Resource-Dependent Complexity of Quantum Channels
Roy Araiza, Yidong Chen, Marius Junge, Peixue Wu
TL;DR
The paper develops a resource-dependent framework for quantum channel complexity, defining $C_S$ and $C_S^{cb}$ via a Lipschitz semi-norm determined by a resource set $S$ and connecting it to Earth Mover/Wasserstein geometry. By choosing $S$ appropriately, the authors recover and relate multiple complexity notions (gate-count, Wasserstein, geometric) and prove universal lower bounds on simulation and gate costs, including linear growth in random circuits up to a Brown-Susskind threshold and lower bounds for Hamiltonian simulation costs. The framework yields concrete tools such as the quantum expected length $\\kappa(S)$, conditional expectations, and index theory, enabling both upper-bound certificates and nontrivial lower bounds across closed and open quantum dynamics. This resource-flexible approach advances the understanding of quantum circuit complexity and its scaling, with implications for certifying computational resources and the feasibility of simulating quantum systems under given constraints.
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.