quantum gate directory
Controlled $y$ Rotation
Symbol
$\mathrm{C}R_y$
Qubits: $2$ Parameters: $1$
Description:
Rotates the target about the $y$ axis by $\theta$ when the control qubit is in the $|1\rangle$ state.
Alternate notations:
- $\mathrm{CR}_y(\theta)$
- $\Lambda(R_y(\theta))$
Controlled version of: $y$ Rotation
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.CRYGate
|
| PennyLane |
pennylane.CRY
|
| Cirq | — ⓘ |
| Q# | — ⓘ |
| PyQuil | — ⓘ |
| Braket | — ⓘ |
| BQSKit |
bqskit.ir.gates.CRYGate
|
| Qibo |
qibo.gates.CRY
|
| pytket |
pytket.circuit.OpType.CRy
ⓘ
|
| Stim | — |
| OpenQASM |
stdgates.inc: cry
|
Groups
This gate is contained in the following groups:
The controlled-$R_y$ gate applies the single-qubit rotation $R_y(\theta)$ to the target, conditioned on the control.
$$ \mathrm{C}R_y(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ 0 & 0 & \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{bmatrix} $$
Special values
| $\theta$ | Gate |
|---|---|
| $0$ | Identity |
| $\pi$ | Controlled-$Y$, up to a phase of $-i$ on the control's $ |
Properties
- Real, so it lies in the orthogonal group — the only one of the three controlled rotations with this property.
- Inverse: $\mathrm{C}R_y(\theta)^\dagger = \mathrm{C}R_y(-\theta)$.
- Related to $\mathrm{C}R_x$ by conjugating the target with $S$: $\mathrm{C}R_y(\theta) = (I \otimes S) \, \mathrm{C}R_x(\theta) \, (I \otimes S^\dagger)$.
Decompositions
- Two CNOTs and two rotations: $\mathrm{C}R_y(\theta) = (I \otimes R_y(\theta/2)) \, \mathrm{CNOT} \, (I \otimes R_y(-\theta/2)) \, \mathrm{CNOT}$.
Usage
- Ubiquitous in state preparation: cascades of $\mathrm{C}R_y$ gates load arbitrary real amplitudes, and it is the standard entangler in many variational ansätze.
Back to home