Z gate

The Z gate leaves $\vert 0 \rangle$ alone and flips the sign of $\vert 1 \rangle$. It acts on one qubit and takes no parameters. The Circuit panel draws it as Z. On its own it changes nothing a measurement can see; the sign 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

Matrix

On the Bloch sphere this is a half-turn about the Z axis. Z is the special case p(π) of the P gate, and two S gates make one Z.

Effect on basis states

A measurement straight after Z shows no change on $\vert 0 \rangle$ or $\vert 1 \rangle$. The sign flip shows once the qubit is in superposition: Z turns $\vert + \rangle$ into $\vert - \rangle$, which an H gate then reads out as a 1.

Inverse

Z is its own inverse. Two Z gates in a row on the same qubit cancel.

Usage

from qiskit import QuantumCircuit

qc = QuantumCircuit(1, 1)
qc.h(0)
qc.z(0)
qc.h(0)
qc.measure(0, 0)

import { QuantumCircuit } from 'qiskit';

const circuit = QuantumCircuit(1, 1);
circuit.h(0);
circuit.z(0);
circuit.h(0);
circuit.measure(0, 0);

Every shot reads 1: the two H gates would cancel on their own, and the Z between them turns the pair into an X.

On IonQ hardware

Z is a virtual Z. The hardware has no physical Z gate; the compiler folds the half-turn into the phase of whatever pulse comes next, so Z costs no gate and no time on the machine. See How your circuit is compiled.

Related

• S gate
• P gate
• RZ gate
• CZ gate
• Gate reference