A posteriori error estimates for the Lindblad master equation

TL;DR

This paper develops explicit a posteriori error bounds for simulating Lindblad master equations in infinite-dimensional quantum systems, enabling adaptive truncation and time-stepping for more efficient and accurate numerical solutions.

Contribution

It introduces a method to dynamically adjust Hilbert space truncation and time steps, providing guaranteed error bounds for Lindblad equation simulations.

Findings

Explicit bounds for truncation and discretization errors

Demonstrated efficiency and tightness of bounds through numerical examples

Adaptive truncation significantly reduces computational cost

Abstract

We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation. Furthermore, as a special case, our method naturally applies to Hamiltonian (unitary) dynamics.