Adaptive low-rank approximation and denoised Monte-Carlo approach for high-dimensional Lindblad equations

TL;DR

This paper introduces adaptive low-rank and denoised Monte Carlo methods for efficiently simulating high-dimensional Lindblad equations, combining deterministic and stochastic approaches with error control and variance reduction.

Contribution

It proposes a novel adaptive low-rank approximation with dynamic rank adjustment and a variance reduction technique using low-rank control variates for Lindblad equations.

Findings

Efficient simulation of quantum collapse and revivals.

Complementary effectiveness of low-rank and Monte Carlo methods.

Demonstrated error control and variance reduction in numerical tests.

Abstract

We present a twofold contribution to the numerical simulation of Lindblad equations. First, an adaptive numerical approach to approximate Lindblad equations using low-rank dynamics is described: a deterministic low-rank approximation of the density operator is computed, and its rank is adjusted dynamically, using an on-the-fly estimator of the error committed when reducing the dimension. On the other hand, when the intrinsic dimension of the Lindblad equation is too high to allow for such a deterministic approximation, we combine classical ensemble averages of quantum Monte Carlo trajectories and a denoising technique. Specifically, a variance reduction method based upon the consideration of a low-rank dynamics as a control variate is developed. Numerical tests for quantum collapse and revivals show the efficiency of each approach, along with the complementarity of the two approaches.