quantum gate directory
Controlled $z$ Rotation
Symbol
$\mathrm{C}R_z$
Qubits: $2$ Parameters: $1$
Description:
Rotates the target about the $z$ axis by $\theta$ when the control qubit is in the $|1\rangle$ state.
Alternate notations:
- $\mathrm{CR}_z(\theta)$
- $\Lambda(R_z(\theta))$
Controlled version of: $z$ Rotation
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.CRZGate
|
| PennyLane |
pennylane.CRZ
|
| Cirq | — ⓘ |
| Q# | — ⓘ |
| PyQuil | — ⓘ |
| Braket | — ⓘ |
| BQSKit |
bqskit.ir.gates.CRZGate
|
| Qibo |
qibo.gates.CRZ
|
| pytket |
pytket.circuit.OpType.CRz
ⓘ
|
| Stim | — |
| OpenQASM |
stdgates.inc: crz
|
Groups
This gate is contained in the following groups:
The controlled-$R_z$ gate applies the single-qubit rotation $R_z(\theta)$ to the target, conditioned on the control.
$$ \mathrm{C}R_z(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \mathrm{e}^{-i\theta/2} & 0 \\ 0 & 0 & 0 & \mathrm{e}^{i\theta/2} \end{bmatrix} $$
Properties
- Diagonal, so it commutes with all other diagonal gates ($Z$, $S$, $T$, $\mathrm{C}Z$, ...).
- Not the same as controlled phase, a frequent source of confusion. The two differ by a phase on the control: $$\mathrm{CP}(\theta) = (P(\theta/2) \otimes I) \, \mathrm{C}R_z(\theta)$$ In particular $\mathrm{C}R_z(\pi) \neq \mathrm{C}Z$; it equals $\mathrm{C}Z$ only up to a $-i$ phase on the control's $|1\rangle$ subspace.
- Inverse: $\mathrm{C}R_z(\theta)^\dagger = \mathrm{C}R_z(-\theta)$.
Decompositions
- Two CNOTs and two rotations: $\mathrm{C}R_z(\theta) = (I \otimes R_z(\theta/2)) \, \mathrm{CNOT} \, (I \otimes R_z(-\theta/2)) \, \mathrm{CNOT}$.
Usage
- The entangling block of many variational ansätze, and the conditional-evolution primitive in quantum simulation circuits.
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