On the Approximation Complexity of Matrix Product Operator Born Machines

2026-05-11

AI Summary

This work investigates the approximation capabilities of Matrix Product Operator Born Machines (MPO-BMs) in probabilistic modeling. We prove that, in the general continuous setting, approximating a target distribution—measured by KL divergence—is NP-hard. However, when the target distribution exhibits locality and a spectral gap in its associated Hamiltonian, we construct an MPO-BM with polynomial bond dimension that achieves efficient approximation with provable error bounds. Moreover, such a model can be learned using only a polynomial number of score queries. This study establishes, for the first time, a precise boundary between computational intractability and learnability for MPO-BMs, integrating tools from computational complexity theory, tensor network approximation, score matching, and spectral analysis of Hamiltonians to highlight the critical role of structural priors in enabling efficient modeling.

Abstract

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.

Problem

• Matrix Product Operator Born Machines
• Approximation Complexity
• Probabilistic Modeling
• KL Approximation
• Tensor Networks

Innovation

• Matrix Product Operator Born Machines
• Approximation Complexity
• Score-based Variational Inference
• Locality
• Spectral Gap