qudits 1 qubit dimension $d$
Clock
Qudit generalization of Pauli-$Z$ that tags each basis state with a root-of-unity phase, $|j\rangle \mapsto \omega^j |j\rangle$.
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:
$$ Z_d |j\rangle = \omega^j |j\rangle, \qquad \omega = \mathrm{e}^{2\pi i / d} $$
For a qutrit ($d = 3$):
$$ Z_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \omega & 0 \\ 0 & 0 & \omega^2 \end{bmatrix} $$
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.
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, shown for $d = 3$.
Phase
Disc area $= |U_{jk}|^2$. Hover an entry for its value, or take the JSON.
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 | — |
| PennyLane | pennylane.TClock ⓘ |
| Cirq | — ⓘ |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | bqskit.ir.gates.ClockGate ⓘ |
| Qibo | — |
| pytket | — |
| Stim | — |
| OpenQASM | — |