Accurate ground state energy estimation with noise and imperfect state preparation

Abstract

We introduce a classical estimator for the post-processing of quantum phase estimation (QPE) data when a single target phase is isolated within a known interval, as is typical of ground state energy estimation of gapped systems. Our estimator filters the QPE signal within this promise region and recovers the phase through a moment-projection routine, which is robust to both external spurious phases and experimental noise. In the noiseless case this achieves an exponential suppression of bias with respect to a naive mean estimator. In the presence of global depolarizing noise the bias is exponentially small in the circuit depth $t$, and the variance is $O(t^{-2}F^{-2})$ for circuit fidelity $F$. This improves by a factor of $t^2$ over a 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]. This yields an overhead interpolating between the $F^{-4}$ scaling typical of explicitly unbiased error mitigation and a reduced $F^{-2}$ scaling when the noise samples fall outside the promise interval. We validate our estimators on a small-scale simulation of the Ising model, observing better-than-expected performance for a global depolarizing noise approximation. This robustness to both multiple eigenvalues and realistic noise makes limited-depth phase estimation practical for early fault tolerant quantum experiments.