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We introduce a notion of complexity for quantum channels, depending on a
suitably chosen resource set. This new class of convex functions on quantum
channels is suitable to study the complexity of both open and closed quantum
systems. Crucial properties of this class of complexities are derived from
Lipschitz norms motivated by noncommutative geometry. We prove linear growth of
our complexity for Hamiltonian simulation and random circuits, up to a
Brown-S\"usskind threshold.
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