Phase estimation, due to Kitaev [arXiv'95], is one of the most fundamental subroutines in quantum computing. In the basic scenario, one is given black-box access to a unitary $U$, and an eigenstate $\lvert ψ\rangle$ of $U$ with unknown eigenvalue $e^{iθ}$, and the task is to estimate the eigenphase $θ$ within $\pmδ$, with high probability. The cost of an algorithm for us is the number of applications of $U$ and $U^{-1}$. We tightly characterize the cost of several variants of phase estimation where we are no longer given an eigenstate, but are required to estimate the maximum eigenphase of $U$, aided by advice in the form of states (or a unitary preparing those states) which are promised to have at least a certain overlap $γ$ with the top eigenspace. We give algorithms and nearly matching lower bounds for all ranges of parameters. We show that a small number of copies of the advice state (or of an advice-preparing unitary) are not significantly better than having no advice at all. We also show that having lots of advice (applications of the advice-preparing unitary) does not significantly reduce cost, and neither does knowledge of the eigenbasis of $U$. We immediately obtain a lower bound on the complexity of the Unitary recurrence time problem, resolving an open question of She and Yuen~[ITCS'23]. Lastly, we study how efficiently one can reduce the error probability in the basic phase-estimation scenario. We show that a phase-estimation algorithm with precision $δ$ and error probability $ε$ has cost $Ω\left(\frac{1}δ\log\frac{1}ε\right)$, matching an easy upper bound. This contrasts with some other scenarios in quantum computing (e.g., search) where error-probability reduction costs only a factor $O(\sqrt{\log(1/ε)})$. Our lower bound uses a variant of the polynomial method with trigonometric polynomials.

翻译：相位估计由Kitaev提出[arXiv'95]，是量子计算中最基本的子程序之一。在基本场景中，算法具备对酉算子$U$的黑盒访问权限，以及$U$的一个特征态$\lvert ψ\rangle$（其未知特征值为$e^{iθ}$），任务是以高概率在$\pmδ$精度内估计特征相位$θ$。算法的代价定义为$U$和$U^{-1}$的调用次数。我们严格刻画了多种相位估计变体的代价：这些变体不再给定特征态，而是需要估计$U$的最大特征相位，并借助承诺与最高特征空间至少具有重叠度$γ$的态（或制备这些态的酉算子）作为辅助。我们给出了所有参数范围内的算法及近乎匹配的下界。研究表明，少量副本的辅助态（或辅助态制备酉算子）并不比无辅助情况显著更优；大量辅助（即辅助态制备酉算子的多次调用）不会显著降低代价，且对$U$特征基的认知亦无此效果。由此我们直接得到酉递归时间问题复杂度的下界，解决了She与Yuen [ITCS'23]的一个公开问题。最后，我们研究了基本相位估计场景中误差概率的降低效率。我们证明：精度为$δ$、误差概率为$ε$的相位估计算法需付出$Ω\left(\frac{1}δ\log\frac{1}ε\right)$的代价，这与一个简易上界匹配。这一结果与量子计算中某些其他场景（如搜索）形成对比——后者将误差概率降低仅需$O(\sqrt{\log(1/ε)})$的因子。我们的下界证明采用了三角多项式的多项式方法变体。