RZZ gate

The RZZ gate, a two-qubit rotation about Z⊗Z: matrix, effect on the basis states, the rzz() call in Python and JS, and its IonQ native-gate form.

The RZZ gate rotates a pair of qubits together about the $Z \otimes Z$ axis by an angle $\theta$ : it adds a phase that depends on whether the two qubits agree. It acts on two qubits and takes one parameter, $\theta$ in radians. The two qubits are interchangeable. The Circuit panel draws a Zz(θ) box on each wire, with the angle shown as a multiple of π. On its own it changes nothing a measurement can see, whatever the angle; the phases show 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.rzz(θ, qubit1, qubit2)`
JS	`circuit.rzz(θ, qubit1, qubit2)`
Qiskit class	`RZZGate`

Matrix

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

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

That is $e^{-i\frac{\theta}{2} Z \otimes Z}$ . It is the same as a CX, an RZ of θ on the target, and a second CX, and the CP gate is an RZZ with single-qubit phases added.

Effect on basis states

Input	Output
$\vert 00 \rangle$	$e^{-i\theta/2}\vert 00 \rangle$
$\vert 01 \rangle$	$e^{i\theta/2}\vert 01 \rangle$
$\vert 10 \rangle$	$e^{i\theta/2}\vert 10 \rangle$
$\vert 11 \rangle$	$e^{-i\theta/2}\vert 11 \rangle$

A measurement straight after RZZ shows no change on any basis state, whatever the angle. The phases show once the qubits are in superposition: this gate is the building block of Ising-model and QAOA circuits.

Inverse

The inverse of rzz(θ) is rzz(-θ).

Usage

from math import pi
from qiskit import QuantumCircuit

qc = QuantumCircuit(2, 2)
qc.h(0)
qc.h(1)
qc.rzz(pi, 0, 1)
qc.h(0)
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.rzz(Math.PI, 0, 1);
circuit.h(0);
circuit.h(1);
circuit.measure([0, 1], [0, 1]);

On a noiseless simulator every shot reads 11: between the H gates, rzz(π) flips the sign of both qubits' $\vert 1 \rangle$ components together.

On IonQ hardware

RZZ is the abstract form of Forte's native entangler: rzz(θ) is exactly ZZ(θ/2π), one native gate with no single-qubit pulses at all, and Forte is the hardware behind Qollab's QPU backends. On Aria it is one MS gate with single-qubit pulses around it. See How your circuit is compiled.

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