Paper
Tight Bounds for Quantum Phase Estimation and Related Problems
Authors
Nikhil S. Mande, Ronald de Wolf
Abstract
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 , and an eigenstate of with unknown eigenvalue , and the task is to estimate the eigenphase within , with high probability. The cost of an algorithm for us is the number of applications of and .
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 , aided by advice in the form of states (or a unitary preparing those states) which are promised to have at least a certain overlap 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 . 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 and error probability has cost , 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 . Our lower bound uses a variant of the polynomial method with trigonometric polynomials.