Controlled-RX Gate (CRX)

A two-qubit gate that applies an X-axis rotation to the target only when the control qubit is set.

Definition

The controlled-RX gate, written CRX(θ), pairs a control qubit with a target qubit. When the control is |0⟩ the target is left alone; when the control is |1⟩ the target undergoes RX(θ) = exp(-i θ X / 2), a rotation of angle θ about the X axis of the Bloch sphere. At θ = π it reduces (up to a global phase on the target subspace) to a controlled-X.

Matrix

The upper-left 2×2 block is the identity (control |0⟩); the lower-right block is exactly RX(θ) (control |1⟩). The off-diagonal factor -i·sin(θ/2) carries the characteristic phase of an X rotation.

Decomposition

CRX decomposes into single-qubit rotations and two CNOTs. A standard identity uses the controlled-U pattern with the ABC construction: RX(θ) = A X B X C with suitable RZ/RY factors, so CRX = (I⊗A)·CNOT·(I⊗B)·CNOT·(I⊗C) plus a control-line phase. See ABC decomposition.

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

Use

Parametric controlled rotations are the workhorse of variational circuits and of quantum simulation, where a continuous angle encodes an interaction strength or an evolution time. They are near-universal building blocks: any two-qubit controlled operation reduces to controlled rotations plus entangling CNOTs.

In modeling work such as Kronos plasma-transport surrogates, parametric controlled rotations let a small circuit represent a tunable coupling without changing its structure — the angle becomes the trained parameter.