RZ gate

The RZ gate, a rotation about Z by any angle: matrix, effect on the basis states, the rz() call in Python and JS, and why it costs nothing on IonQ hardware.

The RZ gate rotates a qubit about the Z axis of the Bloch sphere by an angle $\phi$ . It acts on one qubit and takes one parameter, $\phi$ in radians. The Circuit panel draws it as Rz(φ) with the angle shown as a multiple of π. On its own it changes nothing a measurement can see, whatever the angle; the rotation 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.rz(φ, qubit)`
JS	`circuit.rz(φ, qubit)`
Qiskit class	`RZGate`
Controlled form	`crz(φ, control, target)`

Matrix

RZ(\phi) = \begin{pmatrix} e^{-i\phi/2} & 0 \\ 0 & e^{i\phi/2} \end{pmatrix}

RZ and the P gate are the same operation up to a global phase: rz(φ) and p(φ) give identical measurement statistics in any circuit. rz(π) is a Z gate up to a global phase.

Effect on basis states

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

A measurement straight after RZ shows no change on $\vert 0 \rangle$ or $\vert 1 \rangle$ , whatever the angle. The rotation shows once the qubit is in superposition.

Inverse

The inverse of rz(φ) is rz(-φ).

Usage

from math import pi
from qiskit import QuantumCircuit

qc = QuantumCircuit(1, 1)
qc.h(0)
qc.rz(pi, 0)
qc.h(0)
qc.measure(0, 0)

import { QuantumCircuit } from 'qiskit';

const circuit = QuantumCircuit(1, 1);
circuit.h(0);
circuit.rz(Math.PI, 0);
circuit.h(0);
circuit.measure(0, 0);

Every shot reads 1: rz(π) acts as a Z, and H Z H is an X.

On IonQ hardware

RZ is the virtual Z itself. The hardware performs no physical operation for it; the compiler adds the angle to the phase of the pulses that follow, so RZ costs no gate and no time on the machine, whatever the angle. A circuit heavy in Z rotations can compile smaller than its written gate count suggests. See How your circuit is compiled.

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