,67426,67428],{"class":37,"line":67427},496,[35,67429,6280],{"class":113},[35,67431,67433,67435],{"class":37,"line":67432},497,[35,67434,6637],{"class":241},[35,67436,903],{"class":113},[35,67438,67440],{"class":37,"line":67439},498,[35,67441,67442],{"class":234},"        \"Detuned generalized Rabi frequency: \"\n",[35,67444,67446,67448,67450,67452,67454,67457,67459,67461,67463],{"class":37,"line":67445},499,[35,67447,67189],{"class":109},[35,67449,2986],{"class":234},[35,67451,1681],{"class":245},[35,67453,66093],{"class":113},[35,67455,67456],{"class":234},"'generalized_rabi_mhz'",[35,67458,16788],{"class":113},[35,67460,67062],{"class":109},[35,67462,1697],{"class":245},[35,67464,67465],{"class":234}," MHz\"\n",[35,67467,67469],{"class":37,"line":67468},500,[35,67470,6280],{"class":113},[4995,67472],{"alt":5001,"src":67473},"\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-02.webp",[25,67475,67478],{"className":67476,"code":67477,"language":4991,"meta":30},[4989],"Bloch-sphere trajectory summary\nNatural transition frequency f0: 5.000000 GHz\nResonant drive frequency: 5.000000 GHz\nDetuned drive frequency: 5.040000 GHz\nOn-resonance Rabi rate: 20.000 MHz\nPlotted duration: 25.000 ns\nResonant control axis n_res: (1.000000, 0.000000, 0.000000)\nDetuned control axis n_det: (0.447214, 0.000000, 0.894427)\nResonant final excited-state probability: 1.000000\nDetuned final excited-state probability: 0.026264\nDetuned generalized Rabi frequency: 44.721360 MHz\n",[32,67479,67477],{"__ignoreMap":30},[25,67481,67483],{"className":27,"code":67482,"language":29,"meta":30,"style":30},"# --- High-resolution 3D surface with colormap & legend  ---\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# Controls for surface quality and appearance\nUPSAMPLE_FACTOR: int = 2           # 2=~4× more faces; try 3 for even smoother\nSURFACE_CMAP: str = 'inferno'      # e.g., 'viridis', 'plasma', 'magma', 'inferno'\nZ_MIN, Z_MAX = 0.0, 1.0            # clamp\u002Fnormalize expected Pe range\n\n# --- Bilinear upsampling (SciPy-free) ---\n# 1) Upsample frequency axis for each fixed τ\nfreq_hi = np.linspace(freq_ghz.min(), freq_ghz.max(),\n                      UPSAMPLE_FACTOR * (len(freq_ghz) - 1) + 1)\nPe_tau_f = np.empty((len(dur_ns), len(freq_hi)), dtype=float)\nfor i in range(len(dur_ns)):\n    Pe_tau_f[i, :] = np.interp(freq_hi, freq_ghz, Pe[i, :])\n\n# 2) Upsample τ axis for each fixed frequency\ndur_hi = np.linspace(dur_ns.min(), dur_ns.max(),\n                     UPSAMPLE_FACTOR * (len(dur_ns) - 1) + 1)\nPe_hi = np.empty((len(dur_hi), len(freq_hi)), dtype=float)\nfor j in range(len(freq_hi)):\n    Pe_hi[:, j] = np.interp(dur_hi, dur_ns, Pe_tau_f[:, j])\n\n# Mesh for plotting\nF_hi, T_hi = np.meshgrid(freq_hi, dur_hi, indexing='xy')\n\n# --- Plot ---\nfig = plt.figure(figsize=(8, 6), dpi=200)\nax = fig.add_subplot(111, projection='3d')\n\nsurf = ax.plot_surface(\n    F_hi, T_hi, Pe_hi,\n    rstride=1, cstride=1,             # use every row\u002Fcolumn (after upsampling)\n    linewidth=0, antialiased=True,\n    cmap=SURFACE_CMAP,\n    vmin=Z_MIN, vmax=Z_MAX,\n)\n\nax.set_xlabel('f (GHz)')\nax.set_ylabel(r'$\\tau$ (ns)')\nax.set_zlabel(r'$P_e$')\nax.set_title('Rabi $P_e$ surface')\n\n# Colorbar (legend for the colormap)\ncb = fig.colorbar(surf, ax=ax, shrink=0.7, pad=0.08)\ncb.set_label(r'$P_e$')\n\nfig.tight_layout()\nplt.show()\n",[32,67484,67485,67490,67502,67512,67522,67526,67531,67547,67564,67585,67589,67594,67599,67623,67649,67680,67697,67712,67716,67721,67744,67770,67798,67815,67829,67833,67838,67860,67864,67869,67904,67933,67937,67951,67956,67980,68000,68011,68031,68035,68039,68052,68082,68097,68110,68114,68119,68158,68174,68178,68186],{"__ignoreMap":30},[35,67486,67487],{"class":37,"line":38},[35,67488,67489],{"class":41},"# --- High-resolution 3D surface with colormap & legend  ---\n",[35,67491,67492,67494,67497,67499],{"class":37,"line":45},[35,67493,128],{"class":109},[35,67495,67496],{"class":113}," mpl_toolkits.mplot3d ",[35,67498,110],{"class":109},[35,67500,67501],{"class":113}," Axes3D\n",[35,67503,67504,67506,67508,67510],{"class":37,"line":51},[35,67505,110],{"class":109},[35,67507,162],{"class":113},[35,67509,165],{"class":109},[35,67511,168],{"class":113},[35,67513,67514,67516,67518,67520],{"class":37,"line":57},[35,67515,110],{"class":109},[35,67517,189],{"class":113},[35,67519,165],{"class":109},[35,67521,194],{"class":113},[35,67523,67524],{"class":37,"line":63},[35,67525,103],{"emptyLinePlaceholder":102},[35,67527,67528],{"class":37,"line":69},[35,67529,67530],{"class":41},"# Controls for surface quality and appearance\n",[35,67532,67533,67536,67538,67540,67542,67544],{"class":37,"line":75},[35,67534,67535],{"class":245},"UPSAMPLE_FACTOR",[35,67537,282],{"class":113},[35,67539,285],{"class":241},[35,67541,308],{"class":241},[35,67543,424],{"class":245},[35,67545,67546],{"class":41},"           # 2=~4× more faces; try 3 for even smoother\n",[35,67548,67549,67552,67554,67556,67558,67561],{"class":37,"line":81},[35,67550,67551],{"class":245},"SURFACE_CMAP",[35,67553,282],{"class":113},[35,67555,1831],{"class":241},[35,67557,308],{"class":241},[35,67559,67560],{"class":234}," 'inferno'",[35,67562,67563],{"class":41},"      # e.g., 'viridis', 'plasma', 'magma', 'inferno'\n",[35,67565,67566,67569,67571,67574,67576,67578,67580,67582],{"class":37,"line":87},[35,67567,67568],{"class":245},"Z_MIN",[35,67570,288],{"class":113},[35,67572,67573],{"class":245},"Z_MAX",[35,67575,308],{"class":241},[35,67577,724],{"class":245},[35,67579,288],{"class":113},[35,67581,381],{"class":245},[35,67583,67584],{"class":41},"            # clamp\u002Fnormalize expected Pe range\n",[35,67586,67587],{"class":37,"line":93},[35,67588,103],{"emptyLinePlaceholder":102},[35,67590,67591],{"class":37,"line":99},[35,67592,67593],{"class":41},"# --- Bilinear upsampling (SciPy-free) ---\n",[35,67595,67596],{"class":37,"line":106},[35,67597,67598],{"class":41},"# 1) Upsample frequency axis for each fixed τ\n",[35,67600,67601,67604,67606,67608,67610,67613,67615,67618,67620],{"class":37,"line":117},[35,67602,67603],{"class":113},"freq_hi ",[35,67605,242],{"class":241},[35,67607,2412],{"class":113},[35,67609,62414],{"class":271},[35,67611,67612],{"class":113},"(freq_ghz.",[35,67614,570],{"class":271},[35,67616,67617],{"class":113},"(), freq_ghz.",[35,67619,554],{"class":271},[35,67621,67622],{"class":113},"(),\n",[35,67624,67625,67628,67630,67632,67634,67637,67639,67641,67643,67645,67647],{"class":37,"line":125},[35,67626,67627],{"class":245},"                      UPSAMPLE_FACTOR",[35,67629,463],{"class":241},[35,67631,373],{"class":113},[35,67633,2420],{"class":241},[35,67635,67636],{"class":113},"(freq_ghz) ",[35,67638,564],{"class":241},[35,67640,6079],{"class":245},[35,67642,427],{"class":113},[35,67644,445],{"class":241},[35,67646,6079],{"class":245},[35,67648,384],{"class":113},[35,67650,67651,67654,67656,67658,67661,67663,67665,67668,67670,67673,67675,67678],{"class":37,"line":139},[35,67652,67653],{"class":113},"Pe_tau_f ",[35,67655,242],{"class":241},[35,67657,2412],{"class":113},[35,67659,67660],{"class":271},"empty",[35,67662,4046],{"class":113},[35,67664,2420],{"class":241},[35,67666,67667],{"class":113},"(dur_ns), ",[35,67669,2420],{"class":241},[35,67671,67672],{"class":113},"(freq_hi)), ",[35,67674,6723],{"class":822},[35,67676,67677],{"class":241},"=float",[35,67679,384],{"class":113},[35,67681,67682,67684,67686,67688,67690,67692,67694],{"class":37,"line":152},[35,67683,1874],{"class":109},[35,67685,2958],{"class":113},[35,67687,1663],{"class":109},[35,67689,12746],{"class":241},[35,67691,275],{"class":113},[35,67693,2420],{"class":241},[35,67695,67696],{"class":113},"(dur_ns)):\n",[35,67698,67699,67702,67704,67706,67709],{"class":37,"line":157},[35,67700,67701],{"class":113},"    Pe_tau_f[i, :] ",[35,67703,242],{"class":241},[35,67705,2412],{"class":113},[35,67707,67708],{"class":271},"interp",[35,67710,67711],{"class":113},"(freq_hi, freq_ghz, Pe[i, :])\n",[35,67713,67714],{"class":37,"line":171},[35,67715,103],{"emptyLinePlaceholder":102},[35,67717,67718],{"class":37,"line":184},[35,67719,67720],{"class":41},"# 2) Upsample τ axis for each fixed frequency\n",[35,67722,67723,67726,67728,67730,67732,67735,67737,67740,67742],{"class":37,"line":197},[35,67724,67725],{"class":113},"dur_hi ",[35,67727,242],{"class":241},[35,67729,2412],{"class":113},[35,67731,62414],{"class":271},[35,67733,67734],{"class":113},"(dur_ns.",[35,67736,570],{"class":271},[35,67738,67739],{"class":113},"(), dur_ns.",[35,67741,554],{"class":271},[35,67743,67622],{"class":113},[35,67745,67746,67749,67751,67753,67755,67758,67760,67762,67764,67766,67768],{"class":37,"line":210},[35,67747,67748],{"class":245},"                     UPSAMPLE_FACTOR",[35,67750,463],{"class":241},[35,67752,373],{"class":113},[35,67754,2420],{"class":241},[35,67756,67757],{"class":113},"(dur_ns) ",[35,67759,564],{"class":241},[35,67761,6079],{"class":245},[35,67763,427],{"class":113},[35,67765,445],{"class":241},[35,67767,6079],{"class":245},[35,67769,384],{"class":113},[35,67771,67772,67775,67777,67779,67781,67783,67785,67788,67790,67792,67794,67796],{"class":37,"line":223},[35,67773,67774],{"class":113},"Pe_hi ",[35,67776,242],{"class":241},[35,67778,2412],{"class":113},[35,67780,67660],{"class":271},[35,67782,4046],{"class":113},[35,67784,2420],{"class":241},[35,67786,67787],{"class":113},"(dur_hi), ",[35,67789,2420],{"class":241},[35,67791,67672],{"class":113},[35,67793,6723],{"class":822},[35,67795,67677],{"class":241},[35,67797,384],{"class":113},[35,67799,67800,67802,67804,67806,67808,67810,67812],{"class":37,"line":228},[35,67801,1874],{"class":109},[35,67803,13198],{"class":113},[35,67805,1663],{"class":109},[35,67807,12746],{"class":241},[35,67809,275],{"class":113},[35,67811,2420],{"class":241},[35,67813,67814],{"class":113},"(freq_hi)):\n",[35,67816,67817,67820,67822,67824,67826],{"class":37,"line":249},[35,67818,67819],{"class":113},"    Pe_hi[:, j] ",[35,67821,242],{"class":241},[35,67823,2412],{"class":113},[35,67825,67708],{"class":271},[35,67827,67828],{"class":113},"(dur_hi, dur_ns, Pe_tau_f[:, j])\n",[35,67830,67831],{"class":37,"line":254},[35,67832,103],{"emptyLinePlaceholder":102},[35,67834,67835],{"class":37,"line":260},[35,67836,67837],{"class":41},"# Mesh for plotting\n",[35,67839,67840,67843,67845,67847,67849,67852,67854,67856,67858],{"class":37,"line":265},[35,67841,67842],{"class":113},"F_hi, T_hi ",[35,67844,242],{"class":241},[35,67846,2412],{"class":113},[35,67848,62139],{"class":271},[35,67850,67851],{"class":113},"(freq_hi, dur_hi, ",[35,67853,62145],{"class":822},[35,67855,242],{"class":241},[35,67857,62150],{"class":234},[35,67859,384],{"class":113},[35,67861,67862],{"class":37,"line":326},[35,67863,103],{"emptyLinePlaceholder":102},[35,67865,67866],{"class":37,"line":332},[35,67867,67868],{"class":41},"# --- Plot ---\n",[35,67870,67871,67874,67876,67878,67880,67882,67884,67886,67888,67890,67892,67894,67896,67898,67900,67902],{"class":37,"line":338},[35,67872,67873],{"class":113},"fig ",[35,67875,242],{"class":241},[35,67877,62691],{"class":113},[35,67879,2381],{"class":271},[35,67881,275],{"class":113},[35,67883,2386],{"class":822},[35,67885,242],{"class":241},[35,67887,275],{"class":113},[35,67889,40225],{"class":245},[35,67891,288],{"class":113},[35,67893,2398],{"class":245},[35,67895,685],{"class":113},[35,67897,2767],{"class":822},[35,67899,242],{"class":241},[35,67901,2772],{"class":245},[35,67903,384],{"class":113},[35,67905,67906,67909,67911,67913,67916,67918,67921,67923,67926,67928,67931],{"class":37,"line":344},[35,67907,67908],{"class":113},"ax ",[35,67910,242],{"class":241},[35,67912,62747],{"class":113},[35,67914,67915],{"class":271},"add_subplot",[35,67917,275],{"class":113},[35,67919,67920],{"class":245},"111",[35,67922,288],{"class":113},[35,67924,67925],{"class":822},"projection",[35,67927,242],{"class":241},[35,67929,67930],{"class":234},"'3d'",[35,67932,384],{"class":113},[35,67934,67935],{"class":37,"line":349},[35,67936,103],{"emptyLinePlaceholder":102},[35,67938,67939,67942,67944,67946,67949],{"class":37,"line":367},[35,67940,67941],{"class":113},"surf ",[35,67943,242],{"class":241},[35,67945,62721],{"class":113},[35,67947,67948],{"class":271},"plot_surface",[35,67950,903],{"class":113},[35,67952,67953],{"class":37,"line":387},[35,67954,67955],{"class":113},"    F_hi, T_hi, Pe_hi,\n",[35,67957,67958,67961,67963,67965,67967,67970,67972,67974,67977],{"class":37,"line":404},[35,67959,67960],{"class":822},"    rstride",[35,67962,242],{"class":241},[35,67964,500],{"class":245},[35,67966,288],{"class":113},[35,67968,67969],{"class":822},"cstride",[35,67971,242],{"class":241},[35,67973,500],{"class":245},[35,67975,67976],{"class":113},",             ",[35,67978,67979],{"class":41},"# use every row\u002Fcolumn (after upsampling)\n",[35,67981,67982,67985,67987,67989,67991,67994,67996,67998],{"class":37,"line":434},[35,67983,67984],{"class":822},"    linewidth",[35,67986,242],{"class":241},[35,67988,5312],{"class":245},[35,67990,288],{"class":113},[35,67992,67993],{"class":822},"antialiased",[35,67995,242],{"class":241},[35,67997,828],{"class":245},[35,67999,912],{"class":113},[35,68001,68002,68005,68007,68009],{"class":37,"line":474},[35,68003,68004],{"class":822},"    cmap",[35,68006,242],{"class":241},[35,68008,67551],{"class":245},[35,68010,912],{"class":113},[35,68012,68013,68016,68018,68020,68022,68025,68027,68029],{"class":37,"line":546},[35,68014,68015],{"class":822},"    vmin",[35,68017,242],{"class":241},[35,68019,67568],{"class":245},[35,68021,288],{"class":113},[35,68023,68024],{"class":822},"vmax",[35,68026,242],{"class":241},[35,68028,67573],{"class":245},[35,68030,912],{"class":113},[35,68032,68033],{"class":37,"line":584},[35,68034,384],{"class":113},[35,68036,68037],{"class":37,"line":589},[35,68038,103],{"emptyLinePlaceholder":102},[35,68040,68041,68043,68045,68047,68050],{"class":37,"line":594},[35,68042,62777],{"class":113},[35,68044,62780],{"class":271},[35,68046,275],{"class":113},[35,68048,68049],{"class":234},"'f (GHz)'",[35,68051,384],{"class":113},[35,68053,68054,68056,68058,68060,68062,68065,68068,68071,68073,68075,68077,68080],{"class":37,"line":643},[35,68055,62777],{"class":113},[35,68057,62794],{"class":271},[35,68059,275],{"class":113},[35,68061,6778],{"class":109},[35,68063,68064],{"class":3409},"'$",[35,68066,68067],{"class":241},"\\t",[35,68069,68070],{"class":3409},"au$ ",[35,68072,275],{"class":245},[35,68074,53156],{"class":3409},[35,68076,1694],{"class":245},[35,68078,68079],{"class":3409},"'",[35,68081,384],{"class":113},[35,68083,68084,68086,68089,68091,68093,68095],{"class":37,"line":648},[35,68085,62777],{"class":113},[35,68087,68088],{"class":271},"set_zlabel",[35,68090,275],{"class":113},[35,68092,6778],{"class":109},[35,68094,62770],{"class":3409},[35,68096,384],{"class":113},[35,68098,68099,68101,68103,68105,68108],{"class":37,"line":654},[35,68100,62777],{"class":113},[35,68102,62808],{"class":271},[35,68104,275],{"class":113},[35,68106,68107],{"class":234},"'Rabi $P_e$ surface'",[35,68109,384],{"class":113},[35,68111,68112],{"class":37,"line":660},[35,68113,103],{"emptyLinePlaceholder":102},[35,68115,68116],{"class":37,"line":665},[35,68117,68118],{"class":41},"# Colorbar (legend for the colormap)\n",[35,68120,68121,68123,68125,68127,68129,68132,68134,68136,68138,68141,68143,68146,68148,68151,68153,68156],{"class":37,"line":692},[35,68122,62742],{"class":113},[35,68124,242],{"class":241},[35,68126,62747],{"class":113},[35,68128,62750],{"class":271},[35,68130,68131],{"class":113},"(surf, ",[35,68133,62756],{"class":822},[35,68135,242],{"class":241},[35,68137,62761],{"class":113},[35,68139,68140],{"class":822},"shrink",[35,68142,242],{"class":241},[35,68144,68145],{"class":245},"0.7",[35,68147,288],{"class":113},[35,68149,68150],{"class":822},"pad",[35,68152,242],{"class":241},[35,68154,68155],{"class":245},"0.08",[35,68157,384],{"class":113},[35,68159,68160,68163,68166,68168,68170,68172],{"class":37,"line":714},[35,68161,68162],{"class":113},"cb.",[35,68164,68165],{"class":271},"set_label",[35,68167,275],{"class":113},[35,68169,6778],{"class":109},[35,68171,62770],{"class":3409},[35,68173,384],{"class":113},[35,68175,68176],{"class":37,"line":729},[35,68177,103],{"emptyLinePlaceholder":102},[35,68179,68180,68182,68184],{"class":37,"line":737},[35,68181,62830],{"class":113},[35,68183,2740],{"class":271},[35,68185,2101],{"class":113},[35,68187,68188,68190,68192],{"class":37,"line":768},[35,68189,2378],{"class":113},[35,68191,2792],{"class":271},[35,68193,2101],{"class":113},[4995,68195],{"alt":68196,"src":68197},"Output 3 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-03.webp",[25,68199,68201],{"className":27,"code":68200,"language":29,"meta":30,"style":30},"# @title Cross-sections\n# Slicing controls\nF_CROSS_GHZ: float = 5.000  # drive frequency slice [GHz]\nT_CROSS_NS: float = 100.0   # duration slice [ns]\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef _nearest_index(vec: np.ndarray, value: float, name: str) -> int:\n    \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n    vmin, vmax = float(vec.min()), float(vec.max())\n    if value \u003C vmin or value > vmax:\n        print(f\"[Note] {name}={value} is outside sweep; clamping to [{vmin}, {vmax}].\")\n        value = np.clip(value, vmin, vmax)\n    return int(np.argmin(np.abs(vec - value)))\n\n# --- Slice at f = F_CROSS_GHZ ---\nj = _nearest_index(freq_ghz, F_CROSS_GHZ, \"F_CROSS_GHZ\")\nPe_vs_tau = Pe[:, j]\n\nfig, ax = plt.subplots(figsize=(6, 3.5))\nax.plot(dur_ns, Pe_vs_tau, color='blue')\nax.set_xlabel(r'$\\tau$ (ns)')\nax.set_ylabel(r'$P_e$')\n# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\nax.set_title(r'$P_e(\\tau)$ at f = ' + f'{freq_ghz[j]:.6f} GHz')\nax.set_ylim(0.0, 1.0)\nfig.tight_layout()\nplt.show()\n\n# --- Slice at τ = T_CROSS_NS ---\ni = _nearest_index(dur_ns, T_CROSS_NS, \"T_CROSS_NS\")\nPe_vs_f = Pe[i, :]\n\nfig, ax = plt.subplots(figsize=(6, 3.5))\nax.plot(freq_ghz, Pe_vs_f, color='blue')\nax.set_xlabel('f (GHz)')\nax.set_ylabel(r'$P_e$')\nax.set_title(r'$P_e(f)$ at $\\tau$ = ' + f'{dur_ns[i]:.3f} ns')\nax.set_ylim(0.0, 1.0)\nfig.tight_layout()\nplt.show()\n",[32,68202,68203,68208,68213,68228,68244,68248,68258,68268,68272,68307,68312,68337,68359,68408,68422,68445,68449,68454,68475,68485,68489,68516,68534,68560,68574,68579,68624,68641,68649,68657,68661,68666,68687,68697,68701,68727,68744,68756,68770,68816,68832,68840],{"__ignoreMap":30},[35,68204,68205],{"class":37,"line":38},[35,68206,68207],{"class":41},"# @title Cross-sections\n",[35,68209,68210],{"class":37,"line":45},[35,68211,68212],{"class":41},"# Slicing controls\n",[35,68214,68215,68217,68219,68221,68223,68225],{"class":37,"line":51},[35,68216,61265],{"class":245},[35,68218,282],{"class":113},[35,68220,305],{"class":241},[35,68222,308],{"class":241},[35,68224,61757],{"class":245},[35,68226,68227],{"class":41},"  # drive frequency slice [GHz]\n",[35,68229,68230,68232,68234,68236,68238,68241],{"class":37,"line":57},[35,68231,61364],{"class":245},[35,68233,282],{"class":113},[35,68235,305],{"class":241},[35,68237,308],{"class":241},[35,68239,68240],{"class":245}," 100.0",[35,68242,68243],{"class":41},"   # duration slice [ns]\n",[35,68245,68246],{"class":37,"line":63},[35,68247,103],{"emptyLinePlaceholder":102},[35,68249,68250,68252,68254,68256],{"class":37,"line":69},[35,68251,110],{"class":109},[35,68253,162],{"class":113},[35,68255,165],{"class":109},[35,68257,168],{"class":113},[35,68259,68260,68262,68264,68266],{"class":37,"line":75},[35,68261,110],{"class":109},[35,68263,189],{"class":113},[35,68265,165],{"class":109},[35,68267,194],{"class":113},[35,68269,68270],{"class":37,"line":81},[35,68271,103],{"emptyLinePlaceholder":102},[35,68273,68274,68276,68279,68281,68284,68286,68289,68291,68293,68295,68297,68299,68301,68303,68305],{"class":37,"line":87},[35,68275,268],{"class":109},[35,68277,68278],{"class":271}," _nearest_index",[35,68280,275],{"class":113},[35,68282,68283],{"class":278},"vec",[35,68285,1397],{"class":113},[35,68287,68288],{"class":278},"value",[35,68290,282],{"class":113},[35,68292,305],{"class":241},[35,68294,288],{"class":113},[35,68296,26518],{"class":278},[35,68298,282],{"class":113},[35,68300,1831],{"class":241},[35,68302,636],{"class":113},[35,68304,285],{"class":241},[35,68306,364],{"class":113},[35,68308,68309],{"class":37,"line":93},[35,68310,68311],{"class":234},"    \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n",[35,68313,68314,68317,68319,68321,68324,68326,68329,68331,68333,68335],{"class":37,"line":99},[35,68315,68316],{"class":113},"    vmin, vmax ",[35,68318,242],{"class":241},[35,68320,1505],{"class":241},[35,68322,68323],{"class":113},"(vec.",[35,68325,570],{"class":271},[35,68327,68328],{"class":113},"()), ",[35,68330,305],{"class":241},[35,68332,68323],{"class":113},[35,68334,554],{"class":271},[35,68336,1456],{"class":113},[35,68338,68339,68341,68344,68346,68349,68352,68354,68356],{"class":37,"line":106},[35,68340,352],{"class":109},[35,68342,68343],{"class":113}," value ",[35,68345,10205],{"class":241},[35,68347,68348],{"class":113}," vmin ",[35,68350,68351],{"class":109},"or",[35,68353,68343],{"class":113},[35,68355,1497],{"class":241},[35,68357,68358],{"class":113}," vmax:\n",[35,68360,68361,68364,68366,68368,68371,68373,68375,68377,68379,68381,68383,68385,68388,68390,68393,68395,68397,68399,68401,68403,68406],{"class":37,"line":117},[35,68362,68363],{"class":241},"        print",[35,68365,275],{"class":113},[35,68367,1675],{"class":109},[35,68369,68370],{"class":234},"\"[Note] ",[35,68372,1681],{"class":245},[35,68374,26518],{"class":113},[35,68376,1697],{"class":245},[35,68378,242],{"class":234},[35,68380,1681],{"class":245},[35,68382,68288],{"class":113},[35,68384,1697],{"class":245},[35,68386,68387],{"class":234}," is outside sweep; clamping to [",[35,68389,1681],{"class":245},[35,68391,68392],{"class":113},"vmin",[35,68394,1697],{"class":245},[35,68396,288],{"class":234},[35,68398,1681],{"class":245},[35,68400,68024],{"class":113},[35,68402,1697],{"class":245},[35,68404,68405],{"class":234},"].