A posteriori error estimates for the Lindblad master equation

TL;DR

This work develops computable a posteriori error bounds for simulating open quantum systems described by the Lindblad master equation in infinite-dimensional spaces, with a focus on bosonic modes. It provides a principled two-step framework—Hilbert-space truncation plus time discretization—and derives contraction-based bounds that are evaluable from the computed truncated trajectory. The key contributions include space-only and space-time error estimators, a space-adaptive truncation strategy, and extensive numerical validation across polynomial-bosonic, unitary, and dissipative scenarios, including complex GKP-type dynamics. The results enable fully adaptive simulations that automatically balance accuracy and computational cost, and they are implemented in the open-source dynamiqs_adaptive library to support scalable quantum dynamics research.

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.

Figures (8)

• Figure 1: $\|\boldsymbol{\rho}(T) - {\boldsymbol{\rho}_{(N)}}(T)\|_1$ is the truncation error at the end of the simulation described in \ref{['ex:cat', 'ex:squeezecat']}, and $\xi(T)$ is the estimate output. Errors and estimates change only at odd numbers due to the preservation of the parity operator $e^{i \pi {\bf a^\dag} {\bf a}}$. Saturation occurs below $10^{-13}$ due to the precision of the time solvers.
• Figure 2: 3D plots in log scale of the truncation error $\|\boldsymbol{\rho}(T) - {\boldsymbol{\rho}_{(N)}}(T)\|_1$ (left) and the estimate $\xi(T)$ (right). Slices of these plots are reproduced in \ref{['fig:estimator_2D_slice']}.
• Figure 3: Slices of the 3D plot in \ref{['fig:estimator_2D']} with fixed $N_2$.
• Figure 4: $\|\boldsymbol{\rho}_{(5000)}(T) - \boldsymbol{\rho}_N(T)\|_1$ for various $N$ compared to the estimate $\xi(T)$ using \ref{['eq:estimate']}, for the simulations described at the end of \ref{['subsec:GKP']}.
• Figure 5: Evolution of the size of ${\boldsymbol{\rho}_{(N)}}(t)$ and the value of the estimator $\xi$ when following \ref{['algo_1mode']} on the example described in \ref{['ex:cat']} with two different initial truncations. Parameters of the simulations are $w=5$, $\text{space\_tol}=$1e-11, $n_+=4$, $n_-=4$, the time solver is an adaptive $\text{4}^{th}$ order Runge-Kutta with $\text{time\_tol}$ set at 1e-14. The red plots are associated to a simulation starting with a small truncation. In this case, the space estimates hit the upper bound for $\text{space\_tol}*t$ multiple times, leading to an enlargement of the truncation until the estimator is stabilized with a truncation size of 31. On the blue plots, associated with initial large truncation, we observe that the truncation size shrinks until its estimator hits the upper bound $\text{space\_tol}*t$, then it is enlarged one more time before stabilizing again to a size of 31.

Theorems & Definitions (22)

• Proposition 1
• proof
• Lemma 1
• proof
• Remark 1
• Definition 1
• Proposition 2
• proof
• Example 1
• Corollary 1