Optimal Universal Quantum Circuits for Unitary Complex Conjugation
Daniel Ebler, Michał Horodecki, Marcin Marciniak, Tomasz Młynik, Marco Túlio Quintino, Michał Studziński · IEEE Transactions on Information Theory · 2023
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$\overline {U_{d}}$. 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. This extends previous works which considered the scenario of a single call ($k=1$) of the operation$U_{d}$, and the special case of$k=d-1$calls. We then show that our results encompass optimal transformations from$k$calls of$U_{d}$to$f(U_{d})$for any arbitrary homomorphism$f$from the group of$d$-dimensional unitary operators to itself, since complex conjugation is the only non-trivial automorphism on the group of unitary operators. Finally, we apply our optimal complex conjugation implementation to design a probabilistic circuit for reversing arbitrary quantum evolutions.