CP gate

The controlled-phase gate: matrix, effect on the basis states, the cp() call in Python and JS, and what it becomes on IonQ hardware.

The CP gate multiplies the $\vert 11 \rangle$ amplitude by $e^{i\lambda}$ and leaves the other three basis states alone. It acts on two qubits and takes one parameter, the angle $\lambda$ in radians. Control and target are interchangeable. The Circuit panel draws a dot on the control wire joined to a P(λ) box on the target wire, with the angle shown as a multiple of π. On its own it changes nothing a measurement can see, whatever the angle; the phase shows once the qubits are in superposition.

Kets such as $\vert 0 \rangle$ name the basis states and rotations are described on the Bloch sphere; the qubits lesson introduces both.

Call

Where	Call
Python	`qc.cp(λ, control, target)`
JS	`circuit.cp(λ, control, target)`
Qiskit class	`CPhaseGate`
More controls	`mcp(λ, [controls], target)`

Matrix

The matrix uses the basis order $\vert 00 \rangle, \vert 01 \rangle, \vert 10 \rangle, \vert 11 \rangle$ .

CP(\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & e^{i\lambda} \end{pmatrix}

cp(π) is a CZ gate. CP is the controlled form of the P gate.

Effect on basis states

Input	Output
$\vert 00 \rangle$	$\vert 00 \rangle$
$\vert 01 \rangle$	$\vert 01 \rangle$
$\vert 10 \rangle$	$\vert 10 \rangle$
$\vert 11 \rangle$	$e^{i\lambda}\vert 11 \rangle$

A measurement straight after CP shows no change on any basis state, whatever the angle. The phase shows once the qubits are in superposition, which is what the quantum Fourier transform and phase-estimation circuits rely on.

Inverse

The inverse of cp(λ) is cp(-λ).

Usage

from math import pi
from qiskit import QuantumCircuit

qc = QuantumCircuit(2, 2)
qc.h(0)
qc.h(1)
qc.cp(pi, 0, 1)
qc.h(1)
qc.measure([0, 1], [0, 1])

import { QuantumCircuit } from 'qiskit';

const circuit = QuantumCircuit(2, 2);
circuit.h(0);
circuit.h(1);
circuit.cp(Math.PI, 0, 1);
circuit.h(1);
circuit.measure([0, 1], [0, 1]);

On a noiseless simulator every shot reads 00 or 11, in close to equal numbers: at $\lambda = \pi$ the CP is a CZ, and H, CZ, H on the target is a CX.

On IonQ hardware

CP is not a native gate, but it maps onto one entangling gate for any angle. Up to a global phase, cp(λ) is an rzz(-λ/2) with an rz(λ/2) on each qubit, and the rz rotations are virtual, so on Forte it is a single arbitrary-angle ZZ gate with no single-qubit pulses; on Aria it is one partially entangling MS gate with single-qubit pulses around it. See How your circuit is compiled.

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