Approximation does not help in quantum unitary time-reversal

Abstract

Access to the time-reverse $U^{-1}$ of an unknown quantum unitary process $U$ is widely assumed in quantum learning, metrology, and many-body physics. The fundamental task of unitary time-reversal dictates implementing $U^{-1}$ to within diamond-norm error $ε$ using black-box queries to the $d$-dimensional unitary $U$. Although the query complexity of this task has been extensively studied, existing lower bounds either hold only for the exact case (i.e., $ε=0$) or are suboptimal in $d$. This raises a central question: does approximation help reduce the query complexity of unitary time-reversal? We settle this question in the negative by establishing a robust and tight lower bound $Ω((1-ε)d^2)$ with explicit dependence on the error $ε$. This implies that unitary time-reversal retains optimal exponential hardness (in the number of qubits) even when constant error is allowed. Our bound applies to adaptive and coherent algorithms with unbounded ancillas and holds even when $ε$ is an average-case distance error.