Convergence Analysis of Galerkin Approximations for the Lindblad Master Equation
Rémi Robin, Pierre Rouchon
TL;DR
The paper addresses convergence of Galerkin approximations for the Lindblad master equation on infinite-dimensional Hilbert spaces by developing Sobolev-type a priori estimates and establishing explicit $\mathcal{O}(N^{-(k-d)/2})$ convergence rates in trace norm for the Galerkin truncations ${\boldsymbol{\rho}_{(N)}}$. It shows that, under suitable regularity ${\boldsymbol{\rho}}_0\in{\mathcal W}^{k,1}$ with $k>d$, the discretized solution converges to the exact solution with a rate governed by the Sobolev index $k$ and operator-dimension parameter $d=\max(d_H,2d_j)$ for a single bosonic mode, and extends the approach to multi-mode settings with a general reference operator ${\boldsymbol{\Lambda}}$. The results are illustrated through explicit examples, including the quantum Ornstein–Uhlenbeck process and the dissipative Cat-Qubit, and further generalized to systems with a buffer cavity, highlighting the method’s applicability to autonomous quantum error correction. This work provides rigorous convergence guarantees for numerically simulating open quantum dynamics in infinite dimensions, informing stable and accurate discretization design for practical quantum technologies.
Abstract
This paper analyzes the numerical approximation of the Lindblad master equation on infinite-dimensional Hilbert spaces. We employ a classical Galerkin approach for spatial discretization and investigate the convergence of the discretized solution to the exact solution. Using \textit{a priori} estimates, we derive explicit convergence rates and demonstrate the effectiveness of our method through examples motivated by autonomous quantum error correction.