Tight Bounds for Quantum Phase Estimation and Related Problems

Quantum 10, 2140 (2026).

https://doi.org/10.22331/q-2026-06-15-2140

Phase estimation, due to Kitaev [17], 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 $e^{itheta}$, and the task is to estimate the eigenphase $theta$ within $pmdelta$, 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 $gamma$ 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 [29].
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 $delta$ and error probability $epsilon$ has cost $Omegaleft(frac{1}{delta}logfrac{1}{epsilon}right)$, 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 $O(sqrt{log(1/epsilon)})$. Our lower bound uses a variant of the polynomial method with trigonometric polynomials.