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

We develop a mathematically rigorous framework for simulating multiscale physical systems using quantum computational resources, by translating the 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^-1cL_fast + cL_slow with eps → 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}\(epsΔt + Δt²\) independent of stiffness. We establish precise resource-complexity bounds showing that superlinear gate-count savings Ω\(κcdot(d_tot/d_slow\)^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.