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arXiv:2503.01712

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

Rémi Robin, Pierre Rouchon, Lev-Arcady Sellem

Published Mar 3, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2503.01712

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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.
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.
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 semipositiveness 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.
We develop a mathematically rigorous framework for simulating \emph{multiscale physical systems} using quantum computational resources, by translating the \emph{language of asymptotic-preserving (AP) schemes} into the formalism of quantum channels and Lindbladian dynamics. For stiff open quantum systems governed by singularly perturbed generators $\cL_\eps = \eps^{-1}\cL_{\mathrm{fast}} + \cL_{\mathrm{slow}}$ with $\eps \to 0$, we prove that layered quantum protocols, which implement fast-scale relaxation via native analog evolution or analytic manifold projection, converge uniformly in the diamond norm to consistent discretizations of the limiting slow dynamics, with explicit error bound $\mathcal{O}(\epsΔt + Δt^2)$ independent of stiffness. We establish precise resource-complexity bounds showing that superlinear gate-count savings Ω(κ⋅(dtot/dslow)c)Ω(κ\cdot(d_{\mathrm{tot}}/d_{\mathrm{slow}})^c)Ω(κ⋅(dtot​/dslow​)c) arise if and only if fast dynamics are resolved via (i) hardware-native analog evolution, or (ii) analytic adiabatic elimination reducing effective Hilbert space dimension. The framework is illustrated through cavity QED in the bad-cavity limit and a quantum-inspired AP discretization of kinetic equations converging to fluid limits, with quantified error propagation in trace and diamond norms. This work provides a principled mathematical bridge between classical multiscale numerical analysis and quantum simulation algorithms.
Many physical systems of interest that are encountered in practice are input-output open quantum systems described by quantum stochastic differential equations and defined on an infinite-dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with the error in the finite-dimensional approximation of input-output open quantum systems defined on an infinite-dimensional underlying Hilbert space. We present explicit error bounds between the time evolution of the state of a class of infinite-dimensional quantum systems and its approximation on a finite-dimensional subspace of the original, when both are initialized in the latter subspace. Application to a physical example drawn from the literature is provided to illustrate our results.
Nonlinear stochastic differential equations describe dynamics under uncertainty, but their nonlinear coefficients and noise averaging complicate quantum representations. We develop a Kolmogorov--Lindblad mapping that encodes their probability laws as the position diagonals of trace-one quantum density operators. For each Brownian realisation, a stochastic flow transports the initial ensemble; the square-root Jacobian makes its action on half-densities unitary. Averaging the resulting pure-state projectors gives a Lindblad equation with Hermitian jump operators. Its diagonal reproduces the Fokker--Planck density, while forward and backward intertwining identities recover bounded observables and time correlations independently of the initial coherences. A Galerkin approximation obtained by projecting the Stratonovich generators preserves the Lindblad structure. We give a residual-based error estimate and conditional quantum costs that display the dimension dependence of approximation constants, operator normalisations, state preparation and readout. Numerical experiments for double-well Langevin dynamics and noisy Lorenz--63 show rapid convergence of selected statistics at fixed dimension. The construction provides an exact bridge from flow-regular nonlinear diffusions to quantum channels; any computational advantage additionally requires controlled approximation and coherent access for the chosen problem family.