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.

翻译：在量子学习、计量学与多体物理等领域中，广泛假设可获取未知量子幺正过程$U$的时间反演$U^{-1}$。幺正时间反演这一基本任务要求：通过对$d$维幺正算符$U$进行黑盒查询，以金刚石范数误差$ε$实现$U^{-1}$。尽管该任务的查询复杂度已得到广泛研究，但现有下界要么仅适用于精确情形（即$ε=0$），要么在维度$d$的依赖关系上未达最优。这引出一个核心问题：近似能否降低幺正时间反演的查询复杂度？我们通过建立具有显式误差$ε$依赖关系的鲁棒紧致下界$Ω((1-ε)d^2)$，对此问题给出了否定回答。该结果表明：即使允许常数误差，幺正时间反演仍保持最优的指数级难度（以量子比特数计）。我们的下界适用于具有无限辅助系统的自适应相干算法，且对$ε$为平均距离误差的情形同样成立。