Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments

TL;DR

This paper explores low-cost quantum phase estimation methods for multiple eigenvalues in noisy and noise-free circuits, introducing a new classical post-processing technique based on time-series analysis to improve eigenvalue extraction.

Contribution

It develops a novel classical post-processing approach for quantum phase estimation data, enhancing eigenvalue determination in noisy quantum experiments.

Findings

Variance in eigenvalue estimation scales linearly with the number of experiments.

Time-series analysis performs comparably to Bayesian methods in variance scaling.

Classical post-processing improves eigenvalue extraction under depolarizing noise.

Abstract

Quantum phase estimation is the workhorse behind any quantum algorithm and a promising method for determining ground state energies of strongly correlated quantum systems. Low-cost quantum phase estimation techniques make use of circuits which only use a single ancilla qubit, requiring classical post-processing to extract eigenvalue details of the system. We investigate choices for phase estimation for a unitary matrix with low-depth noise-free or noisy circuits, varying both the phase estimation circuits themselves as well as the classical post-processing to determine the eigenvalue phases. We work in the scenario when the input state is not an eigenstate of the unitary matrix. We develop a new post-processing technique to extract eigenvalues from phase estimation data based on a classical time-series (or frequency) analysis and contrast this to an analysis via Bayesian methods. We calculate the variance in estimating single eigenvalues via the time-series analysis analytically, finding that it scales to first order in the number of experiments performed, and to first or second order (depending on the experiment design) in the circuit depth. Numerical simulations confirm this scaling for both estimators. We attempt to compensate for the noise with both classical post-processing techniques, finding good results in the presence of depolarizing noise, but smaller improvements in $9$-qubit circuit-level simulations of superconducting qubits aimed at resolving the electronic ground-state of a $H_4$-molecule.