\"",[35,68407,384],{"class":113},[35,68409,68410,68413,68415,68417,68419],{"class":37,"line":125},[35,68411,68412],{"class":113},"        value ",[35,68414,242],{"class":241},[35,68416,2412],{"class":113},[35,68418,62386],{"class":271},[35,68420,68421],{"class":113},"(value, vmin, vmax)\n",[35,68423,68424,68426,68428,68430,68433,68435,68437,68440,68442],{"class":37,"line":139},[35,68425,549],{"class":109},[35,68427,1434],{"class":241},[35,68429,64321],{"class":113},[35,68431,68432],{"class":271},"argmin",[35,68434,64321],{"class":113},[35,68436,2543],{"class":271},[35,68438,68439],{"class":113},"(vec ",[35,68441,564],{"class":241},[35,68443,68444],{"class":113}," value)))\n",[35,68446,68447],{"class":37,"line":152},[35,68448,103],{"emptyLinePlaceholder":102},[35,68450,68451],{"class":37,"line":157},[35,68452,68453],{"class":41},"# --- Slice at f = F_CROSS_GHZ ---\n",[35,68455,68456,68459,68461,68463,68466,68468,68470,68473],{"class":37,"line":171},[35,68457,68458],{"class":113},"j ",[35,68460,242],{"class":241},[35,68462,68278],{"class":271},[35,68464,68465],{"class":113},"(freq_ghz, ",[35,68467,61265],{"class":245},[35,68469,288],{"class":113},[35,68471,68472],{"class":234},"\"F_CROSS_GHZ\"",[35,68474,384],{"class":113},[35,68476,68477,68480,68482],{"class":37,"line":184},[35,68478,68479],{"class":113},"Pe_vs_tau ",[35,68481,242],{"class":241},[35,68483,68484],{"class":113}," Pe[:, j]\n",[35,68486,68487],{"class":37,"line":197},[35,68488,103],{"emptyLinePlaceholder":102},[35,68490,68491,68493,68495,68497,68499,68501,68503,68505,68507,68509,68511,68514],{"class":37,"line":210},[35,68492,62686],{"class":113},[35,68494,242],{"class":241},[35,68496,62691],{"class":113},[35,68498,62694],{"class":271},[35,68500,275],{"class":113},[35,68502,2386],{"class":822},[35,68504,242],{"class":241},[35,68506,275],{"class":113},[35,68508,2398],{"class":245},[35,68510,288],{"class":113},[35,68512,68513],{"class":245},"3.5",[35,68515,2401],{"class":113},[35,68517,68518,68520,68522,68525,68527,68529,68532],{"class":37,"line":223},[35,68519,62777],{"class":113},[35,68521,29869],{"class":271},[35,68523,68524],{"class":113},"(dur_ns, Pe_vs_tau, ",[35,68526,2576],{"class":822},[35,68528,242],{"class":241},[35,68530,68531],{"class":234},"'blue'",[35,68533,384],{"class":113},[35,68535,68536,68538,68540,68542,68544,68546,68548,68550,68552,68554,68556,68558],{"class":37,"line":228},[35,68537,62777],{"class":113},[35,68539,62780],{"class":271},[35,68541,275],{"class":113},[35,68543,6778],{"class":109},[35,68545,68064],{"class":3409},[35,68547,68067],{"class":241},[35,68549,68070],{"class":3409},[35,68551,275],{"class":245},[35,68553,53156],{"class":3409},[35,68555,1694],{"class":245},[35,68557,68079],{"class":3409},[35,68559,384],{"class":113},[35,68561,68562,68564,68566,68568,68570,68572],{"class":37,"line":249},[35,68563,62777],{"class":113},[35,68565,62794],{"class":271},[35,68567,275],{"class":113},[35,68569,6778],{"class":109},[35,68571,62770],{"class":3409},[35,68573,384],{"class":113},[35,68575,68576],{"class":37,"line":254},[35,68577,68578],{"class":41},"# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\n",[35,68580,68581,68583,68585,68587,68589,68592,68594,68596,68599,68601,68604,68606,68608,68610,68612,68615,68617,68619,68622],{"class":37,"line":260},[35,68582,62777],{"class":113},[35,68584,62808],{"class":271},[35,68586,275],{"class":113},[35,68588,6778],{"class":109},[35,68590,68591],{"class":3409},"'$P_e",[35,68593,275],{"class":245},[35,68595,68067],{"class":241},[35,68597,68598],{"class":3409},"au",[35,68600,1694],{"class":245},[35,68602,68603],{"class":3409},"$ at f = '",[35,68605,415],{"class":241},[35,68607,2159],{"class":109},[35,68609,68079],{"class":234},[35,68611,1681],{"class":245},[35,68613,68614],{"class":113},"freq_ghz[j]",[35,68616,67062],{"class":109},[35,68618,1697],{"class":245},[35,68620,68621],{"class":234}," GHz'",[35,68623,384],{"class":113},[35,68625,68626,68628,68631,68633,68635,68637,68639],{"class":37,"line":265},[35,68627,62777],{"class":113},[35,68629,68630],{"class":271},"set_ylim",[35,68632,275],{"class":113},[35,68634,376],{"class":245},[35,68636,288],{"class":113},[35,68638,381],{"class":245},[35,68640,384],{"class":113},[35,68642,68643,68645,68647],{"class":37,"line":326},[35,68644,62830],{"class":113},[35,68646,2740],{"class":271},[35,68648,2101],{"class":113},[35,68650,68651,68653,68655],{"class":37,"line":332},[35,68652,2378],{"class":113},[35,68654,2792],{"class":271},[35,68656,2101],{"class":113},[35,68658,68659],{"class":37,"line":338},[35,68660,103],{"emptyLinePlaceholder":102},[35,68662,68663],{"class":37,"line":344},[35,68664,68665],{"class":41},"# --- Slice at τ = T_CROSS_NS ---\n",[35,68667,68668,68671,68673,68675,68678,68680,68682,68685],{"class":37,"line":349},[35,68669,68670],{"class":113},"i ",[35,68672,242],{"class":241},[35,68674,68278],{"class":271},[35,68676,68677],{"class":113},"(dur_ns, ",[35,68679,61364],{"class":245},[35,68681,288],{"class":113},[35,68683,68684],{"class":234},"\"T_CROSS_NS\"",[35,68686,384],{"class":113},[35,68688,68689,68692,68694],{"class":37,"line":367},[35,68690,68691],{"class":113},"Pe_vs_f ",[35,68693,242],{"class":241},[35,68695,68696],{"class":113}," Pe[i, :]\n",[35,68698,68699],{"class":37,"line":387},[35,68700,103],{"emptyLinePlaceholder":102},[35,68702,68703,68705,68707,68709,68711,68713,68715,68717,68719,68721,68723,68725],{"class":37,"line":404},[35,68704,62686],{"class":113},[35,68706,242],{"class":241},[35,68708,62691],{"class":113},[35,68710,62694],{"class":271},[35,68712,275],{"class":113},[35,68714,2386],{"class":822},[35,68716,242],{"class":241},[35,68718,275],{"class":113},[35,68720,2398],{"class":245},[35,68722,288],{"class":113},[35,68724,68513],{"class":245},[35,68726,2401],{"class":113},[35,68728,68729,68731,68733,68736,68738,68740,68742],{"class":37,"line":434},[35,68730,62777],{"class":113},[35,68732,29869],{"class":271},[35,68734,68735],{"class":113},"(freq_ghz, Pe_vs_f, ",[35,68737,2576],{"class":822},[35,68739,242],{"class":241},[35,68741,68531],{"class":234},[35,68743,384],{"class":113},[35,68745,68746,68748,68750,68752,68754],{"class":37,"line":474},[35,68747,62777],{"class":113},[35,68749,62780],{"class":271},[35,68751,275],{"class":113},[35,68753,68049],{"class":234},[35,68755,384],{"class":113},[35,68757,68758,68760,68762,68764,68766,68768],{"class":37,"line":546},[35,68759,62777],{"class":113},[35,68761,62794],{"class":271},[35,68763,275],{"class":113},[35,68765,6778],{"class":109},[35,68767,62770],{"class":3409},[35,68769,384],{"class":113},[35,68771,68772,68774,68776,68778,68780,68782,68784,68786,68788,68791,68793,68796,68798,68800,68802,68804,68807,68809,68811,68814],{"class":37,"line":584},[35,68773,62777],{"class":113},[35,68775,62808],{"class":271},[35,68777,275],{"class":113},[35,68779,6778],{"class":109},[35,68781,68591],{"class":3409},[35,68783,275],{"class":245},[35,68785,1675],{"class":3409},[35,68787,1694],{"class":245},[35,68789,68790],{"class":3409},"$ at $",[35,68792,68067],{"class":241},[35,68794,68795],{"class":3409},"au$ = '",[35,68797,415],{"class":241},[35,68799,2159],{"class":109},[35,68801,68079],{"class":234},[35,68803,1681],{"class":245},[35,68805,68806],{"class":113},"dur_ns[i]",[35,68808,67137],{"class":109},[35,68810,1697],{"class":245},[35,68812,68813],{"class":234}," ns'",[35,68815,384],{"class":113},[35,68817,68818,68820,68822,68824,68826,68828,68830],{"class":37,"line":589},[35,68819,62777],{"class":113},[35,68821,68630],{"class":271},[35,68823,275],{"class":113},[35,68825,376],{"class":245},[35,68827,288],{"class":113},[35,68829,381],{"class":245},[35,68831,384],{"class":113},[35,68833,68834,68836,68838],{"class":37,"line":594},[35,68835,62830],{"class":113},[35,68837,2740],{"class":271},[35,68839,2101],{"class":113},[35,68841,68842,68844,68846],{"class":37,"line":643},[35,68843,2378],{"class":113},[35,68845,2792],{"class":271},[35,68847,2101],{"class":113},[4995,68849],{"alt":68850,"src":68851},"Output 4 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-04.webp",[4995,68853],{"alt":68854,"src":68855},"Output 5 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-05.webp",[25,68857,68859],{"className":27,"code":68858,"language":29,"meta":30,"style":30},"# @title FFT of Rabi oscillations along tau (controls + computation)\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# -------- Control knobs (FFT & display) --------\nFFT_PAD: int = 8                 # zero-padding factor (2, 4, 8, ...)\nWINDOW: str = \"hann\"             # \"hann\", \"boxcar\"\nDETREND_MEAN: bool = True        # subtract mean along τ before FFT\nAMP_MODE: str = \"magnitude\"      # \"magnitude\" | \"power\" | \"db\"\nNORM_MODE: str = \"global\"        # \"global\" | \"per_f\" (column-wise)\nFREQ_UNITS: str = \"MHz\"          # \"MHz\" or \"GHz\" for Fourier axis\n\n# -------- Helpers --------\ndef _make_window(n: int, kind: str) -> np.ndarray:\n    if kind.lower() == \"hann\":\n        return np.hanning(n)\n    return np.ones(n, dtype=float)  # boxcar\n\ndef rfft_spectrogram(\n    Pe: np.ndarray,\n    dur_ns: np.ndarray,\n    pad: int = 8,\n    window: str = \"hann\",\n    detrend_mean: bool = True,\n    amp_mode: str = \"magnitude\",\n    norm_mode: str = \"global\",\n    freq_units: str = \"MHz\",\n) -> tuple[np.ndarray, np.ndarray]:\n    \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n\n    Args:\n        Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n        dur_ns: τ grid [ns], length N_tau, evenly spaced.\n        pad: zero-padding factor (≥1).\n        window: \"hann\" or \"boxcar\".\n        detrend_mean: subtract column-wise mean before FFT.\n        amp_mode: \"magnitude\", \"power\", or \"db\".\n        norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n        freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n\n    Returns:\n        f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n        S:     2D nonnegative spectrogram (N_w, N_f).\n    \"\"\"\n    Pe = np.asarray(Pe, float)\n    dur_ns = np.asarray(dur_ns, float)\n    assert Pe.shape[0] == dur_ns.size, \"Pe must be shaped (N_tau, N_f).