qudits 1 qubit dimension $d$
Shift
Qudit generalization of Pauli-$X$ that cyclically increments the basis state, $|j\rangle \mapsto |j+1 \bmod d\rangle$.
The shift gate generalizes Pauli-$X$ from qubits to $d$-level qudits. Instead of flipping between two states, it cycles through all $d$ of them:
$$ X_d |j\rangle = |j + 1 \bmod d\rangle, \qquad X_3 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} $$
Properties
- Order $d$: $X_d^d = I$, and for $d > 2$ the gate is not Hermitian — its inverse is the down-shift $X_d^{d-1}$. At $d = 2$ it reduces exactly to Pauli-$X$.
- A permutation matrix, hence real and orthogonal.
- Weyl commutation relation with the clock gate: $Z_d X_d = \omega \, X_d Z_d$ where $\omega = \mathrm{e}^{2\pi i/d}$. Together, products $\omega^a X_d^b Z_d^c$ form the generalized Pauli (Weyl–Heisenberg) group.
- Diagonalized by the Fourier transform: $F X_d F^\dagger = Z_d$, so its eigenvalues are the $d$-th roots of unity and its eigenvectors are the Fourier basis states.
Usage
- Together with the clock gate, this is the starting point for qudit stabilizer codes, qudit teleportation, and discrete phase-space (Wigner function) constructions.
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.TShift ⓘ |
| Cirq | — ⓘ |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | bqskit.ir.gates.ShiftGate ⓘ |
| Qibo | — |
| pytket | — |
| Stim | — |
| OpenQASM | — |