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

arXiv:2605.18886v2 Announce Type: replace 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 a protocol starting in the slow manifold (with the general state case sketched via an initial boundary layer) followed by effective evolution converges uniformly to the reduced slow state of the true stiff dynamics, with an explicit global error bound $\mathcal{O}(\eps^2(1+T) + T\Delta t^p)$ that rigorously captures $\mathcal{O}(\eps)$ physical effects (e.g., Purcell decay) without unphysical dissipation. We establish resource-complexity scaling showing that avoiding the $\Omega(1/\eps)$ stiffness penalty while capturing $\mathcal{O}(\eps)$ dynamics is efficiently achieved via analytic adiabatic elimination to precompute the effective generator, providing dimensional savings amortized against classical precomputation. The framework is illustrated through cavity QED in the bad-cavity limit and a heuristic quantum-inspired AP discretization of linear kinetic equations. This work provides a principled mathematical bridge between classical multiscale numerical analysis and quantum simulation algorithms.