Tight Bounds for Quantum Phase Estimation and Related Problems
This work establishes tight query complexity bounds—up to logarithmic factors—for quantum phase estimation (QPE) and its variants across all parameter regimes. We consider three problems: standard QPE, QPE with a prior auxiliary state overlapping the target eigenspace by at least γ, and maximum eigenphase estimation. Using techniques including trigonometric polynomial analysis, information-theoretic lower bound derivation, constructive algorithm design, and error amplification, we prove that achieving precision δ with failure probability ε requires Ω((1/δ) log(1/ε)) queries—matching the best-known upper bounds. Our results precisely quantify the utility of auxiliary states and prior knowledge, revealing fundamental limits on their effectiveness. Moreover, we fully resolve the query complexity of the Unitary recurrence time problem.