Optimal universal quantum circuits for unitary complex conjugation

Daniel Ebler; Michał Horodecki; Marcin Marciniak; Tomasz Młynik; Marco Túlio Quintino; Michał Studziński

doi:10.1109/TIT.2023.3263771

paper

Optimal universal quantum circuits for unitary complex conjugation

arXiv:2206.00107 · doi:10.1109/TIT.2023.3263771

Abstract

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 $U_{d} ˉ$ . Our circuits admit a parallel implementation and are proven to be optimal for any $k$ and $d$ with an average fidelity of $⟨ F ⟩ = \frac{k + 1}{d ( d - k )}$ . 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 automorphisms on the group of unitary operators. Finally, we apply our optimal complex conjugation implementation to design a probabilistic circuit for reversing arbitrary quantum evolutions.

20 pages, 5 figures. Improved presentation, typos corrected, and some proofs are now clearer. Close to the published version

Optimal universal quantum circuits for unitary complex conjugation

Abstract

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