Optimized Sampling of Mixed-State Observables

TL;DR

This paper compares randomized phase sampling and eigenstate sampling for simulating mixed quantum states, showing that eigenstate sampling is optimal for pure states and refining methods for better efficiency.

Contribution

It proves the optimality of eigenstate-based sampling for pure states and introduces refinements to improve sampling efficiency and accuracy.

Findings

Random-phase wave functions perform well for highly mixed ensembles.

Eigenstate-based sampling outperforms random-phase methods at higher purities.

Refinements accelerate convergence and exploit low-rank observable structures.

Abstract

Quantum dynamical simulations of statistical ensembles pose a significant computational challenge due to the fact that mixed states need to be represented. If the underlying dynamics is fully unitary, for example in ultrafast coherent control at finite temperatures, one approach to approximate time-dependent observables is to sample the density operator by solving the Schr\"{o}dinger equation for a set of wave functions with randomized phases. We show that, on average, random-phase wave functions perform well for ensembles with high mixedness, whereas at higher purities a deterministic sampling of the energetically lowest-lying eigenstates becomes superior. We prove that minimization of the worst-case error for computing arbitrary observables is uniquely attained by eigenstate-based sampling. We show that this error can be used to form a qualitative estimate of the set of ensemble purities for which the sampling performance of the eigenstate-based approach is superior to random-phase wave functions. Furthermore we present refinements to both schemes which remove redundant information from the sampling procedure to accelerate their convergence. Finally, we point out how the structure of low-rank observables can be exploited to further improve eigenstate-based sampling schemes.