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 $\lvert \psi \rangle$ of $U$ with unknown eigenvalue $e^{i\theta}$, and the task is to estimate the eigenphase $\theta$ within $\pm\delta$, with high probability. The cost of an algorithm for us will be 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 arbitrary 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 $\gamma$ with the top eigenspace. We give algorithms and matching lower bounds (up to logarithmic factors) 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 $U$. As an immediate consequence we obtain a lower bound on the complexity of the Unitary recurrence time problem, matching an upper bound of She and Yuen~[ITCS'23] and resolving one of their open questions. Lastly, we show that a phase-estimation algorithm with precision $\delta$ and error probability $\epsilon$ has cost $\Omega\left(\frac{1}{\delta}\log\frac{1}{\epsilon}\right)$, matching an easy upper bound. This contrasts with some other scenarios in quantum computing (e.g., search) where error-reduction costs only a factor $O(\sqrt{\log(1/\epsilon)})$. Our lower bound technique uses a variant of the polynomial method with trigonometric polynomials.

翻译：相位估计由 Kitaev [arXiv'95] 提出，是量子计算中最基础的子程序之一。在基本场景中，给定对酉算子 $U$ 的黑盒访问权限，以及 $U$ 的一个本征态 $\lvert \psi \rangle$（具有未知本征值 $e^{i\theta}$），任务是以高概率在 $\pm\delta$ 范围内估计本征相位 $\theta$。算法的代价将由 $U$ 和 $U^{-1}$ 的应用次数来衡量。我们严格刻画了相位估计若干变体的代价，在这些变体中，我们不再被任意给定一个本征态，而是需要估计 $U$ 的最大本征相位，并辅以状态（或制备这些状态的酉算子）形式的建议，这些状态被承诺与最大本征空间至少有一定重叠 $\gamma$。我们为所有参数范围给出了算法以及匹配的下界（至多对数因子）。研究表明，少量副本的建议状态（或建议制备酉算子）并不比无建议情况有显著优势。同时，大量建议（建议制备酉算子的多次应用）也不会显著降低代价，且对 $U$ 的本征基的了解同样无济于事。作为直接推论，我们得到了酉重现时间问题复杂性的下界，匹配了 She 和 Yuen [ITCS'23] 的上界，并解决了他们的一个开放问题。最后，我们证明精度为 $\delta$、错误概率为 $\epsilon$ 的相位估计算法代价为 $\Omega\left(\frac{1}{\delta}\log\frac{1}{\epsilon}\right)$，与一个简单上界匹配。这与量子计算中的其他场景（如搜索）形成对比，在后一类场景中，错误降低仅需因子 $O(\sqrt{\log(1/\epsilon)})$。我们的下界技术采用了一种使用三角多项式的多项式方法变体。