quantum gate directory

The Chrestenson gate is the $d$-level generalization of the Hadamard gate. Let $\omega = \mathrm{e}^{2\pi i/d}$. It maps a computational-basis state $|j\rangle$ to an equal superposition whose phases depend on $j$:

$$ C_d |j\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} \omega^{jk}|k\rangle, \qquad (C_d)_{kj} = \frac{\omega^{jk}}{\sqrt{d}}. $$

For a four-level qudit, $\omega=i$ and

$$ C_4 = \frac{1}{2} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & i & -1 & -i \\ 1 & -1 & 1 & -1 \\ 1 & -i & -1 & i \end{bmatrix}. $$

Properties

• $C_2 = H$: in dimension two the Chrestenson gate is exactly the Hadamard gate.
• Unitary and symmetric: $C_d^\dagger C_d = I$ and $C_d^\mathsf{T} = C_d$. Its inverse replaces $\omega$ by $\omega^{-1}$.
• It is the one-qudit, $d$-dimensional quantum Fourier transform under the same positive-exponent convention. Consequently, $C_d^2|j\rangle = |-j \bmod d\rangle$ and $C_d^4 = I$.
• It exchanges the generalized Pauli bases: $C_d X_d C_d^\dagger = Z_d$ and $C_d Z_d C_d^\dagger = X_d^{-1}$, where $X_d$ and $Z_d$ are the shift and clock gates.

Usage

• Creates a maximal superposition from $|0\rangle$: $C_d|0\rangle = \frac{1}{\sqrt{d}}\sum_k |k\rangle$.
• In generalized Pauli-check sandwiching, $C_d$ and $C_d^\dagger$ prepare and unprepare qudit ancilla syndrome registers. This use is described in Agyeman-Bobie et al. (2026).