T gate

The T gate, an eighth-turn phase: matrix, effect on the basis states, the t() call in Python and JS, and why it costs nothing on IonQ hardware.

The T gate multiplies the $\vert 1 \rangle$ amplitude by $e^{i\pi/4}$ , an eighth turn of phase, and leaves $\vert 0 \rangle$ alone. It acts on one qubit and takes no parameters. The Circuit panel draws it as T. On its own it changes nothing a measurement can see; the phase shows once the qubit is 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.t(qubit)`
JS	`circuit.t(qubit)`
Qiskit class	`TGate`

Matrix

T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}

On the Bloch sphere this is an eighth-turn about the Z axis. Two T gates make an S, four make a Z. It is the special case p(π/4) of the P gate.

Effect on basis states

Input	Output
$\vert 0 \rangle$	$\vert 0 \rangle$
$\vert 1 \rangle$	$e^{i\pi/4}\vert 1 \rangle$
$\vert + \rangle$	$\tfrac{1}{\sqrt{2}}(\vert 0 \rangle + e^{i\pi/4}\vert 1 \rangle)$

A measurement straight after T shows no change on $\vert 0 \rangle$ or $\vert 1 \rangle$ .

Inverse

The inverse of T is Tdg, tdg(qubit).

Usage

from qiskit import QuantumCircuit

qc = QuantumCircuit(1, 1)
qc.h(0)
for _ in range(4):
    qc.t(0)
qc.h(0)
qc.measure(0, 0)

import { QuantumCircuit } from 'qiskit';

const circuit = QuantumCircuit(1, 1);
circuit.h(0);
for (let i = 0; i < 4; i++) circuit.t(0);
circuit.h(0);
circuit.measure(0, 0);

Every shot reads 1: four T gates make a Z, and H Z H is an X.

On IonQ hardware

T is a virtual Z. The compiler folds the eighth-turn into the phase of the next pulse, so it costs no gate and no time on the machine. See How your circuit is compiled.

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