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  • Open Access

Analytical Lower Bound on Query Complexity for Transformations of Unknown Unitary Operations

Tatsuki Odake1, Satoshi Yoshida1,*, and Mio Murao1,2,†

  • 1Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
  • 2Trans-Scale Quantum Science Institute, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan

  • *Contact author: satoshiyoshida.phys@gmail.com
  • †Contact author: murao@phys.s.u-tokyo.ac.jp
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Phys. Rev. Lett. 135, 230603 – Published 5 December, 2025

DOI: https://doi.org/10.1103/drp2-rzzw

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 Letter, we establish analytical lower bounds for the query complexity of unitary inversion, transposition, and complex conjugation, which hold even if the input unitary is an unknown logarithmic-depth unitary. Specifically, our lower bound of d2 for unitary inversion demonstrates the asymptotic optimality of the deterministic exact inversion protocol, which operates with O(d2) queries. We introduce a novel framework utilizing differentiation to derive these lower bounds on query complexity for general differentiable functions f:  SU(d)→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 partially known setting, where the input unitary operation is promised to be within a subgroup of SU(d) and the probabilistic setting, where transformations succeed probabilistically.

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Information

Phys. Rev. Lett. 135, 230603– Published 5 December, 2025
  • Received 10 November 2024
  • Revised 22 September 2025
  • Accepted 5 November 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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