Accurate ground state energy estimation with noise and imperfect state preparation

We introduce a classical estimator for the post-processing of quantum phase estimation (QPE) data generated either by quantum-Fourier-transform-based or quantum-signal-processing-based methods. These methods sample from a distribution supported on $[0, 2\pi]$, depending on the location of the target phase within this window. We focus on the case where the target phase lies within a smaller promise region, where no other phases are present, which is typical of e.g. ground state energy estimation of gapped quantum systems. Our estimation technique is based on 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. We observe in the noiseless case our estimator achieves an exponential improvement over naive mean estimation in certain parameter regimes. In the presence of global depolarizing noise, no robust state of the art is known, and 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 $T^4$ over the naive approach. To mitigate realistic circuit-level noise, we combine our method with the explicit unbiasing scheme described in Ref.~\cite{dutkiewiczError2025}. As an illustrative example, we implement these estimators on a small-scale simulation of the Ising model. We observe that the moment-projection estimator retains some residual bias removable by the explicit unbiasing scheme, but this was significantly smaller than the variance in the system for the parameter range tested. The robustness of the moment-projection estimator in the presence of both multiple eigenvalues and noise makes phase estimation with limited depth practical for early fault tolerant quantum experiments.