quantum gate directory
Clock
Symbol
$Z_d$
Qubits: $1$ Dimension: $d$
Description:
Qudit generalization of Pauli-$Z$ that tags each basis state with a root-of-unity phase, $|j\rangle \mapsto \omega^j |j\rangle$.
Alternate notations:
- $Z_d$
- $\Sigma_z$
- $\sigma$
Controlled variant: Controlled Clock
SDK Support
| SDK | Name |
|---|---|
| Qiskit | — |
| PennyLane |
pennylane.TClock
ⓘ
|
| Cirq | — ⓘ |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit |
bqskit.ir.gates.ClockGate
ⓘ
|
| Qibo | — |
| pytket | — |
| Stim | — |
| OpenQASM | — |
Groups
This gate is contained in the following groups:
The clock gate generalizes Pauli-$Z$ from qubits to $d$-level qudits, advancing the phase of each basis state like the hand of a clock:
$$ Z_d |j\rangle = \omega^j |j\rangle, \quad \omega = \mathrm{e}^{2\pi i / d}, \qquad Z_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \omega & 0 \\ 0 & 0 & \omega^2 \end{bmatrix} $$
Properties
- Order $d$: $Z_d^d = I$, and for $d > 2$ the gate is not Hermitian — its inverse is $Z_d^{d-1}$. At $d = 2$ it reduces exactly to Pauli-$Z$.
- Diagonal, with the $d$-th roots of unity as eigenvalues.
- Weyl commutation relation with the shift gate: $Z_d X_d = \omega \, X_d Z_d$ — the finite-dimensional analog of the position–momentum relation.
- The Fourier transform exchanges clock and shift: $F Z_d F^\dagger = X_d^{-1}$, mirroring how it exchanges the two mutually unbiased bases.
Usage
- With the shift gate it generates the generalized Pauli group, the foundation of qudit stabilizer codes and qudit magic-state theory.
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