quantum gate directory
Controlled $x$ Rotation
Symbol
$\mathrm{C}R_x$
Qubits: $2$ Parameters: $1$
Description:
Rotates the target about the $x$ axis by $\theta$ when the control qubit is in the $|1\rangle$ state.
Alternate notations:
- $\mathrm{CR}_x(\theta)$
- $\Lambda(R_x(\theta))$
Controlled version of: $x$ Rotation
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.CRXGate
|
| PennyLane |
pennylane.CRX
|
| Cirq | — ⓘ |
| Q# | — ⓘ |
| PyQuil | — ⓘ |
| Braket | — ⓘ |
| BQSKit |
bqskit.ir.gates.CRXGate
|
| Qibo |
qibo.gates.CRX
|
| pytket |
pytket.circuit.OpType.CRx
ⓘ
|
| Stim | — |
| OpenQASM |
stdgates.inc: crx
|
The controlled-$R_x$ gate applies the single-qubit rotation $R_x(\theta)$ to the target, conditioned on the control.
$$ \mathrm{C}R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos\frac{\theta}{2} & -i\sin\frac{\theta}{2} \\ 0 & 0 & -i\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{bmatrix} $$
Special values
| $\theta$ | Gate |
|---|---|
| $0$ | Identity |
| $\pi$ | CNOT, up to a phase of $-i$ on the control's $ |
| $4\pi$ | Identity (the period is $4\pi$, not $2\pi$) |
Properties
- Inverse: $\mathrm{C}R_x(\theta)^\dagger = \mathrm{C}R_x(-\theta)$.
- Related to $\mathrm{C}R_z$ by a basis change on the target: $\mathrm{C}R_x(\theta) = (I \otimes H) \, \mathrm{C}R_z(\theta) \, (I \otimes H)$.
- Note the $4\pi$ periodicity: $\mathrm{C}R_x(2\pi) = Z \otimes I$, not the identity, since $R_x(2\pi) = -I$ and the sign is only applied when the control is set — where it acts as a $Z$ on the control.
Usage
- A standard entangling block in hardware-efficient variational ansätze.
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