Low rank approximation for the numerical simulation of high dimensional Lindblad and Riccati equations

TL;DR

This paper presents a systematic low-rank numerical method for approximating high-dimensional Lindblad and Riccati equations, demonstrating efficiency and potential for quantum and control applications.

Contribution

It introduces a deterministic rank m approximation scheme for high-dimensional density operators and Riccati equations, with a demonstrated numerical scheme and efficiency in complex quantum systems.

Findings

Efficient approximation of high-dimensional Lindblad equations with rank 12.

Successful application to quantum oscillation revivals in 50 atoms and 200 photons.

Adaptation of the method for Riccati differential equations in Kalman filtering.

Abstract

A systematic numerical approach to approximate high dimensional Lindblad equations is described. It is based on a deterministic rank m approximation of the density operator, the rank m being the only parameter to adjust. From a known initial value, this rank m approximation gives at each time-step an estimate of the largest m eigen-values with their eigen-vectors of the density operator. A numerical scheme is proposed. Its numerical efficiency in the case of a rank 12 approximation is demonstrated for oscillation revivals of 50 atoms interacting resonantly with a slightly damped coherent quantized field of 200 photons. The approach may be employed for other similar equations. We in particularly show how to adapt such low-rank approximation for Riccati differential equations appearing in Kalman filtering.