quantum gate directory

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 $d$:

$$ \begin{align*} \mathrm{SUM}_d |c,t\rangle & = |c, t+c \bmod d\rangle, \\ \mathrm{SUM}_d & = \sum_{c=0}^{d-1}|c\rangle\langle c| \otimes X_d^c, \end{align*} $$

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 $\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}|k,k\rangle$.

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.