\"\n\n    # Uniform sampling assumption (true for np.linspace)\n    dt = float(dur_ns[1] - dur_ns[0]) * 1e-9               # [s]\n    n_tau = dur_ns.size\n    nfft = int(2 ** int(np.ceil(np.log2(n_tau))) * max(1, pad))\n    w = _make_window(n_tau, window)\n\n    # Prepare output\n    S_list = []\n    X = Pe.copy()\n    if detrend_mean:\n        X = X - X.mean(axis=0, keepdims=True)\n    X *= w[:, None]\n\n    # Real FFT along τ (axis=0)\n    Xf = np.fft.rfft(X, n=nfft, axis=0)                     # shape (N_w, N_f)\n    f_fft_hz = np.fft.rfftfreq(nfft, d=dt)                  # [Hz]\n\n    # Amplitude selection\n    if amp_mode.lower() == \"power\":\n        S = np.abs(Xf) ** 2\n    else:\n        S = np.abs(Xf)\n\n    # Normalization\n    if norm_mode.lower() == \"per_f\":\n        col_max = np.maximum(S.max(axis=0, keepdims=True), 1e-15)\n        S = S \u002F col_max\n    else:\n        S = S \u002F max(S.max(), 1e-15)\n\n    if amp_mode.lower() == \"db\":\n        S = 20.0 * np.log10(np.maximum(S, 1e-12))  # dB, clipped floor\n\n    # Units\n    if freq_units.upper() == \"GHz\":\n        f_fft = f_fft_hz * 1e-9\n    else:\n        f_fft = f_fft_hz * 1e-6\n\n    return f_fft, S\n\n# Compute spectrogram with current grid\nf_fft_axis, S_fft = rfft_spectrogram(\n    Pe, dur_ns,\n    pad=FFT_PAD,\n    window=WINDOW,\n    detrend_mean=DETREND_MEAN,\n    amp_mode=AMP_MODE,\n    norm_mode=NORM_MODE,\n    freq_units=FREQ_UNITS,\n)\nprint(\"FFT grid:\", S_fft.shape, \"| Fourier freq range:\", f_fft_axis[0], \"→\", f_fft_axis[-1], FREQ_UNITS)\n",[32,68860,68861,68866,68876,68886,68890,68895,68911,68927,68943,68959,68975,68991,68995,69000,69026,69044,69056,69077,69081,69090,69097,69104,69119,69134,69149,69164,69179,69194,69199,69204,69208,69212,69217,69222,69227,69232,69237,69242,69247,69252,69256,69260,69265,69270,69274,69291,69308,69328,69332,69337,69370,69380,69421,69433,69437,69442,69451,69466,69473,69510,69524,69528,69533,69568,69593,69597,69602,69620,69638,69644,69657,69661,69666,69684,69724,69738,69744,69766,69770,69787,69818,69822,69827,69846,69860,69866,69879,69883,69890,69894,69899,69910,69915,69925,69935,69945,69955,69965,69975,69979],{"__ignoreMap":30},[35,68862,68863],{"class":37,"line":38},[35,68864,68865],{"class":41},"# @title FFT of Rabi oscillations along tau (controls + computation)\n",[35,68867,68868,68870,68872,68874],{"class":37,"line":45},[35,68869,110],{"class":109},[35,68871,162],{"class":113},[35,68873,165],{"class":109},[35,68875,168],{"class":113},[35,68877,68878,68880,68882,68884],{"class":37,"line":51},[35,68879,110],{"class":109},[35,68881,189],{"class":113},[35,68883,165],{"class":109},[35,68885,194],{"class":113},[35,68887,68888],{"class":37,"line":57},[35,68889,103],{"emptyLinePlaceholder":102},[35,68891,68892],{"class":37,"line":63},[35,68893,68894],{"class":41},"# -------- Control knobs (FFT & display) --------\n",[35,68896,68897,68899,68901,68903,68905,68908],{"class":37,"line":69},[35,68898,61481],{"class":245},[35,68900,282],{"class":113},[35,68902,285],{"class":241},[35,68904,308],{"class":241},[35,68906,68907],{"class":245}," 8",[35,68909,68910],{"class":41},"                 # zero-padding factor (2, 4, 8, ...)\n",[35,68912,68913,68915,68917,68919,68921,68924],{"class":37,"line":75},[35,68914,61495],{"class":245},[35,68916,282],{"class":113},[35,68918,1831],{"class":241},[35,68920,308],{"class":241},[35,68922,68923],{"class":234}," \"hann\"",[35,68925,68926],{"class":41},"             # \"hann\", \"boxcar\"\n",[35,68928,68929,68931,68933,68935,68937,68940],{"class":37,"line":81},[35,68930,61510],{"class":245},[35,68932,282],{"class":113},[35,68934,6091],{"class":241},[35,68936,308],{"class":241},[35,68938,68939],{"class":245}," True",[35,68941,68942],{"class":41},"        # subtract mean along τ before FFT\n",[35,68944,68945,68947,68949,68951,68953,68956],{"class":37,"line":87},[35,68946,61524],{"class":245},[35,68948,282],{"class":113},[35,68950,1831],{"class":241},[35,68952,308],{"class":241},[35,68954,68955],{"class":234}," \"magnitude\"",[35,68957,68958],{"class":41},"      # \"magnitude\" | \"power\" | \"db\"\n",[35,68960,68961,68963,68965,68967,68969,68972],{"class":37,"line":93},[35,68962,61539],{"class":245},[35,68964,282],{"class":113},[35,68966,1831],{"class":241},[35,68968,308],{"class":241},[35,68970,68971],{"class":234}," \"global\"",[35,68973,68974],{"class":41},"        # \"global\" | \"per_f\" (column-wise)\n",[35,68976,68977,68979,68981,68983,68985,68988],{"class":37,"line":99},[35,68978,61554],{"class":245},[35,68980,282],{"class":113},[35,68982,1831],{"class":241},[35,68984,308],{"class":241},[35,68986,68987],{"class":234}," \"MHz\"",[35,68989,68990],{"class":41},"          # \"MHz\" or \"GHz\" for Fourier axis\n",[35,68992,68993],{"class":37,"line":106},[35,68994,103],{"emptyLinePlaceholder":102},[35,68996,68997],{"class":37,"line":117},[35,68998,68999],{"class":41},"# -------- Helpers --------\n",[35,69001,69002,69004,69007,69009,69011,69013,69015,69017,69020,69022,69024],{"class":37,"line":125},[35,69003,268],{"class":109},[35,69005,69006],{"class":271}," _make_window",[35,69008,275],{"class":113},[35,69010,291],{"class":278},[35,69012,282],{"class":113},[35,69014,285],{"class":241},[35,69016,288],{"class":113},[35,69018,69019],{"class":278},"kind",[35,69021,282],{"class":113},[35,69023,1831],{"class":241},[35,69025,61983],{"class":113},[35,69027,69028,69030,69033,69036,69038,69040,69042],{"class":37,"line":139},[35,69029,352],{"class":109},[35,69031,69032],{"class":113}," kind.",[35,69034,69035],{"class":271},"lower",[35,69037,2519],{"class":113},[35,69039,358],{"class":241},[35,69041,68923],{"class":234},[35,69043,364],{"class":113},[35,69045,69046,69048,69050,69053],{"class":37,"line":152},[35,69047,370],{"class":109},[35,69049,2412],{"class":113},[35,69051,69052],{"class":271},"hanning",[35,69054,69055],{"class":113},"(n)\n",[35,69057,69058,69060,69062,69065,69068,69070,69072,69074],{"class":37,"line":157},[35,69059,549],{"class":109},[35,69061,2412],{"class":113},[35,69063,69064],{"class":271},"ones",[35,69066,69067],{"class":113},"(n, ",[35,69069,6723],{"class":822},[35,69071,67677],{"class":241},[35,69073,27221],{"class":113},[35,69075,69076],{"class":41},"# boxcar\n",[35,69078,69079],{"class":37,"line":171},[35,69080,103],{"emptyLinePlaceholder":102},[35,69082,69083,69085,69088],{"class":37,"line":184},[35,69084,268],{"class":109},[35,69086,69087],{"class":271}," rfft_spectrogram",[35,69089,903],{"class":113},[35,69091,69092,69095],{"class":37,"line":197},[35,69093,69094],{"class":278},"    Pe",[35,69096,6047],{"class":113},[35,69098,69099,69102],{"class":37,"line":210},[35,69100,69101],{"class":278},"    dur_ns",[35,69103,6047],{"class":113},[35,69105,69106,69109,69111,69113,69115,69117],{"class":37,"line":223},[35,69107,69108],{"class":278},"    pad",[35,69110,282],{"class":113},[35,69112,285],{"class":241},[35,69114,308],{"class":241},[35,69116,68907],{"class":245},[35,69118,912],{"class":113},[35,69120,69121,69124,69126,69128,69130,69132],{"class":37,"line":228},[35,69122,69123],{"class":278},"    window",[35,69125,282],{"class":113},[35,69127,1831],{"class":241},[35,69129,308],{"class":241},[35,69131,68923],{"class":234},[35,69133,912],{"class":113},[35,69135,69136,69139,69141,69143,69145,69147],{"class":37,"line":249},[35,69137,69138],{"class":278},"    detrend_mean",[35,69140,282],{"class":113},[35,69142,6091],{"class":241},[35,69144,308],{"class":241},[35,69146,68939],{"class":245},[35,69148,912],{"class":113},[35,69150,69151,69154,69156,69158,69160,69162],{"class":37,"line":254},[35,69152,69153],{"class":278},"    amp_mode",[35,69155,282],{"class":113},[35,69157,1831],{"class":241},[35,69159,308],{"class":241},[35,69161,68955],{"class":234},[35,69163,912],{"class":113},[35,69165,69166,69169,69171,69173,69175,69177],{"class":37,"line":260},[35,69167,69168],{"class":278},"    norm_mode",[35,69170,282],{"class":113},[35,69172,1831],{"class":241},[35,69174,308],{"class":241},[35,69176,68971],{"class":234},[35,69178,912],{"class":113},[35,69180,69181,69184,69186,69188,69190,69192],{"class":37,"line":265},[35,69182,69183],{"class":278},"    freq_units",[35,69185,282],{"class":113},[35,69187,1831],{"class":241},[35,69189,308],{"class":241},[35,69191,68987],{"class":234},[35,69193,912],{"class":113},[35,69195,69196],{"class":37,"line":326},[35,69197,69198],{"class":113},") -> tuple[np.ndarray, np.ndarray]:\n",[35,69200,69201],{"class":37,"line":332},[35,69202,69203],{"class":234},"    \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n",[35,69205,69206],{"class":37,"line":338},[35,69207,103],{"emptyLinePlaceholder":102},[35,69209,69210],{"class":37,"line":344},[35,69211,6131],{"class":234},[35,69213,69214],{"class":37,"line":349},[35,69215,69216],{"class":234},"        Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n",[35,69218,69219],{"class":37,"line":367},[35,69220,69221],{"class":234},"        dur_ns: τ grid [ns], length N_tau, evenly spaced.\n",[35,69223,69224],{"class":37,"line":387},[35,69225,69226],{"class":234},"        pad: zero-padding factor (≥1).\n",[35,69228,69229],{"class":37,"line":404},[35,69230,69231],{"class":234},"        window: \"hann\" or \"boxcar\".\n",[35,69233,69234],{"class":37,"line":434},[35,69235,69236],{"class":234},"        detrend_mean: subtract column-wise mean before FFT.\n",[35,69238,69239],{"class":37,"line":474},[35,69240,69241],{"class":234},"        amp_mode: \"magnitude\", \"power\", or \"db\".\n",[35,69243,69244],{"class":37,"line":546},[35,69245,69246],{"class":234},"        norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n",[35,69248,69249],{"class":37,"line":584},[35,69250,69251],{"class":234},"        freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n",[35,69253,69254],{"class":37,"line":589},[35,69255,103],{"emptyLinePlaceholder":102},[35,69257,69258],{"class":37,"line":594},[35,69259,6190],{"class":234},[35,69261,69262],{"class":37,"line":643},[35,69263,69264],{"class":234},"        f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n",[35,69266,69267],{"class":37,"line":648},[35,69268,69269],{"class":234},"        S:     2D nonnegative spectrogram (N_w, N_f).\n",[35,69271,69272],{"class":37,"line":654},[35,69273,329],{"class":234},[35,69275,69276,69279,69281,69283,69285,69287,69289],{"class":37,"line":660},[35,69277,69278],{"class":113},"    Pe ",[35,69280,242],{"class":241},[35,69282,2412],{"class":113},[35,69284,64324],{"class":271},[35,69286,62389],{"class":113},[35,69288,305],{"class":241},[35,69290,384],{"class":113},[35,69292,69293,69296,69298,69300,69302,69304,69306],{"class":37,"line":665},[35,69294,69295],{"class":113},"    dur_ns ",[35,69297,242],{"class":241},[35,69299,2412],{"class":113},[35,69301,64324],{"class":271},[35,69303,68677],{"class":113},[35,69305,305],{"class":241},[35,69307,384],{"class":113},[35,69309,69310,69313,69316,69318,69320,69322,69325],{"class":37,"line":692},[35,69311,69312],{"class":109},"    assert",[35,69314,69315],{"class":113}," Pe.shape[",[35,69317,5312],{"class":245},[35,69319,238],{"class":113},[35,69321,358],{"class":241},[35,69323,69324],{"class":113}," dur_ns.size, ",[35,69326,69327],{"class":234},"\"Pe must be shaped (N_tau, N_f).