Optimal universal quantum circuits for unitary complex conjugation

• Published in: arXiv.org 2024-08
• Main Authors: Ebler, Daniel, Horodecki, Michał, Marciniak, Marcin, Młynik, Tomasz, Quintino, Marco Túlio, Studziński, Michał
• Format: Article
• Language: English
• Subjects: Accuracy; Automorphisms; Circuit design; Circuits; Conjugation; Homomorphisms; Operators; Optimization
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2206.00107

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.