rotations 2 qubits 1 parameter
Controlled $z$ Rotation
Rotates the target about the $z$ axis by $\theta$ when the control qubit is in the $|1\rangle$ state.
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.
The unitary, drawn
Every entry of the matrix as a disc. The area is the transition probability $|U_{jk}|^2$; the colour and the hand are the phase. Rows are outputs $\langle j|$, columns inputs $|k\rangle$, in the computational basis. Each parameter has a slider; play runs through the family along the path of the Quirk link ($\theta = 4\pi t$).
Phase
Disc area $= |U_{jk}|^2$. Hover an entry for its value, or take the JSON.
Weyl chamber
Where this gate sits among all two-qubit gates, up to single-qubit rotations before and after.
Coordinates:
not entangling entangling perfect entangler
$(a, b, c)$ means the gate equals $e^{i(a\,XX + b\,YY + c\,ZZ)}$ up to single-qubit gates, reduced to the chamber $\pi/4 \ge a \ge b \ge |c|$ (the convention of the canonical gate, Qiskit and Cirq). The $a = \pi/4$ face is a mirror: $(\pi/4, b, c)$ and $(\pi/4, b, -c)$ are the same gate. Drag the figure to rotate it, double-click to reset.
SDK support
How to call this gate in each framework. Hover a ⓘ for a note; the support matrix compares every gate at once.
| 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 |