RZX Gate

A parametric ZX-interaction gate, the ideal operation realized by the cross-resonance drive and a flexible pulse-efficient primitive.

Definition

The RZX(θ) gate implements exp(-i θ (Z⊗X) / 2): a rotation of the target about X whose sense depends on the control's Z value. It is the ideal, calibration-target interaction produced by the cross-resonance effect, and it can be run at any angle θ rather than only at the CNOT-equivalent value.

Matrix

When the control is |0⟩ the target sees RX(θ); when the control is |1⟩ it sees RX(-θ). At θ = π/2 the gate is locally equivalent to CNOT, so a pair of single-qubit gates converts it into a standard CNOT.

Pulse-efficient compilation

Because the physical cross-resonance pulse can be shortened to produce a smaller θ, RZX at a partial angle costs proportionally less time than a full CNOT. Compilers that resynthesize two-qubit blocks directly into scaled RZX pulses — rather than into fixed CNOTs — cut circuit duration and error for gates whose native interaction angle is small.

import numpy as np
def rzx(theta):
    c, s = np.cos(theta/2), -1j*np.sin(theta/2)
    m = np.eye(4, dtype=complex)
    m[0,0]=c; m[1,1]=c; m[0,1]=s; m[1,0]=s
    m[2,2]=c; m[3,3]=c; m[2,3]=-s; m[3,2]=-s
    return m

Uses

RZX layers are used in pulse-efficient variational circuits and in simulating Hamiltonians whose interaction terms are ZX-like. Its local equivalence to the Ising gates means any RXX or RYY can be relabeled into an RZX by single-qubit basis changes. See cross-resonance and RXX.