\"\n",[35,69329,69330],{"class":37,"line":714},[35,69331,103],{"emptyLinePlaceholder":102},[35,69333,69334],{"class":37,"line":729},[35,69335,69336],{"class":41},"    # Uniform sampling assumption (true for np.linspace)\n",[35,69338,69339,69342,69344,69346,69349,69351,69353,69355,69358,69360,69363,69365,69367],{"class":37,"line":737},[35,69340,69341],{"class":113},"    dt ",[35,69343,242],{"class":241},[35,69345,1505],{"class":241},[35,69347,69348],{"class":113},"(dur_ns[",[35,69350,500],{"class":245},[35,69352,238],{"class":113},[35,69354,564],{"class":241},[35,69356,69357],{"class":113}," dur_ns[",[35,69359,5312],{"class":245},[35,69361,69362],{"class":113},"]) ",[35,69364,484],{"class":241},[35,69366,62357],{"class":245},[35,69368,69369],{"class":41},"               # [s]\n",[35,69371,69372,69375,69377],{"class":37,"line":768},[35,69373,69374],{"class":113},"    n_tau ",[35,69376,242],{"class":241},[35,69378,69379],{"class":113}," dur_ns.size\n",[35,69381,69382,69385,69387,69389,69391,69393,69396,69398,69400,69402,69404,69407,69410,69412,69414,69416,69418],{"class":37,"line":793},[35,69383,69384],{"class":113},"    nfft ",[35,69386,242],{"class":241},[35,69388,1434],{"class":241},[35,69390,275],{"class":113},[35,69392,460],{"class":245},[35,69394,69395],{"class":241}," **",[35,69397,1434],{"class":241},[35,69399,64321],{"class":113},[35,69401,787],{"class":271},[35,69403,64321],{"class":113},[35,69405,69406],{"class":271},"log2",[35,69408,69409],{"class":113},"(n_tau))) ",[35,69411,484],{"class":241},[35,69413,773],{"class":241},[35,69415,275],{"class":113},[35,69417,500],{"class":245},[35,69419,69420],{"class":113},", pad))\n",[35,69422,69423,69426,69428,69430],{"class":37,"line":798},[35,69424,69425],{"class":113},"    w ",[35,69427,242],{"class":241},[35,69429,69006],{"class":271},[35,69431,69432],{"class":113},"(n_tau, window)\n",[35,69434,69435],{"class":37,"line":803},[35,69436,103],{"emptyLinePlaceholder":102},[35,69438,69439],{"class":37,"line":809},[35,69440,69441],{"class":41},"    # Prepare output\n",[35,69443,69444,69447,69449],{"class":37,"line":814},[35,69445,69446],{"class":113},"    S_list ",[35,69448,242],{"class":241},[35,69450,1850],{"class":113},[35,69452,69453,69456,69458,69461,69464],{"class":37,"line":833},[35,69454,69455],{"class":113},"    X ",[35,69457,242],{"class":241},[35,69459,69460],{"class":113}," Pe.",[35,69462,69463],{"class":271},"copy",[35,69465,2101],{"class":113},[35,69467,69468,69470],{"class":37,"line":845},[35,69469,352],{"class":109},[35,69471,69472],{"class":113}," detrend_mean:\n",[35,69474,69475,69478,69480,69483,69485,69488,69491,69493,69495,69497,69499,69501,69504,69506,69508],{"class":37,"line":854},[35,69476,69477],{"class":113},"        X ",[35,69479,242],{"class":241},[35,69481,69482],{"class":113}," X ",[35,69484,564],{"class":241},[35,69486,69487],{"class":113}," X.",[35,69489,69490],{"class":271},"mean",[35,69492,275],{"class":113},[35,69494,65439],{"class":822},[35,69496,242],{"class":241},[35,69498,5312],{"class":245},[35,69500,288],{"class":113},[35,69502,69503],{"class":822},"keepdims",[35,69505,242],{"class":241},[35,69507,828],{"class":245},[35,69509,384],{"class":113},[35,69511,69512,69514,69517,69520,69522],{"class":37,"line":865},[35,69513,69455],{"class":113},[35,69515,69516],{"class":241},"*=",[35,69518,69519],{"class":113}," w[:, ",[35,69521,61240],{"class":245},[35,69523,1074],{"class":113},[35,69525,69526],{"class":37,"line":873},[35,69527,103],{"emptyLinePlaceholder":102},[35,69529,69530],{"class":37,"line":878},[35,69531,69532],{"class":41},"    # Real FFT along τ (axis=0)\n",[35,69534,69535,69538,69540,69543,69546,69549,69551,69553,69556,69558,69560,69562,69565],{"class":37,"line":883},[35,69536,69537],{"class":113},"    Xf ",[35,69539,242],{"class":241},[35,69541,69542],{"class":113}," np.fft.",[35,69544,69545],{"class":271},"rfft",[35,69547,69548],{"class":113},"(X, ",[35,69550,291],{"class":822},[35,69552,242],{"class":241},[35,69554,69555],{"class":113},"nfft, ",[35,69557,65439],{"class":822},[35,69559,242],{"class":241},[35,69561,5312],{"class":245},[35,69563,69564],{"class":113},")                     ",[35,69566,69567],{"class":41},"# shape (N_w, N_f)\n",[35,69569,69570,69573,69575,69577,69580,69583,69585,69587,69590],{"class":37,"line":897},[35,69571,69572],{"class":113},"    f_fft_hz ",[35,69574,242],{"class":241},[35,69576,69542],{"class":113},[35,69578,69579],{"class":271},"rfftfreq",[35,69581,69582],{"class":113},"(nfft, ",[35,69584,9487],{"class":822},[35,69586,242],{"class":241},[35,69588,69589],{"class":113},"dt)                  ",[35,69591,69592],{"class":41},"# [Hz]\n",[35,69594,69595],{"class":37,"line":906},[35,69596,103],{"emptyLinePlaceholder":102},[35,69598,69599],{"class":37,"line":915},[35,69600,69601],{"class":41},"    # Amplitude selection\n",[35,69603,69604,69606,69609,69611,69613,69615,69618],{"class":37,"line":923},[35,69605,352],{"class":109},[35,69607,69608],{"class":113}," amp_mode.",[35,69610,69035],{"class":271},[35,69612,2519],{"class":113},[35,69614,358],{"class":241},[35,69616,69617],{"class":234}," \"power\"",[35,69619,364],{"class":113},[35,69621,69622,69625,69627,69629,69631,69634,69636],{"class":37,"line":929},[35,69623,69624],{"class":113},"        S ",[35,69626,242],{"class":241},[35,69628,2412],{"class":113},[35,69630,2543],{"class":271},[35,69632,69633],{"class":113},"(Xf) ",[35,69635,421],{"class":241},[35,69637,62258],{"class":245},[35,69639,69640,69642],{"class":37,"line":935},[35,69641,6403],{"class":109},[35,69643,364],{"class":113},[35,69645,69646,69648,69650,69652,69654],{"class":37,"line":942},[35,69647,69624],{"class":113},[35,69649,242],{"class":241},[35,69651,2412],{"class":113},[35,69653,2543],{"class":271},[35,69655,69656],{"class":113},"(Xf)\n",[35,69658,69659],{"class":37,"line":950},[35,69660,103],{"emptyLinePlaceholder":102},[35,69662,69663],{"class":37,"line":958},[35,69664,69665],{"class":41},"    # Normalization\n",[35,69667,69668,69670,69673,69675,69677,69679,69682],{"class":37,"line":964},[35,69669,352],{"class":109},[35,69671,69672],{"class":113}," norm_mode.",[35,69674,69035],{"class":271},[35,69676,2519],{"class":113},[35,69678,358],{"class":241},[35,69680,69681],{"class":234}," \"per_f\"",[35,69683,364],{"class":113},[35,69685,69686,69689,69691,69693,69696,69699,69701,69703,69705,69707,69709,69711,69713,69715,69717,69719,69722],{"class":37,"line":969},[35,69687,69688],{"class":113},"        col_max ",[35,69690,242],{"class":241},[35,69692,2412],{"class":113},[35,69694,69695],{"class":271},"maximum",[35,69697,69698],{"class":113},"(S.",[35,69700,554],{"class":271},[35,69702,275],{"class":113},[35,69704,65439],{"class":822},[35,69706,242],{"class":241},[35,69708,5312],{"class":245},[35,69710,288],{"class":113},[35,69712,69503],{"class":822},[35,69714,242],{"class":241},[35,69716,828],{"class":245},[35,69718,685],{"class":113},[35,69720,69721],{"class":245},"1e-15",[35,69723,384],{"class":113},[35,69725,69726,69728,69730,69733,69735],{"class":37,"line":976},[35,69727,69624],{"class":113},[35,69729,242],{"class":241},[35,69731,69732],{"class":113}," S ",[35,69734,398],{"class":241},[35,69736,69737],{"class":113}," col_max\n",[35,69739,69740,69742],{"class":37,"line":984},[35,69741,6403],{"class":109},[35,69743,364],{"class":113},[35,69745,69746,69748,69750,69752,69754,69756,69758,69760,69762,69764],{"class":37,"line":992},[35,69747,69624],{"class":113},[35,69749,242],{"class":241},[35,69751,69732],{"class":113},[35,69753,398],{"class":241},[35,69755,773],{"class":241},[35,69757,69698],{"class":113},[35,69759,554],{"class":271},[35,69761,2672],{"class":113},[35,69763,69721],{"class":245},[35,69765,384],{"class":113},[35,69767,69768],{"class":37,"line":998},[35,69769,103],{"emptyLinePlaceholder":102},[35,69771,69772,69774,69776,69778,69780,69782,69785],{"class":37,"line":1003},[35,69773,352],{"class":109},[35,69775,69608],{"class":113},[35,69777,69035],{"class":271},[35,69779,2519],{"class":113},[35,69781,358],{"class":241},[35,69783,69784],{"class":234}," \"db\"",[35,69786,364],{"class":113},[35,69788,69789,69791,69793,69795,69797,69799,69802,69804,69806,69809,69812,69815],{"class":37,"line":1010},[35,69790,69624],{"class":113},[35,69792,242],{"class":241},[35,69794,61773],{"class":245},[35,69796,463],{"class":241},[35,69798,2412],{"class":113},[35,69800,69801],{"class":271},"log10",[35,69803,64321],{"class":113},[35,69805,69695],{"class":271},[35,69807,69808],{"class":113},"(S, ",[35,69810,69811],{"class":245},"1e-12",[35,69813,69814],{"class":113},"))  ",[35,69816,69817],{"class":41},"# dB, clipped floor\n",[35,69819,69820],{"class":37,"line":1018},[35,69821,103],{"emptyLinePlaceholder":102},[35,69823,69824],{"class":37,"line":1026},[35,69825,69826],{"class":41},"    # Units\n",[35,69828,69829,69831,69834,69837,69839,69841,69844],{"class":37,"line":1032},[35,69830,352],{"class":109},[35,69832,69833],{"class":113}," freq_units.",[35,69835,69836],{"class":271},"upper",[35,69838,2519],{"class":113},[35,69840,358],{"class":241},[35,69842,69843],{"class":234}," \"GHz\"",[35,69845,364],{"class":113},[35,69847,69848,69851,69853,69856,69858],{"class":37,"line":1037},[35,69849,69850],{"class":113},"        f_fft ",[35,69852,242],{"class":241},[35,69854,69855],{"class":113}," f_fft_hz ",[35,69857,484],{"class":241},[35,69859,62097],{"class":245},[35,69861,69862,69864],{"class":37,"line":1044},[35,69863,6403],{"class":109},[35,69865,364],{"class":113},[35,69867,69868,69870,69872,69874,69876],{"class":37,"line":1052},[35,69869,69850],{"class":113},[35,69871,242],{"class":241},[35,69873,69855],{"class":113},[35,69875,484],{"class":241},[35,69877,69878],{"class":245}," 1e-6\n",[35,69880,69881],{"class":37,"line":1060},[35,69882,103],{"emptyLinePlaceholder":102},[35,69884,69885,69887],{"class":37,"line":1066},[35,69886,549],{"class":109},[35,69888,69889],{"class":113}," f_fft, S\n",[35,69891,69892],{"class":37,"line":1071},[35,69893,103],{"emptyLinePlaceholder":102},[35,69895,69896],{"class":37,"line":1077},[35,69897,69898],{"class":41},"# Compute spectrogram with current grid\n",[35,69900,69901,69904,69906,69908],{"class":37,"line":1082},[35,69902,69903],{"class":113},"f_fft_axis, S_fft ",[35,69905,242],{"class":241},[35,69907,69087],{"class":271},[35,69909,903],{"class":113},[35,69911,69912],{"class":37,"line":1088},[35,69913,69914],{"class":113},"    Pe, dur_ns,\n",[35,69916,69917,69919,69921,69923],{"class":37,"line":1093},[35,69918,69108],{"class":822},[35,69920,242],{"class":241},[35,69922,61481],{"class":245},[35,69924,912],{"class":113},[35,69926,69927,69929,69931,69933],{"class":37,"line":1107},[35,69928,69123],{"class":822},[35,69930,242],{"class":241},[35,69932,61495],{"class":245},[35,69934,912],{"class":113},[35,69936,69937,69939,69941,69943],{"class":37,"line":1120},[35,69938,69138],{"class":822},[35,69940,242],{"class":241},[35,69942,61510],{"class":245},[35,69944,912],{"class":113},[35,69946,69947,69949,69951,69953],{"class":37,"line":1133},[35,69948,69153],{"class":822},[35,69950,242],{"class":241},[35,69952,61524],{"class":245},[35,69954,912],{"class":113},[35,69956,69957,69959,69961,69963],{"class":37,"line":1161},[35,69958,69168],{"class":822},[35,69960,242],{"class":241},[35,69962,61539],{"class":245},[35,69964,912],{"class":113},[35,69966,69967,69969,69971,69973],{"class":37,"line":1170},[35,69968,69183],{"class":822},[35,69970,242],{"class":241},[35,69972,61554],{"class":245},[35,69974,912],{"class":113},[35,69976,69977],{"class":37,"line":1183},[35,69978,384],{"class":113},[35,69980,69981,69983,69985,69988,69991,69994,69997,69999,70001,70004,70006,70008,70010,70012,70014],{"class":37,"line":1194},[35,69982,1861],{"class":241},[35,69984,275],{"class":113},[35,69986,69987],{"class":234},"\"FFT grid:\"",[35,69989,69990],{"class":113},", S_fft.shape, ",[35,69992,69993],{"class":234},"\"| Fourier freq range:\"",[35,69995,69996],{"class":113},", f_fft_axis[",[35,69998,5312],{"class":245},[35,70000,3033],{"class":113},[35,70002,70003],{"class":234},"\"→\"",[35,70005,69996],{"class":113},[35,70007,564],{"class":241},[35,70009,500],{"class":245},[35,70011,3033],{"class":113},[35,70013,61554],{"class":245},[35,70015,384],{"class":113},[25,70017,70020],{"className":70018,"code":70019,"language":4991,"meta":30},[4989],"FFT grid: (2049, 401) | Fourier freq range: 0.0 → 999.9999999999999 MHz\n",[32,70021,70019],{"__ignoreMap":30},[25,70023,70025],{"className":27,"code":70024,"language":29,"meta":30,"style":30},"# @title Theoretical Rabi ridge\nimport math\ntwo_pi = 2.0 * math.pi\nf0 = F0_GHZ * 1e9\ndelta_hz = two_pi * (freq_ghz*1e9 - f0)   # rad\u002Fs detuning\nomega_onres = two_pi * OMEGA_ONRESONANCE_MHZ * 1e6  # rad\u002Fs\nOmega_R = np.sqrt(omega_onres**2 + delta_hz**2)     # rad\u002Fs\nridge_hz = Omega_R \u002F (2.0 * math.pi)                # Hz\nif