We introduce a framework for quantifying the minimal resources required for quantum simulations based on the Lipschitz dual picture of non-commutative Wasserstein metric. This approach naturally leads to rigorous lower bounds on the circuit depth and volume necessary to implement quantum operations and prepare quantum states. In particular, we show that simulating a quantum channel whose Lipschitz constant scales linearly with the system size $n$ requires a circuit depth lower bounded by $\Omega(\log n)$. Moreover, applying this framework to Lindbladian-based algorithms for Gibbs or ground state preparation, we show that even for systems engineered to exhibit rapid mixing, the required circuit volume for implementing such algorithms scales at least linearly with $n$.
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