# CCRz gate (Controlled-Controlled-RZ)

**CCRz** applies Z rotation by $\theta$ to the target if both controls are $|1\rangle$. Parametric doubly-controlled rotation for variational algorithms.

Action: $|c_1 c_2 t\rangle \to |c_1 c_2 (R_Z(\theta)|t\rangle)\rangle$ where rotation is applied only when both controls are $|1\rangle$.

$$\text{CCRz}(\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & e^{-i\theta/2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & e^{i\theta/2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & e^{-i\theta/2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\theta/2} \end{pmatrix}$$

## Properties

- **Parametric**: rotation angle $\theta$ is tunable
- **Asymmetric**: control and target qubits are distinct
- **Diagonal**: only applies phases, like RZ
- **Exponential form**: $\text{CCRz}(\theta) = e^{-i\theta Z/2}$ on target when controls are $|11\rangle$

## Uses

- **Variational quantum algorithms**: QAOA and VQE ansätze with doubly-controlled rotations
- **Conditional phase accumulation**: rotate target phase based on control qubits
- **Quantum simulation**: tunable Z interactions (Ising-like) for Hamiltonians
- **Parameterized circuits**: hybrid classical-quantum optimization

## Decomposition

CCRz can be decomposed using CX and single-qubit phase rotations:

$$\text{CCRz}(\theta) = (I \otimes I \otimes R_Z(\theta/2)) \text{CNOT}_{12} (I \otimes I \otimes R_Z(\theta/2)) \text{CNOT}_{12}$$

More efficient decompositions exist; often combined with CCP or other parametric gates.

## Implementation

- **Superconducting qubits**: typically virtual (reference frame adjustment); very low cost if decomposition is free
- **Trapped ions**: controlled phase via detuned laser interactions
- **Photonic**: challenging; usually decomposed

## Relations

- [[quantum-gate-ccrx|CCRx]], [[quantum-gate-ccry|CCRy]], [[quantum-gate-ccrz|CCRz]]: doubly-controlled rotations around three axes
- [[quantum-gate-rz|RZ gate]]: single-qubit version; target rotation
- [[quantum-gate-ccp|CCP]]: related parametric gate; CCP applies global phase instead of rotation
- [[quantum-gate-rotation|Rotation gates]]: general parametric rotation category
- [[quantum-gate-three-qubit|Three-qubit gates]]: general category