FREQ_UNITS.upper() == \"GHz\":\n    ridge_axis = ridge_hz * 1e-9\nelse:\n    ridge_axis = ridge_hz * 1e-6  # MHz\n",[32,70026,70027,70032,70038,70051,70065,70092,70112,70144,70167,70186,70200,70206],{"__ignoreMap":30},[35,70028,70029],{"class":37,"line":38},[35,70030,70031],{"class":41},"# @title Theoretical Rabi ridge\n",[35,70033,70034,70036],{"class":37,"line":45},[35,70035,110],{"class":109},[35,70037,122],{"class":113},[35,70039,70040,70043,70045,70047,70049],{"class":37,"line":51},[35,70041,70042],{"class":113},"two_pi ",[35,70044,242],{"class":241},[35,70046,62048],{"class":245},[35,70048,463],{"class":241},[35,70050,62053],{"class":113},[35,70052,70053,70056,70058,70061,70063],{"class":37,"line":57},[35,70054,70055],{"class":113},"f0 ",[35,70057,242],{"class":241},[35,70059,70060],{"class":245}," F0_GHZ",[35,70062,463],{"class":241},[35,70064,62068],{"class":245},[35,70066,70067,70070,70072,70074,70076,70079,70081,70084,70086,70089],{"class":37,"line":63},[35,70068,70069],{"class":113},"delta_hz ",[35,70071,242],{"class":241},[35,70073,62107],{"class":113},[35,70075,484],{"class":241},[35,70077,70078],{"class":113}," (freq_ghz",[35,70080,484],{"class":241},[35,70082,70083],{"class":245},"1e9",[35,70085,503],{"class":241},[35,70087,70088],{"class":113}," f0)   ",[35,70090,70091],{"class":41},"# rad\u002Fs detuning\n",[35,70093,70094,70097,70099,70101,70103,70106,70108,70110],{"class":37,"line":69},[35,70095,70096],{"class":113},"omega_onres ",[35,70098,242],{"class":241},[35,70100,62107],{"class":113},[35,70102,484],{"class":241},[35,70104,70105],{"class":245}," OMEGA_ONRESONANCE_MHZ",[35,70107,463],{"class":241},[35,70109,62117],{"class":245},[35,70111,62120],{"class":41},[35,70113,70114,70117,70119,70121,70123,70126,70128,70130,70132,70135,70137,70139,70142],{"class":37,"line":75},[35,70115,70116],{"class":113},"Omega_R ",[35,70118,242],{"class":241},[35,70120,2412],{"class":113},[35,70122,490],{"class":271},[35,70124,70125],{"class":113},"(omega_onres",[35,70127,421],{"class":241},[35,70129,460],{"class":245},[35,70131,415],{"class":241},[35,70133,70134],{"class":113}," delta_hz",[35,70136,421],{"class":241},[35,70138,460],{"class":245},[35,70140,70141],{"class":113},")     ",[35,70143,62174],{"class":41},[35,70145,70146,70149,70151,70153,70155,70157,70159,70161,70164],{"class":37,"line":81},[35,70147,70148],{"class":113},"ridge_hz ",[35,70150,242],{"class":241},[35,70152,62273],{"class":113},[35,70154,398],{"class":241},[35,70156,373],{"class":113},[35,70158,63978],{"class":245},[35,70160,463],{"class":241},[35,70162,70163],{"class":113}," math.pi)                ",[35,70165,70166],{"class":41},"# Hz\n",[35,70168,70169,70171,70174,70176,70178,70180,70182,70184],{"class":37,"line":87},[35,70170,1492],{"class":109},[35,70172,70173],{"class":245}," FREQ_UNITS",[35,70175,5122],{"class":113},[35,70177,69836],{"class":271},[35,70179,2519],{"class":113},[35,70181,358],{"class":241},[35,70183,69843],{"class":234},[35,70185,364],{"class":113},[35,70187,70188,70191,70193,70196,70198],{"class":37,"line":93},[35,70189,70190],{"class":113},"    ridge_axis ",[35,70192,242],{"class":241},[35,70194,70195],{"class":113}," ridge_hz ",[35,70197,484],{"class":241},[35,70199,62097],{"class":245},[35,70201,70202,70204],{"class":37,"line":99},[35,70203,65649],{"class":109},[35,70205,364],{"class":113},[35,70207,70208,70210,70212,70214,70216,70219],{"class":37,"line":106},[35,70209,70190],{"class":113},[35,70211,242],{"class":241},[35,70213,70195],{"class":113},[35,70215,484],{"class":241},[35,70217,70218],{"class":245}," 1e-6",[35,70220,70221],{"class":41},"  # MHz\n",[25,70223,70225],{"className":27,"code":70224,"language":29,"meta":30,"style":30},"# @title 2D heatmap\nfig, ax = plt.subplots(figsize=(7.5, 5.5))\nim = ax.pcolormesh(freq_ghz, f_fft_axis, S_fft, shading=\"auto\", cmap=\"inferno\")\ncb = fig.colorbar(im, ax=ax)\ncb.set_label(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\n\nax.plot(freq_ghz, ridge_axis, \"--\", lw=1.5, label=r\"$\\Omega_R\u002F2\\pi$\")\nax.set_xlabel(\"Drive frequency f (GHz)\")\nax.set_ylabel(f\"Fourier frequency ({FREQ_UNITS})\")\nax.set_title(\"Fourier transform along pulse length: $|\\\\mathcal{F}_\\\\tau\\\\{P_e\\\\}|$\")\nax.legend(loc=\"upper right\")\nfig.tight_layout()\nplt.show()\n",[32,70226,70227,70232,70260,70293,70313,70356,70360,70405,70418,70439,70477,70495,70503],{"__ignoreMap":30},[35,70228,70229],{"class":37,"line":38},[35,70230,70231],{"class":41},"# @title 2D heatmap\n",[35,70233,70234,70236,70238,70240,70242,70244,70246,70248,70250,70253,70255,70258],{"class":37,"line":45},[35,70235,62686],{"class":113},[35,70237,242],{"class":241},[35,70239,62691],{"class":113},[35,70241,62694],{"class":271},[35,70243,275],{"class":113},[35,70245,2386],{"class":822},[35,70247,242],{"class":241},[35,70249,275],{"class":113},[35,70251,70252],{"class":245},"7.5",[35,70254,288],{"class":113},[35,70256,70257],{"class":245},"5.5",[35,70259,2401],{"class":113},[35,70261,70262,70265,70267,70269,70271,70274,70276,70278,70281,70283,70286,70288,70291],{"class":37,"line":51},[35,70263,70264],{"class":113},"im ",[35,70266,242],{"class":241},[35,70268,62721],{"class":113},[35,70270,62724],{"class":271},[35,70272,70273],{"class":113},"(freq_ghz, f_fft_axis, S_fft, ",[35,70275,62730],{"class":822},[35,70277,242],{"class":241},[35,70279,70280],{"class":234},"\"auto\"",[35,70282,288],{"class":113},[35,70284,70285],{"class":822},"cmap",[35,70287,242],{"class":241},[35,70289,70290],{"class":234},"\"inferno\"",[35,70292,384],{"class":113},[35,70294,70295,70297,70299,70301,70303,70306,70308,70310],{"class":37,"line":57},[35,70296,62742],{"class":113},[35,70298,242],{"class":241},[35,70300,62747],{"class":113},[35,70302,62750],{"class":271},[35,70304,70305],{"class":113},"(im, ",[35,70307,62756],{"class":822},[35,70309,242],{"class":241},[35,70311,70312],{"class":113},"ax)\n",[35,70314,70315,70317,70319,70321,70324,70326,70328,70331,70334,70337,70339,70341,70344,70346,70349,70351,70354],{"class":37,"line":63},[35,70316,68162],{"class":113},[35,70318,68165],{"class":271},[35,70320,275],{"class":113},[35,70322,70323],{"class":234},"\"FFT amplitude\"",[35,70325,415],{"class":241},[35,70327,373],{"class":113},[35,70329,70330],{"class":234},"\" (dB)\"",[35,70332,70333],{"class":109}," if",[35,70335,70336],{"class":245}," AMP_MODE",[35,70338,5122],{"class":113},[35,70340,69035],{"class":271},[35,70342,70343],{"class":113},"()",[35,70345,358],{"class":241},[35,70347,70348],{"class":234},"\"db\"",[35,70350,1502],{"class":109},[35,70352,70353],{"class":234}," \" (arb.)\"",[35,70355,2401],{"class":113},[35,70357,70358],{"class":37,"line":69},[35,70359,103],{"emptyLinePlaceholder":102},[35,70361,70362,70364,70366,70369,70372,70374,70377,70379,70381,70383,70385,70387,70389,70391,70394,70397,70400,70403],{"class":37,"line":75},[35,70363,62777],{"class":113},[35,70365,29869],{"class":271},[35,70367,70368],{"class":113},"(freq_ghz, ridge_axis, ",[35,70370,70371],{"class":234},"\"--\"",[35,70373,288],{"class":113},[35,70375,70376],{"class":822},"lw",[35,70378,242],{"class":241},[35,70380,2648],{"class":245},[35,70382,288],{"class":113},[35,70384,2966],{"class":822},[35,70386,242],{"class":241},[35,70388,6778],{"class":109},[35,70390,3359],{"class":3409},[35,70392,70393],{"class":241},"\\O",[35,70395,70396],{"class":3409},"mega_R\u002F2",[35,70398,70399],{"class":241},"\\p",[35,70401,70402],{"class":3409},"i$\"",[35,70404,384],{"class":113},[35,70406,70407,70409,70411,70413,70416],{"class":37,"line":81},[35,70408,62777],{"class":113},[35,70410,62780],{"class":271},[35,70412,275],{"class":113},[35,70414,70415],{"class":234},"\"Drive frequency f (GHz)\"",[35,70417,384],{"class":113},[35,70419,70420,70422,70424,70426,70428,70431,70434,70437],{"class":37,"line":87},[35,70421,62777],{"class":113},[35,70423,62794],{"class":271},[35,70425,275],{"class":113},[35,70427,1675],{"class":109},[35,70429,70430],{"class":234},"\"Fourier frequency (",[35,70432,70433],{"class":245},"{FREQ_UNITS}",[35,70435,70436],{"class":234},")\"",[35,70438,384],{"class":113},[35,70440,70441,70443,70445,70447,70450,70452,70454,70457,70460,70462,70465,70467,70470,70472,70475],{"class":37,"line":93},[35,70442,62777],{"class":113},[35,70444,62808],{"class":271},[35,70446,275],{"class":113},[35,70448,70449],{"class":234},"\"Fourier transform along pulse length: $|",[35,70451,62816],{"class":241},[35,70453,7108],{"class":234},[35,70455,70456],{"class":245},"{F}",[35,70458,70459],{"class":234},"_",[35,70461,62816],{"class":241},[35,70463,70464],{"class":234},"tau",[35,70466,62816],{"class":241},[35,70468,70469],{"class":234},"{P_e",[35,70471,62816],{"class":241},[35,70473,70474],{"class":234},"}|$\"",[35,70476,384],{"class":113},[35,70478,70479,70481,70483,70485,70488,70490,70493],{"class":37,"line":99},[35,70480,62777],{"class":113},[35,70482,3108],{"class":271},[35,70484,275],{"class":113},[35,70486,70487],{"class":822},"loc",[35,70489,242],{"class":241},[35,70491,70492],{"class":234},"\"upper right\"",[35,70494,384],{"class":113},[35,70496,70497,70499,70501],{"class":37,"line":106},[35,70498,62830],{"class":113},[35,70500,2740],{"class":271},[35,70502,2101],{"class":113},[35,70504,70505,70507,70509],{"class":37,"line":117},[35,70506,2378],{"class":113},[35,70508,2792],{"class":271},[35,70510,2101],{"class":113},[4995,70512],{"alt":70513,"src":70514},"Output 6 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-06.webp",[25,70516,70518],{"className":27,"code":70517,"language":29,"meta":30,"style":30},"# --- Adaptive high-res 3D surface ---\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# ------ Control knobs ------\nUPSAMPLE_F_FREQ: int = 3       # try 2–3; upsample the drive-frequency axis only\nMAX_FACES: int = 300_000       # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\nMIN_ROWS: int = 128            # keep at least this many Fourier-frequency rows\nMAX_ROWS: int = 1024           # hard cap for rows to keep plotting fast\n\n# Preserve global color limits so the legend stays consistent\nvmin = float(np.nanmin(S_fft))\nvmax = float(np.nanmax(S_fft))\n\n# Shapes\nn_rows, n_cols = S_fft.shape        # rows=Fourier freq (ν), cols=drive freq (f)\n\n# 1) Upsample along the typically smaller axis (drive frequency) only\nn_cols_up = min(UPSAMPLE_F_FREQ * (n_cols - 1) + 1, 1200)  # sensible upper bound\nfreq_hi = np.linspace(freq_ghz.min(), freq_ghz.max(), n_cols_up)\n\nS_up_f = np.empty((n_rows, n_cols_up), dtype=float)\nfor i in range(n_rows):\n    S_up_f[i, :] = np.interp(freq_hi, freq_ghz, S_fft[i, :])\n\n# 2) Determine how many Fourier rows we can afford for MAX_FACES\n# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\nn_rows_cap = int(MAX_FACES \u002F max(n_cols_up - 1, 1)) + 1\nn_rows_plot = int(np.clip(n_rows_cap, MIN_ROWS, min(MAX_ROWS, n_rows)))\n\n# Down-sample ν-axis evenly to n_rows_plot\nrow_idx = np.linspace(0, n_rows - 1, n_rows_plot).astype(int)\nf_fft_plot = f_fft_axis[row_idx]\nS_final = S_up_f[row_idx, :]\n\n# Mesh for plotting\nF2_hi, FFOUR_hi = np.meshgrid(freq_hi, f_fft_plot, indexing=\"xy\")\n\n# 3) Plot (keeps 'inferno')\nfig = plt.figure(figsize=(8, 6), dpi=200)\nax = fig.add_subplot(111, projection=\"3d\")\nsurf = ax.plot_surface(\n    F2_hi, FFOUR_hi, S_final,\n    rstride=1, cstride=1,\n    linewidth=0, antialiased=True,\n    cmap=\"inferno\",\n    vmin=vmin, vmax=vmax\n)\n\nax.set_xlabel(\"f (GHz)\")\nax.set_ylabel(f\"Fourier freq ({FREQ_UNITS})\")\nax.set_zlabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower() == \"db\" else \" (arb.)