Full- and low-rank exponential Euler integrators for the Lindblad equation

TL;DR

This paper introduces new exponential Euler integrators for the Lindblad equation that unconditionally preserve positivity and trace, with theoretical error bounds and demonstrated numerical effectiveness.

Contribution

Develops novel full- and low-rank exponential Euler integrators for the Lindblad equation that guarantee physical properties unconditionally, advancing numerical methods for open quantum systems.

Findings

Integrators preserve positivity and trace unconditionally.

Theoretical error estimates are established.

Numerical experiments show improved performance over existing methods.

Abstract

The Lindblad equation is a widely used quantum master equation to model the dynamical evolution of open quantum systems whose states are described by density matrices. These solution matrices are characterized by semi-positiveness and trace preserving properties, which must be guaranteed in any physically meaningful numerical simulation. In this paper, novel full- and low-rank exponential Euler integrators are developed for approximating the Lindblad equation that preserve positivity and trace unconditionally. Theoretical results are presented that provide sharp error estimates for the two classes of exponential integration methods. Results of numerical experiments are discussed that illustrate the effectiveness of the proposed schemes, beyond present state-of-the-art capabilities.