Resource-Dependent Complexity of Quantum Channels

Abstract: We introduce a new framework for quantifying the complexity of quantum channels, grounded in a suitably chosen resource set. This class of convex functions is designed to analyze the complexity of both open and closed quantum systems. By leveraging Lipschitz norms inspired by quantum optimal transport theory, we rigorously establish the fundamental properties of this complexity measure. The flexibility in selecting the resource set allows us to derive effective lower bounds for gate complexities and simulation costs of both Hamiltonian simulations and dynamics of open quantum systems. Additionally, we demonstrate that this complexity measure exhibits linear growth for random quantum circuits and finite-dimensional quantum simulations, up to the Brown-Susskind threshold.