Analytical lower bound on query complexity for transformations of unknown unitary operations

• URL: http://arxiv.org/abs/2405.07625v2
• Date: Thu, 28 Nov 2024 05:07:31 GMT
• Title: Analytical lower bound on query complexity for transformations of unknown unitary operations
• Authors: Tatsuki Odake, Satoshi Yoshida, Mio Murao,
• Abstract summary: We establish analytical lower bounds for the query complexity of unitary inversion. We extend our framework to the probabilistic setting, where transformations must succeed with a certain probability.
• Score: 0.8192907805418581
• License:
• Abstract: Recent developments have revealed deterministic and exact protocols for performing complex conjugation, inversion, and transposition of a general $d$-dimensional unknown unitary operation using a finite number of queries to a black-box unitary operation. In this work, we establish analytical lower bounds for the query complexity of unitary inversion, transposition, and complex conjugation. Specifically, our lower bound of $d^2$ for unitary inversion demonstrates the asymptotic optimality of the deterministic exact inversion protocol, which operates with $O(d^2)$ queries. We introduce a novel framework utilizing differentiation to derive these lower bounds on query complexity for general differentiable functions $f: \mathrm{SU}(d)\to \mathrm{SU}(d)$. As a corollary, we prove that a catalytic protocol -- a new concept recently noted in the study of exact unitary inversion -- is impossible for unitary complex conjugation. Furthermore, we extend our framework to the probabilistic setting, where transformations must succeed with a certain probability, revealing a potential trade-off between the number of queries and the required success probability.

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