Tight Bound for Quantum Unitary Time-Reversal

Time-reversal of unitary evolution is fundamental in quantum information processing. Many scenarios, particularly those in quantum learning and metrology, assume free access to the time-reverse of an unknown unitary. In this paper, we settle the query complexity of the unitary time-reversal task: approximately implementing U⁻¹ given only black-box access to an unknown d-dimensional unitary U. We provide a tight query lower bound Ω((1-ε)d²) for the unitary time-reversal to within diamond norm error $ε$. Notably, our lower bound applies to general coherent protocols with unbounded ancillas, and holds even when $ε$ is an average-case distance error and access to control-U is available.