\"))\nax.set_title(\"Rabi FFT surface (adaptive resolution)\")\nfig.colorbar(surf, ax=ax, shrink=0.7, pad=0.08, label=\"Amplitude\")\n\n# A clearer default view\nax.view_init(elev=25, azim=-60)\n\nfig.tight_layout()\nplt.show()\n",[32,70519,70520,70525,70535,70545,70555,70559,70564,70581,70598,70615,70632,70636,70641,70658,70674,70678,70683,70696,70700,70705,70743,70764,70768,70788,70801,70815,70819,70824,70829,70864,70893,70897,70902,70935,70945,70955,70959,70963,70989,70993,70998,71032,71057,71069,71079,71097,71115,71125,71141,71145,71149,71162,71181,71217,71230,71269,71273,71278,71305,71309,71317],{"__ignoreMap":30},[35,70521,70522],{"class":37,"line":38},[35,70523,70524],{"class":41},"# --- Adaptive high-res 3D surface ---\n",[35,70526,70527,70529,70531,70533],{"class":37,"line":45},[35,70528,128],{"class":109},[35,70530,67496],{"class":113},[35,70532,110],{"class":109},[35,70534,67501],{"class":113},[35,70536,70537,70539,70541,70543],{"class":37,"line":51},[35,70538,110],{"class":109},[35,70540,162],{"class":113},[35,70542,165],{"class":109},[35,70544,168],{"class":113},[35,70546,70547,70549,70551,70553],{"class":37,"line":57},[35,70548,110],{"class":109},[35,70550,189],{"class":113},[35,70552,165],{"class":109},[35,70554,194],{"class":113},[35,70556,70557],{"class":37,"line":63},[35,70558,103],{"emptyLinePlaceholder":102},[35,70560,70561],{"class":37,"line":69},[35,70562,70563],{"class":41},"# ------ Control knobs ------\n",[35,70565,70566,70569,70571,70573,70575,70578],{"class":37,"line":75},[35,70567,70568],{"class":245},"UPSAMPLE_F_FREQ",[35,70570,282],{"class":113},[35,70572,285],{"class":241},[35,70574,308],{"class":241},[35,70576,70577],{"class":245}," 3",[35,70579,70580],{"class":41},"       # try 2–3; upsample the drive-frequency axis only\n",[35,70582,70583,70586,70588,70590,70592,70595],{"class":37,"line":81},[35,70584,70585],{"class":245},"MAX_FACES",[35,70587,282],{"class":113},[35,70589,285],{"class":241},[35,70591,308],{"class":241},[35,70593,70594],{"class":245}," 300_000",[35,70596,70597],{"class":41},"       # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\n",[35,70599,70600,70603,70605,70607,70609,70612],{"class":37,"line":87},[35,70601,70602],{"class":245},"MIN_ROWS",[35,70604,282],{"class":113},[35,70606,285],{"class":241},[35,70608,308],{"class":241},[35,70610,70611],{"class":245}," 128",[35,70613,70614],{"class":41},"            # keep at least this many Fourier-frequency rows\n",[35,70616,70617,70620,70622,70624,70626,70629],{"class":37,"line":93},[35,70618,70619],{"class":245},"MAX_ROWS",[35,70621,282],{"class":113},[35,70623,285],{"class":241},[35,70625,308],{"class":241},[35,70627,70628],{"class":245}," 1024",[35,70630,70631],{"class":41},"           # hard cap for rows to keep plotting fast\n",[35,70633,70634],{"class":37,"line":99},[35,70635,103],{"emptyLinePlaceholder":102},[35,70637,70638],{"class":37,"line":106},[35,70639,70640],{"class":41},"# Preserve global color limits so the legend stays consistent\n",[35,70642,70643,70646,70648,70650,70652,70655],{"class":37,"line":117},[35,70644,70645],{"class":113},"vmin ",[35,70647,242],{"class":241},[35,70649,1505],{"class":241},[35,70651,64321],{"class":113},[35,70653,70654],{"class":271},"nanmin",[35,70656,70657],{"class":113},"(S_fft))\n",[35,70659,70660,70663,70665,70667,70669,70672],{"class":37,"line":125},[35,70661,70662],{"class":113},"vmax ",[35,70664,242],{"class":241},[35,70666,1505],{"class":241},[35,70668,64321],{"class":113},[35,70670,70671],{"class":271},"nanmax",[35,70673,70657],{"class":113},[35,70675,70676],{"class":37,"line":139},[35,70677,103],{"emptyLinePlaceholder":102},[35,70679,70680],{"class":37,"line":152},[35,70681,70682],{"class":41},"# Shapes\n",[35,70684,70685,70688,70690,70693],{"class":37,"line":157},[35,70686,70687],{"class":113},"n_rows, n_cols ",[35,70689,242],{"class":241},[35,70691,70692],{"class":113}," S_fft.shape        ",[35,70694,70695],{"class":41},"# rows=Fourier freq (ν), cols=drive freq (f)\n",[35,70697,70698],{"class":37,"line":171},[35,70699,103],{"emptyLinePlaceholder":102},[35,70701,70702],{"class":37,"line":184},[35,70703,70704],{"class":41},"# 1) Upsample along the typically smaller axis (drive frequency) only\n",[35,70706,70707,70710,70712,70714,70716,70718,70720,70723,70725,70727,70729,70731,70733,70735,70738,70740],{"class":37,"line":197},[35,70708,70709],{"class":113},"n_cols_up ",[35,70711,242],{"class":241},[35,70713,673],{"class":241},[35,70715,275],{"class":113},[35,70717,70568],{"class":245},[35,70719,463],{"class":241},[35,70721,70722],{"class":113}," (n_cols ",[35,70724,564],{"class":241},[35,70726,6079],{"class":245},[35,70728,427],{"class":113},[35,70730,445],{"class":241},[35,70732,6079],{"class":245},[35,70734,288],{"class":113},[35,70736,70737],{"class":245},"1200",[35,70739,27221],{"class":113},[35,70741,70742],{"class":41},"# sensible upper bound\n",[35,70744,70745,70747,70749,70751,70753,70755,70757,70759,70761],{"class":37,"line":210},[35,70746,67603],{"class":113},[35,70748,242],{"class":241},[35,70750,2412],{"class":113},[35,70752,62414],{"class":271},[35,70754,67612],{"class":113},[35,70756,570],{"class":271},[35,70758,67617],{"class":113},[35,70760,554],{"class":271},[35,70762,70763],{"class":113},"(), n_cols_up)\n",[35,70765,70766],{"class":37,"line":223},[35,70767,103],{"emptyLinePlaceholder":102},[35,70769,70770,70773,70775,70777,70779,70782,70784,70786],{"class":37,"line":228},[35,70771,70772],{"class":113},"S_up_f ",[35,70774,242],{"class":241},[35,70776,2412],{"class":113},[35,70778,67660],{"class":271},[35,70780,70781],{"class":113},"((n_rows, n_cols_up), ",[35,70783,6723],{"class":822},[35,70785,67677],{"class":241},[35,70787,384],{"class":113},[35,70789,70790,70792,70794,70796,70798],{"class":37,"line":249},[35,70791,1874],{"class":109},[35,70793,2958],{"class":113},[35,70795,1663],{"class":109},[35,70797,12746],{"class":241},[35,70799,70800],{"class":113},"(n_rows):\n",[35,70802,70803,70806,70808,70810,70812],{"class":37,"line":254},[35,70804,70805],{"class":113},"    S_up_f[i, :] ",[35,70807,242],{"class":241},[35,70809,2412],{"class":113},[35,70811,67708],{"class":271},[35,70813,70814],{"class":113},"(freq_hi, freq_ghz, S_fft[i, :])\n",[35,70816,70817],{"class":37,"line":260},[35,70818,103],{"emptyLinePlaceholder":102},[35,70820,70821],{"class":37,"line":265},[35,70822,70823],{"class":41},"# 2) Determine how many Fourier rows we can afford for MAX_FACES\n",[35,70825,70826],{"class":37,"line":326},[35,70827,70828],{"class":41},"# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\n",[35,70830,70831,70834,70836,70838,70840,70842,70844,70846,70849,70851,70853,70855,70857,70860,70862],{"class":37,"line":332},[35,70832,70833],{"class":113},"n_rows_cap ",[35,70835,242],{"class":241},[35,70837,1434],{"class":241},[35,70839,275],{"class":113},[35,70841,70585],{"class":245},[35,70843,62349],{"class":241},[35,70845,773],{"class":241},[35,70847,70848],{"class":113},"(n_cols_up ",[35,70850,564],{"class":241},[35,70852,6079],{"class":245},[35,70854,288],{"class":113},[35,70856,500],{"class":245},[35,70858,70859],{"class":113},")) ",[35,70861,445],{"class":241},[35,70863,734],{"class":245},[35,70865,70866,70869,70871,70873,70875,70877,70880,70882,70884,70886,70888,70890],{"class":37,"line":338},[35,70867,70868],{"class":113},"n_rows_plot ",[35,70870,242],{"class":241},[35,70872,1434],{"class":241},[35,70874,64321],{"class":113},[35,70876,62386],{"class":271},[35,70878,70879],{"class":113},"(n_rows_cap, ",[35,70881,70602],{"class":245},[35,70883,288],{"class":113},[35,70885,570],{"class":241},[35,70887,275],{"class":113},[35,70889,70619],{"class":245},[35,70891,70892],{"class":113},", n_rows)))\n",[35,70894,70895],{"class":37,"line":344},[35,70896,103],{"emptyLinePlaceholder":102},[35,70898,70899],{"class":37,"line":349},[35,70900,70901],{"class":41},"# Down-sample ν-axis evenly to n_rows_plot\n",[35,70903,70904,70907,70909,70911,70913,70915,70917,70920,70922,70924,70927,70929,70931,70933],{"class":37,"line":367},[35,70905,70906],{"class":113},"row_idx ",[35,70908,242],{"class":241},[35,70910,2412],{"class":113},[35,70912,62414],{"class":271},[35,70914,275],{"class":113},[35,70916,5312],{"class":245},[35,70918,70919],{"class":113},", n_rows ",[35,70921,564],{"class":241},[35,70923,6079],{"class":245},[35,70925,70926],{"class":113},", n_rows_plot).",[35,70928,1359],{"class":271},[35,70930,275],{"class":113},[35,70932,285],{"class":241},[35,70934,384],{"class":113},[35,70936,70937,70940,70942],{"class":37,"line":387},[35,70938,70939],{"class":113},"f_fft_plot ",[35,70941,242],{"class":241},[35,70943,70944],{"class":113}," f_fft_axis[row_idx]\n",[35,70946,70947,70950,70952],{"class":37,"line":404},[35,70948,70949],{"class":113},"S_final ",[35,70951,242],{"class":241},[35,70953,70954],{"class":113}," S_up_f[row_idx, :]\n",[35,70956,70957],{"class":37,"line":434},[35,70958,103],{"emptyLinePlaceholder":102},[35,70960,70961],{"class":37,"line":474},[35,70962,67837],{"class":41},[35,70964,70965,70968,70971,70973,70975,70977,70980,70982,70984,70987],{"class":37,"line":546},[35,70966,70967],{"class":113},"F2_hi, ",[35,70969,70970],{"class":245},"FFOUR_hi",[35,70972,308],{"class":241},[35,70974,2412],{"class":113},[35,70976,62139],{"class":271},[35,70978,70979],{"class":113},"(freq_hi, f_fft_plot, ",[35,70981,62145],{"class":822},[35,70983,242],{"class":241},[35,70985,70986],{"class":234},"\"xy\"",[35,70988,384],{"class":113},[35,70990,70991],{"class":37,"line":584},[35,70992,103],{"emptyLinePlaceholder":102},[35,70994,70995],{"class":37,"line":589},[35,70996,70997],{"class":41},"# 3) Plot (keeps 'inferno')\n",[35,70999,71000,71002,71004,71006,71008,71010,71012,71014,71016,71018,71020,71022,71024,71026,71028,71030],{"class":37,"line":594},[35,71001,67873],{"class":113},[35,71003,242],{"class":241},[35,71005,62691],{"class":113},[35,71007,2381],{"class":271},[35,71009,275],{"class":113},[35,71011,2386],{"class":822},[35,71013,242],{"class":241},[35,71015,275],{"class":113},[35,71017,40225],{"class":245},[35,71019,288],{"class":113},[35,71021,2398],{"class":245},[35,71023,685],{"class":113},[35,71025,2767],{"class":822},[35,71027,242],{"class":241},[35,71029,2772],{"class":245},[35,71031,384],{"class":113},[35,71033,71034,71036,71038,71040,71042,71044,71046,71048,71050,71052,71055],{"class":37,"line":643},[35,71035,67908],{"class":113},[35,71037,242],{"class":241},[35,71039,62747],{"class":113},[35,71041,67915],{"class":271},[35,71043,275],{"class":113},[35,71045,67920],{"class":245},[35,71047,288],{"class":113},[35,71049,67925],{"class":822},[35,71051,242],{"class":241},[35,71053,71054],{"class":234},"\"3d\"",[35,71056,384],{"class":113},[35,71058,71059,71061,71063,71065,71067],{"class":37,"line":648},[35,71060,67941],{"class":113},[35,71062,242],{"class":241},[35,71064,62721],{"class":113},[35,71066,67948],{"class":271},[35,71068,903],{"class":113},[35,71070,71071,71074,71076],{"class":37,"line":654},[35,71072,71073],{"class":113},"    