CCZ (Controlled-Controlled-Z or Doubly-Controlled-Z) is a three-qubit gate: applies a phase to the target qubit if both control qubits are $|1\rangle$. Like CZ but doubly-controlled, it is symmetric in all three qubits and less commonly used than CCX on most platforms.

Action: $|c_1 c_2 t\rangle \to (-1)^{c_1 c_2 t}|c_1 c_2 t\rangle$ where only the $|111\rangle$ state gets a $-1$ phase.

$$\text{CCZ} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \end{pmatrix}$$

Basis state mapping: identity on all states except $|111\rangle$. Maps $|111\rangle \to -|111\rangle$.

• Self-inverse: $\text{CCZ}^2 = I$
• Symmetric: all three qubits play equivalent roles; $\text{CCZ}$ is invariant under qubit permutations
• Diagonal: only applies phases, like CZ
• Related to CZ: extends the two-qubit CZ to three qubits

• Alternative to CCX: some algorithms use CCZ instead of CCX for phase-based operations
• Quantum simulation: simulating Hamiltonians with three-body interactions
• Stabilizer codes: appears in some error correction schemes
• Graph states: generates certain entangled graph states

CCZ can be decomposed using CCX and Hadamard gates:

$$\text{CCZ} = (I \otimes I \otimes H) \text{CCX} (I \otimes I \otimes H)$$

Alternatively, decompose into CX/CZ chains. Gate count: two Hadamard gates and one CCX, or ~6 CX gates.

• CCZ($\theta$): applies phase $e^{i\theta}$ to $|111\rangle$; $\text{CCZ}(\theta) = \text{diag}(1, 1, 1, 1, 1, 1, 1, e^{i\theta})$
• Use: tunable three-body interactions in parametric systems

• Superconducting qubits: typically decomposed into CX chains or via CCX; native implementation uncommon
• Trapped ions: three-body interactions possible with laser pulse sequences
• Photonic: challenging; usually decomposed

• CCX: doubly-controlled X; related via Hadamard conjugation
• CCY: doubly-controlled Y; related to CCZ by phase gate conjugation
• CZ: two-qubit version; CCZ extends to three qubits
• Pauli Z: the gate applied to target when both controls are $|1\rangle$
• Three-qubit gates: general category