Accurate ground state energy estimation with noise and imperfect state preparation

TL;DR

This paper presents a classical estimator for quantum phase estimation data that is robust to noise and imperfect state preparation, significantly improving bias and variance performance over naive methods, and is practical for early fault-tolerant quantum experiments.

Contribution

The authors introduce a moment-projection estimator for phase estimation that effectively filters signals within a promise region, achieving exponential bias suppression and noise robustness, with practical validation on the Ising model.

Findings

Estimator achieves exponential bias suppression in noiseless case.

Estimator reduces bias exponentially with circuit depth under depolarizing noise.

Validation on Ising model simulation shows improved performance under realistic noise.

Abstract

We introduce a classical estimator for the post-processing of quantum phase estimation data generated either by quantum-Fourier-transform-based or quantum-signal-processing-based methods. We focus on the estimation of a single target phase promised to be within an interval where no other phases are present, which is typical of e.g. ground state energy estimation of gapped quantum systems. This allows us to perform phase estimation by filtering the signal within the promise region and recovering the phase through a moment-projection estimator. We show that our methods are robust in the presence of both additional phases outside the promise region and global depolarizing noise. In the noiseless case our estimator can achieve an exponential suppression of bias with respect to a naive mean estimator. In the presence of global depolarizing noise our estimator achieves a bias exponentially small in the circuit depth $t$ at fixed circuit fidelity $F$, and a variance proportional to $t^{-2}$, improving by a factor of $t^2$ over the naive shifted-and-rescaled-mean approach. To mitigate realistic circuit-level noise, we combine our method with the explicit unbiasing scheme described in [Dutkiewicz et al., 2025]. As an illustrative example, we implement these estimators on a small-scale simulation of the Ising model, validating our theoretical results and finding better-than-expected performance for a global depolarizing noise approximation. The robustness of the moment-projection estimator in the presence of both multiple eigenvalues and realistic noise makes phase estimation with limited depth practical for early fault tolerant quantum experiments.