From Tensor Networks to Tractable Circuits, and back

2026-04-30

AI Summary

This work bridges the theoretical gap between tensor networks and tractable circuits from knowledge compilation in representing pseudo-Boolean functions. By establishing rigorous structural correspondences, it proves for the first time that matrix product states are equivalent to nondeterministic ordered binary decision diagrams, and that tree tensor networks are equivalent to structured decomposable negation normal form circuits. This bidirectional theoretical transfer not only endows tensor networks with formal guarantees of canonicity and computability but also enhances tractable circuits with greater expressive power and a richer algorithmic toolkit. The results foster cross-community synergy by enabling mutual adoption and co-development of methodologies from both fields.

Abstract

Tensor networks and circuits are widely used data structures to represent pseudo-Boolean functions. These two formalisms have been studied primarily in separate communities, and this paper aims to establish equivalences between them. We show that some classes of tensor networks that are appealing in practice correspond to classes of circuits with specific properties that have been studied in knowledge compilation as \emph{tractable circuits}. In particular, we prove that matrix product states (tensor trains) coincide with nondeterministic edge-valued decision diagrams and that tree tensor networks exactly correspond to structured-decomposable circuits. These correspondences enable direct transfer of structural and algorithmic results; for example, canonicity and tractability guarantees known for circuits yield analogous guarantees for the associated tensor networks, and vice versa.

Problem

• tensor networks
• tractable circuits
• pseudo-Boolean functions
• knowledge compilation
• equivalences

Innovation

• tensor networks
• tractable circuits
• matrix product states
• structured-decomposable circuits
• knowledge compilation