Tight Bounds for Quantum Phase Estimation and Related Problems
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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 |ψ⟩ of U with unknown eigenvalue e^iθ, and the task is to estimate the eigenphase θ within ±δ, 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 γ 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 δ and error probability ϵ has cost Ω(1/δlog1/ϵ), 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(√(log(1/ϵ))). Our lower bound technique uses a variant of the polynomial method with trigonometric polynomials.
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