F2_hi, ",[35,71075,70970],{"class":245},[35,71077,71078],{"class":113},", S_final,\n",[35,71080,71081,71083,71085,71087,71089,71091,71093,71095],{"class":37,"line":660},[35,71082,67960],{"class":822},[35,71084,242],{"class":241},[35,71086,500],{"class":245},[35,71088,288],{"class":113},[35,71090,67969],{"class":822},[35,71092,242],{"class":241},[35,71094,500],{"class":245},[35,71096,912],{"class":113},[35,71098,71099,71101,71103,71105,71107,71109,71111,71113],{"class":37,"line":665},[35,71100,67984],{"class":822},[35,71102,242],{"class":241},[35,71104,5312],{"class":245},[35,71106,288],{"class":113},[35,71108,67993],{"class":822},[35,71110,242],{"class":241},[35,71112,828],{"class":245},[35,71114,912],{"class":113},[35,71116,71117,71119,71121,71123],{"class":37,"line":692},[35,71118,68004],{"class":822},[35,71120,242],{"class":241},[35,71122,70290],{"class":234},[35,71124,912],{"class":113},[35,71126,71127,71129,71131,71134,71136,71138],{"class":37,"line":714},[35,71128,68015],{"class":822},[35,71130,242],{"class":241},[35,71132,71133],{"class":113},"vmin, ",[35,71135,68024],{"class":822},[35,71137,242],{"class":241},[35,71139,71140],{"class":113},"vmax\n",[35,71142,71143],{"class":37,"line":729},[35,71144,384],{"class":113},[35,71146,71147],{"class":37,"line":737},[35,71148,103],{"emptyLinePlaceholder":102},[35,71150,71151,71153,71155,71157,71160],{"class":37,"line":768},[35,71152,62777],{"class":113},[35,71154,62780],{"class":271},[35,71156,275],{"class":113},[35,71158,71159],{"class":234},"\"f (GHz)\"",[35,71161,384],{"class":113},[35,71163,71164,71166,71168,71170,71172,71175,71177,71179],{"class":37,"line":793},[35,71165,62777],{"class":113},[35,71167,62794],{"class":271},[35,71169,275],{"class":113},[35,71171,1675],{"class":109},[35,71173,71174],{"class":234},"\"Fourier freq (",[35,71176,70433],{"class":245},[35,71178,70436],{"class":234},[35,71180,384],{"class":113},[35,71182,71183,71185,71187,71189,71191,71193,71195,71197,71199,71201,71203,71205,71207,71209,71211,71213,71215],{"class":37,"line":798},[35,71184,62777],{"class":113},[35,71186,68088],{"class":271},[35,71188,275],{"class":113},[35,71190,70323],{"class":234},[35,71192,415],{"class":241},[35,71194,373],{"class":113},[35,71196,70330],{"class":234},[35,71198,70333],{"class":109},[35,71200,70336],{"class":245},[35,71202,5122],{"class":113},[35,71204,69035],{"class":271},[35,71206,2519],{"class":113},[35,71208,358],{"class":241},[35,71210,69784],{"class":234},[35,71212,1502],{"class":109},[35,71214,70353],{"class":234},[35,71216,2401],{"class":113},[35,71218,71219,71221,71223,71225,71228],{"class":37,"line":803},[35,71220,62777],{"class":113},[35,71222,62808],{"class":271},[35,71224,275],{"class":113},[35,71226,71227],{"class":234},"\"Rabi FFT surface (adaptive resolution)\"",[35,71229,384],{"class":113},[35,71231,71232,71234,71236,71238,71240,71242,71244,71246,71248,71250,71252,71254,71256,71258,71260,71262,71264,71267],{"class":37,"line":809},[35,71233,62830],{"class":113},[35,71235,62750],{"class":271},[35,71237,68131],{"class":113},[35,71239,62756],{"class":822},[35,71241,242],{"class":241},[35,71243,62761],{"class":113},[35,71245,68140],{"class":822},[35,71247,242],{"class":241},[35,71249,68145],{"class":245},[35,71251,288],{"class":113},[35,71253,68150],{"class":822},[35,71255,242],{"class":241},[35,71257,68155],{"class":245},[35,71259,288],{"class":113},[35,71261,2966],{"class":822},[35,71263,242],{"class":241},[35,71265,71266],{"class":234},"\"Amplitude\"",[35,71268,384],{"class":113},[35,71270,71271],{"class":37,"line":814},[35,71272,103],{"emptyLinePlaceholder":102},[35,71274,71275],{"class":37,"line":833},[35,71276,71277],{"class":41},"# A clearer default view\n",[35,71279,71280,71282,71285,71287,71290,71292,71294,71296,71299,71301,71303],{"class":37,"line":845},[35,71281,62777],{"class":113},[35,71283,71284],{"class":271},"view_init",[35,71286,275],{"class":113},[35,71288,71289],{"class":822},"elev",[35,71291,242],{"class":241},[35,71293,2680],{"class":245},[35,71295,288],{"class":113},[35,71297,71298],{"class":822},"azim",[35,71300,18628],{"class":241},[35,71302,63137],{"class":245},[35,71304,384],{"class":113},[35,71306,71307],{"class":37,"line":854},[35,71308,103],{"emptyLinePlaceholder":102},[35,71310,71311,71313,71315],{"class":37,"line":865},[35,71312,62830],{"class":113},[35,71314,2740],{"class":271},[35,71316,2101],{"class":113},[35,71318,71319,71321,71323],{"class":37,"line":873},[35,71320,2378],{"class":113},[35,71322,2792],{"class":271},[35,71324,2101],{"class":113},[4995,71326],{"alt":71327,"src":71328},"Output 7 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-07.webp",[25,71330,71332],{"className":27,"code":71331,"language":29,"meta":30,"style":30},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\nF_STAR_GHZ: float = float(F0_GHZ)   # slice near resonance\nNU_STAR = float(ridge_axis[np.argmin(np.abs(freq_ghz - F_STAR_GHZ))])  # near Ω_R\u002F2π\n\n# Slice at f = F_STAR_GHZ\nj = int(np.argmin(np.abs(freq_ghz - F_STAR_GHZ)))\nfig, ax = plt.subplots(figsize=(6.0, 3.8))\nax.plot(f_fft_axis, S_fft[:, j], color='blue')\nax.axvline(ridge_axis[j], ls=\"--\", lw=1.0, label=r\"$\\Omega_R\u002F2\\pi$\")\nax.set_xlabel(f\"Fourier frequency ({FREQ_UNITS})\")\nax.set_ylabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\nax.set_title(fr\"Slice at $f={freq_ghz[j]:.6f}$ GHz\")\nax.legend()\nfig.tight_layout(); plt.show()\n\n# Slice at ν = NU_STAR\ni = int(np.argmin(np.abs(f_fft_axis - NU_STAR)))\nfig, ax = plt.subplots(figsize=(6.0, 3.8))\nax.plot(freq_ghz, S_fft[i, :], color='blue')\nax.set_xlabel(\"Drive frequency f (GHz)\")\nax.set_ylabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\nax.set_title(fr\"Slice at Fourier freq ≈ {f_fft_axis[i]:.3f} {FREQ_UNITS}\")\nfig.tight_layout(); plt.show()\n",[32,71333,71334,71339,71362,71394,71398,71403,71427,71455,71472,71517,71535,71571,71598,71606,71619,71623,71628,71654,71680,71697,71709,71745,71774],{"__ignoreMap":30},[35,71335,71336],{"class":37,"line":38},[35,71337,71338],{"class":41},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\n",[35,71340,71341,71344,71346,71348,71350,71352,71354,71356,71359],{"class":37,"line":45},[35,71342,71343],{"class":245},"F_STAR_GHZ",[35,71345,282],{"class":113},[35,71347,305],{"class":241},[35,71349,308],{"class":241},[35,71351,1505],{"class":241},[35,71353,275],{"class":113},[35,71355,61004],{"class":245},[35,71357,71358],{"class":113},")   ",[35,71360,71361],{"class":41},"# slice near resonance\n",[35,71363,71364,71367,71369,71371,71374,71376,71378,71380,71383,71385,71388,71391],{"class":37,"line":51},[35,71365,71366],{"class":245},"NU_STAR",[35,71368,308],{"class":241},[35,71370,1505],{"class":241},[35,71372,71373],{"class":113},"(ridge_axis[np.",[35,71375,68432],{"class":271},[35,71377,64321],{"class":113},[35,71379,2543],{"class":271},[35,71381,71382],{"class":113},"(freq_ghz ",[35,71384,564],{"class":241},[35,71386,71387],{"class":245}," F_STAR_GHZ",[35,71389,71390],{"class":113},"))])  ",[35,71392,71393],{"class":41},"# near Ω_R\u002F2π\n",[35,71395,71396],{"class":37,"line":57},[35,71397,103],{"emptyLinePlaceholder":102},[35,71399,71400],{"class":37,"line":63},[35,71401,71402],{"class":41},"# Slice at f = F_STAR_GHZ\n",[35,71404,71405,71407,71409,71411,71413,71415,71417,71419,71421,71423,71425],{"class":37,"line":69},[35,71406,68458],{"class":113},[35,71408,242],{"class":241},[35,71410,1434],{"class":241},[35,71412,64321],{"class":113},[35,71414,68432],{"class":271},[35,71416,64321],{"class":113},[35,71418,2543],{"class":271},[35,71420,71382],{"class":113},[35,71422,564],{"class":241},[35,71424,71387],{"class":245},[35,71426,18579],{"class":113},[35,71428,71429,71431,71433,71435,71437,71439,71441,71443,71445,71448,71450,71453],{"class":37,"line":75},[35,71430,62686],{"class":113},[35,71432,242],{"class":241},[35,71434,62691],{"class":113},[35,71436,62694],{"class":271},[35,71438,275],{"class":113},[35,71440,2386],{"class":822},[35,71442,242],{"class":241},[35,71444,275],{"class":113},[35,71446,71447],{"class":245},"6.0",[35,71449,288],{"class":113},[35,71451,71452],{"class":245},"3.8",[35,71454,2401],{"class":113},[35,71456,71457,71459,71461,71464,71466,71468,71470],{"class":37,"line":81},[35,71458,62777],{"class":113},[35,71460,29869],{"class":271},[35,71462,71463],{"class":113},"(f_fft_axis, S_fft[:, j], ",[35,71465,2576],{"class":822},[35,71467,242],{"class":241},[35,71469,68531],{"class":234},[35,71471,384],{"class":113},[35,71473,71474,71476,71479,71482,71485,71487,71489,71491,71493,71495,71497,71499,71501,71503,71505,71507,71509,71511,71513,71515],{"class":37,"line":87},[35,71475,62777],{"class":113},[35,71477,71478],{"class":271},"axvline",[35,71480,71481],{"class":113},"(ridge_axis[j], ",[35,71483,71484],{"class":822},"ls",[35,71486,242],{"class":241},[35,71488,70371],{"class":234},[35,71490,288],{"class":113},[35,71492,70376],{"class":822},[35,71494,242],{"class":241},[35,71496,381],{"class":245},[35,71498,288],{"class":113},[35,71500,2966],{"class":822},[35,71502,242],{"class":241},[35,71504,6778],{"class":109},[35,71506,3359],{"class":3409},[35,71508,70393],{"class":241},[35,71510,70396],{"class":3409},[35,71512,70399],{"class":241},[35,71514,70402],{"class":3409},[35,71516,384],{"class":113},[35,71518,71519,71521,71523,71525,71527,71529,71531,71533],{"class":37,"line":93},[35,71520,62777],{"class":113},[35,71522,62780],{"class":271},[35,71524,275],{"class":113},[35,71526,1675],{"class":109},[35,71528,70430],{"class":234},[35,71530,70433],{"class":245},[35,71532,70436],{"class":234},[35,71534,384],{"class":113},[35,71536,71537,71539,71541,71543,71545,71547,71549,71551,71553,71555,71557,71559,71561,71563,71565,71567,71569],{"class":37,"line":99},[35,71538,62777],{"class":113},[35,71540,62794],{"class":271},[35,71542,275],{"class":113},[35,71544,70323],{"class":234},[35,71546,415],{"class":241},[35,71548,373],{"class":113},[35,71550,70330],{"class":234},[35,71552,70333],{"class":109},[35,71554,70336],{"class":245},[35,71556,5122],{"class":113},[35,71558,69035],{"class":271},[35,71560,70343],{"class":113},[35,71562,358],{"class":241},[35,71564,70348],{"class":234},[35,71566,1502],{"class":109},[35,71568,70353],{"class":234},[35,71570,2401],{"class":113},[35,71572,71573,71575,71577,71579,71582,71585,71587,71589,71591,71593,71596],{"class":37,"line":106},[35,71574,62777],{"class":113},[35,71576,62808],{"class":271},[35,71578,275],{"class":113},[35,71580,71581],{"class":109},"fr",[35,71583,71584],{"class":234},"\"Slice 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at Fourier freq ≈ ",[35,71759,1681],{"class":245},[35,71761,71762],{"class":113},"f_fft_axis[i]",[35,71764,67137],{"class":109},[35,71766,1697],{"class":245},[35,71768,71769],{"class":245}," {FREQ_UNITS}",[35,71771,2986],{"class":234},[35,71773,384],{"class":113},[35,71775,71776,71778,71780,71782,71784],{"class":37,"line":228},[35,71777,62830],{"class":113},[35,71779,2740],{"class":271},[35,71781,71614],{"class":113},[35,71783,2792],{"class":271},[35,71785,2101],{"class":113},[4995,71787],{"alt":71788,"src":71789},"Output 8 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-08.webp",[4995,71791],{"alt":71792,"src":71793},"Output 9 of the notebook code 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