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On the Approximation Complexity of Matrix Product Operator Born Machines

Matrix product operator Born machines (MPO-BMs) are tractable tensor-network models for probabilistic modeling, but their efficient approximation capability remains unclear. We characterize this boundary from both negative and positive perspectives. First, we prove that KL approximation is NP-hard for MPO-BMs in the continuous setting, ruling out universal efficient approximation in the worst case. Second, for score-based variational inference, we show that, under a locality and spectral-gap conditions on the loss-induced Hamiltonian, structured targets (e.g., path-graph Markov random fields) admit MPO-BM approximations with polynomial bond dimension and provable KL guarantees. Third, under the same locality structure, we prove that polynomially many score queries suffice to estimate the induced Hamiltonian and obtain such guarantees. Our results provide a theoretical characterization of when MPO-BMs are fundamentally hard to approximate and when they become efficiently learnable.