Optimal universal quantum circuits for unitary complex conjugation

• URL: http://arxiv.org/abs/2206.00107v3
• Date: Tue, 27 Aug 2024 11:24:18 GMT
• Title: Optimal universal quantum circuits for unitary complex conjugation
• Authors: Daniel Ebler, Michał Horodecki, Marcin Marciniak, Tomasz Młynik, Marco Túlio Quintino, Michał Studziński,
• Abstract summary: This work presents optimal quantum circuits for transforming a number $k$ of calls of $U_d$ into its complex conjugate $barU_d$. Our circuits admit a parallel implementation and are proven to be optimal for any $k$ and $d$ with an average fidelity of $leftlangleFrightrangle =frack+1d(d-k)$.
• Score: 1.6492989697868894
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\bar{U_d}$. 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. 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.

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