Tight Bounds for Quantum Phase Estimation and Related Problems
For quantum algorithm designers, this paper provides fundamental limits on the cost of phase estimation with imperfect or limited prior knowledge, clarifying the power and limitations of advice in quantum computing.
This work tightly characterizes the query complexity of quantum phase estimation variants where the input is not an eigenstate, but an advice state with known overlap with the top eigenspace. It provides matching upper and lower bounds, showing that small advice is not significantly helpful and that large advice does not drastically reduce cost, and also resolves an open problem on unitary recurrence time.
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.