quantum gate directory

Qubits: $2$ Dimension: $d$

SDK Support

SDK	Name
Qiskit	—
PennyLane	—
Cirq	— ⓘ
Q#	—
PyQuil	—
Braket	—
BQSKit	—
Qibo	—
pytket	—
Stim	—
OpenQASM	—

Groups

This gate is contained in the following groups:

• orthogonal
• permutation

Reference

Generalizing Pauli Checks for Qudit-based Quantum Error Detection and Mitigation (2026)

The controlled shift, commonly called the SUM or modular-addition gate, is the qudit generalization of CNOT. With the control listed first, it adds the control value to the target modulo $d$:

$$ \mathrm{SUM}_d |c,t\rangle = |c, t+c \bmod d\rangle, \qquad \mathrm{SUM}d = \sum^{d-1}|c\rangle\langle c| \otimes X_d^c, $$

where $X_d$ is the shift gate. It permutes the $d^2$ two-qudit computational-basis states.

Properties

• Its inverse subtracts the control value: $\mathrm{SUM}_d^\dagger |c,t\rangle = |c,t-c \bmod d\rangle$. Hence $\mathrm{SUM}_d^d=I$.
• At $d=2$, addition modulo two is XOR, so $\mathrm{SUM}_2$ is CNOT.
• Conjugating the target with the Chrestenson gate produces the controlled clock gate: $\mathrm{CZ}_d = (I \otimes C_d),\mathrm{SUM}_d,(I \otimes C_d^\dagger)$.
• Applying $C_d$ to the control of $|0,0\rangle$, then applying $\mathrm{SUM}d$, prepares the generalized GHZ state $\frac{1}{\sqrt{d}}\sum^{d-1}|k,k\rangle$.

Usage

• Reversible modular arithmetic, qudit teleportation, and generalized GHZ-state preparation.
• In the paper's Pauli-check sandwiching protocol, the shift-type checks condition $X_d^k$ on ancilla state $|k\rangle$; these are the SUM gate and its inverse, with the ancilla as the control.