Unconditionally stable time discretization of Lindblad master equations in infinite dimension using quantum channels

TL;DR

This work addresses the challenge of time-stepping Lindblad master equations in infinite dimensions with unbounded generators by developing explicit quantum-channel schemes that are linear, completely positive and trace-preserving (CPTP). The authors prove first-order convergence of these CPTP schemes to the continuous solution under suitable regularity assumptions and extend the analysis to Galerkin-truncated finite-dimensional approximations. They derive a rigorous a priori framework using a regularity scale built from a reference operator $\Lambda$, establishing stability and contraction properties that enable robust time discretization without a CFL-type restriction. Numerical experiments on bosonic-code-inspired models (multi-photon loss channels) demonstrate that the quantum-channel schemes remain accurate with large time-steps and are competitive with or superior to traditional explicit and Runge–Kutta methods, while preserving physical properties of quantum evolutions. The results have significant practical impact for simulating open quantum systems in infinite dimensions, offering stable, structure-preserving, and computationally efficient time integration tools.

Abstract

We examine the time discretization of Lindblad master equations in infinite-dimensional Hilbert spaces. Our study is motivated by the fact that, with unbounded Lindbladian, projecting the evolution onto a finite-dimensional subspace using a Galerkin approximation inherently introduces stiffness, leading to a Courant--Friedrichs--Lewy type condition for explicit integration schemes. We propose and establish the convergence of a family of explicit numerical schemes for time discretization adapted to infinite dimension. These schemes correspond to quantum channels and thus preserve the physical properties of quantum evolutions on the set of density operators: linearity, complete positivity and trace. Numerical experiments inspired by bosonic quantum codes illustrate the practical interest of this approach when approximating the solution of infinite dimensional problems by that of finite dimensional problems of increasing dimension.

Figures (1)

• Figure 1: Comparison of the Euler, Lu--Cao, Quantum channel and Runge--Kutta solvers. These methods are benchmarked on simulations of the preparation of a dissipative cat qubit (see \ref{['eq_inflation_cat']}) in the top row and a Z-Gate on a dissipative cat qubit (see \ref{['eq_z_gate_cat']}) on the bottom row, for increasing dimension $N_{\mathrm{Fock}}$ of the truncated Hilbert space. For each case, a reference solution is computed using an adaptive Dormand--Prince method with relative and absolute precisions set at $10^{-14}$. The error of each solver is then approximated as the supremum in time of the trace norm distance between the result of the solver and this reference solution. For each plot, we computed the solution for 40 different values of $\Delta t$ and then connected these points with a continuous line as a guide for the eye. The saturation of the error of the Lu--Cao solvers comes from the fact that their error is structurally bounded: indeed, these schemes preserve the convex set of density operators, whose diameter is 2.

Theorems & Definitions (35)

• Remark 1
• Definition 1
• Definition 2
• Theorem 1: Theorem 3.22 of fagnolaQuantumMarkovSemigroups1999
• Example 1: daviesQuantumDynamicalSemigroups1977, Example 3.3
• Proposition 1
• Lemma 1
• proof
• Proposition 2
• proof