Optimal universal quantum circuits for unitary complex conjugation

Understand

Let $U_d$ be a unitary operator representing an arbitrary $d$-dimensional unitary quantum operation.

• This work presents optimal quantum circuits for transforming a number $k$ of calls of $U_d$ into its complex conjugate $\bar{U_d}$.
• Our circuits admit a parallel implementation and are proven to be optimal for any $k$ and $d$ with an average fidelity of $\left\langle{F}\right\rangle =\frac{k+1}{d(d-k)}$.
• Optimality is shown for average fidelity, robustness to noise, and other standard figures of merit.