quantum gate directory

The controlled clock gate, also called qudit controlled-$Z$, is the $d$-level generalization of controlled-$Z$. Let $\omega = \mathrm{e}^{2\pi i/d}$. With the control listed first, it applies a phase determined by both computational-basis values:

$$ \begin{align*} \mathrm{CZ}_d |c,t\rangle & = \omega^{ct}|c,t\rangle, \\ \mathrm{CZ}_d & = \sum_{c=0}^{d-1}|c\rangle\langle c| \otimes Z_d^c, \end{align*} $$

where $Z_d$ is the clock gate. Equivalently, its diagonal element on $|c,t\rangle$ is $\omega^{ct}$.

Properties

• At $d=2$, $\mathrm{CZ}_2$ is the usual controlled-$Z$ gate.
• Diagonal and symmetric under exchanging the two qudits: $\omega^{ct}=\omega^{tc}$.
• Its inverse applies the conjugate phase, $\mathrm{CZ}_d^\dagger|c,t\rangle=\omega^{-ct}|c,t\rangle$. For $d>2$, it is generally not self-inverse, but $\mathrm{CZ}_d^d=I$.
• It is the Chrestenson-basis form of controlled shift: $\mathrm{CZ}_d=(I\otimes C_d)\,\mathrm{SUM}_d\,(I\otimes C_d^\dagger)$.

Usage

• Qudit phase kickback, generalized stabilizer circuits, and entangling controlled-phase operations.
• In the paper's Pauli-check sandwiching protocol, the clock-type right check is $\mathrm{CZ}_d$ and the corresponding left check is $\mathrm{CZ}_d^\dagger$, with an ancilla acting as the control.