Fisher information based lower bounds on the cost of quantum phase estimation

Abstract: Quantum phase estimation (QPE) is a cornerstone of quantum algorithms designed to estimate the eigenvalues of a unitary operator. QPE is typically implemented through two paradigms with distinct circuit structures: quantum Fourier transform-based QPE (QFT-QPE) and Hadamard test-based QPE (HT-QPE). Existing performance assessments fail to separate the statistical information inherent in the quantum circuit from the efficiency of classical post-processing, thereby obscuring the limits intrinsic to the circuit structure itself. In this study, we employ Fisher information and the Cramer-Rao lower bound to formulate the performance limits of circuit designs independent of the efficiency of classical post-processing. Defining the circuit depth as $T$ and the total runtime as $t_{\rm total}$, our results demonstrate that the achievable scaling is constrained by a non-trivial lower bound on their product $T\,t_{\rm total}$, although previous studies have typically treated the circuit depth $T$ and the total runtime $t_{\rm total}$ as separate resources. Notably, QFT-QPE possesses a more favorable scaling with respect to the overlap between the input state and the target eigenstate corresponding to the desired eigenvalue than HT-QPE. Numerical simulations confirm these theoretical findings, demonstrating a clear performance crossover between the two paradigms depending on the overlap. Furthermore, we verify that practical algorithms, specifically the quantum multiple eigenvalue Gaussian filtered search (QMEGS) and curve-fitted QPE, achieve performance levels closely approaching our derived limits. By elucidating the performance limits inherent in quantum circuit structures, this work concludes that the optimal choice of circuit configuration depends significantly on the overlap.