Heisenberg limited multiple eigenvalue estimation via off-the-grid compressed sensing

TL;DR

This paper introduces an off-the-grid compressed sensing method combined with signal classification for efficient, simultaneous estimation of multiple eigenvalues in quantum systems, achieving Heisenberg-limited accuracy with minimal sampling.

Contribution

It presents a novel off-grid compressed sensing protocol for quantum eigenvalue estimation, demonstrating Heisenberg-limited performance and incorporating prior knowledge for improved accuracy.

Findings

Achieves Heisenberg limit in eigenvalue estimation

Requires only a few percent of the autocorrelation function sampling

Develops a modified protocol leveraging prior signal knowledge

Abstract

Quantum phase estimation is the flagship algorithm for quantum simulation on fault-tolerant quantum computers. We demonstrate that an \emph{off-grid} compressed sensing protocol, combined with a state-of-the-art signal classification method, enables the simultaneous estimation of multiple eigenvalues of a unitary matrix using the Hadamard test while sampling only a few percent of the full autocorrelation function. Our numerical evidence indicates that the proposed algorithm achieves the Heisenberg limit in both strongly and weakly correlated regimes and requires very short evolution times to obtain an $\epsilon$-accurate estimate of multiple eigenvalues at once. Additionally -- and of independent interest -- we develop a modified off-grid protocol that leverages prior knowledge of the underlying signal for faster and more accurate recovery. Finally, we argue that this algorithm may offer a potential quantum advantage by analyzing its resilience with respect to the quality of the initial input state.