Optimal Universal Quantum Circuits for Unitary Complex Conjugation

Let Ud 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 Ud into its complex conjugate Ud. Our circuits admit a parallel implementation and are proven to be optimal for any k and d with an average fidelity of {F}=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 Ud, and the special case of k=d-1 calls. We then show that our results encompass optimal transformations from k calls of Ud to f(Ud) 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.