Tight bounds for quantum phase estimation and related problems

Phase estimation, due to Kitaev [arXiv’95], is one of the most fundamental subroutines in quantum computing, used in Shor’s factoring algorithm, optimization algorithms, quantum chemistry algorithms, and many others. In the basic scenario, one is given black-box access to a unitary U, and an eigenstate |ψ〉 of U with unknown eigenvalue eiθ, and the task is to estimate the eigenphase θ within ±δ, with high probability. The repeated application of U and U−1 is typically the most expensive part of phase estimation, so for us the cost of an algorithm will be that number of applications. Motivated by the “guided Hamiltonian problem” in quantum chemistry, 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 a crossover point below which advice is not helpful: o(1/γ2) copies of the advice state (or o(1/γ) applications of an advice-preparing unitary) are not significantly better than having no advice at all. We also show that having knowledge of the eigenbasis of U does not significantly reduce cost. Our upper bounds use the subroutine of generalized maximum-finding of van Apeldoorn, Gilyén, Gribling, and de Wolf [Quantum’20], the state-based Hamiltonian simulation of Lloyd, Mohseni, and Rebentrost [Nature Physics’13], and several other techniques. Our lower bounds follow by reductions from a fractional version of the Boolean OR function with advice, which we lower bound by a simple modification of the adversary method of Ambainis [JCSS’02]. As an immediate consequence we also 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 an open question posed by them. Lastly, we study how efficiently one can reduce the error probability in the basic phase-estimation scenario. We show that an algorithm solving phase estimation to precision δ with error probability at most ε must have cost Ω(1δ log 1ϵ), matching the obvious way to error-reduce the basic constant-error-probability phase estimation algorithm. This contrasts with some other scenarios in quantum computing (e.g. search) where error-reduction costs only a factor O(plog(1/ϵ)). Our lower bound technique uses a variant of the polynomial method with trigonometric polynomials.

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