Tight Bounds for Quantum Phase Estimation and Related Problems

Phase estimation, due to Kitaev \cite{Kit95}, 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 | ψ ⟩ of U with unknown eigenvalue e i θ , and the task is to estimate the eigenphase θ within ± δ , 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 \cite{SY23}.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 ( 1 δ log ⁡ 1 ϵ ) , 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 ( log ⁡ ( 1 / ϵ ) ) . Our lower bound uses a variant of the polynomial method with trigonometric polynomials.