# Tensor Network Contractions for #SAT

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## Abstract

The computational cost of counting the number of solutions satisfying a Boolean formula, which is a problem instance of #SAT, has proven subtle to quantify. Even when finding individual satisfying solutions is computationally easy (e.g. 2-SAT, which is in \(\mathsf{{P}}\)), determining the number of solutions can be #\(\mathsf{{P}}\)-hard. Recently, computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics-inspired techniques. By these methods, we give an algorithm using an axiomatic tensor contraction language for *n*-variable #SAT instances with complexity \(O((g+cd)^{O(1)} 2^c)\) where *c* is the number of COPY-tensors, *g* is the number of gates, and *d* is the maximal degree of any COPY-tensor. Thus, *n*-variable counting problems can be solved efficiently when their tensor network expression has at most \(O(\log n)\) COPY-tensors and polynomial fan-out. This framework also admits an intuitive proof of a variant of the Tovey conjecture (the r,1-SAT instance of the Dubois–Tovey theorem). This study increases the theory, expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science.

## Keywords

Statistical physics Complexity Computational physics Quantum physics## Notes

### Acknowledgments

We thank Tobias Fritz and Eduardo Mucciolo for providing feedback. JDB acknowledges financial support from the Fondazione Compagnia di San Paolo through the Q-ARACNE project and the Foundational Questions Institute (FQXi, under Grant FQXi-RFP3-1322). JM and JT acknowledges the NSF (under Grant NSF-1007808) for financial support.

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