Conjecture on hard tensor-network instances and BP fixed points

Prove or disprove that there exist hard-to-contract tensor-network instances for which belief-propagation fixed points exist but are either algorithmically hard to find or associated with strong violations of loop decay, including for tensor networks encoding partition functions of classical models.

Background

The paper’s guarantees for BP-based expansions hinge on expanding around a fixed point that exhibits loop decay. The authors suspect that, for certain contraction-hard tensor networks, fixed points may exist yet be practically inaccessible to standard message-passing, or, even when found, yield loop corrections that violate the decay conditions required for convergence.

They further note that such phenomena may arise even for tensor networks corresponding to classical partition functions, suggesting this conjecture is relevant beyond purely quantum settings and touches on complexity-theoretic aspects of tensor-network contraction.

References

We conjecture that for some hard to contract TN instances, the fixed points can exist and (i) either be hard to find, or (ii) severely violate decay of loops; these issues arise even in the tensor networks corresponding to partition functions of classical models, as we explore in upcoming work.

— Belief Propagation and Tensor Network Expansions for Many-Body Quantum Systems: Rigorous Results and Fundamental Limits  (2604.03228 - Midha et al., 3 Apr 2026) in Section: Discussions (final paragraphs)