Operator-Algebraic Methods for Asymptotic-Preserving Quantum Simulation of Open Systems

TL;DR

This paper introduces a rigorous quantum simulation framework for multiscale open systems, translating classical asymptotic-preserving schemes into quantum channels and Lindbladian dynamics, with explicit error bounds and resource complexity analysis.

Contribution

It develops a mathematically rigorous quantum simulation framework for multiscale open systems, connecting classical AP schemes with quantum channels and providing explicit error and resource bounds.

Findings

Layered quantum protocols converge uniformly to slow dynamics with explicit error bounds.

Superlinear gate-count savings are achievable when fast dynamics are resolved via native analog evolution or adiabatic elimination.

Framework applied to cavity QED and kinetic equations, with quantified error propagation.

Abstract

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 $\mathcal{O}(\eps\Delta t + \Delta t^2)$ independent of stiffness. We establish precise resource-complexity bounds showing that superlinear gate-count savings $\Omega(\kappa\cdot(d_{\mathrm{tot}}/d_{\mathrm{slow}})^c)$ 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.