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

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.