Tensor Network Contractions for #SAT

Jacob D. Biamonte, Jason Morton, Jacob Turner

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)


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 $$\mathsf{{P}}$$P), determining the number of solutions can be #$$\mathsf{{P}}$$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 $$O((g+cd)^{O(1)} 2^c)$$O((g+cd)O(1) 2c) 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 $$O(\log n)$$O(logn) 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.

Original languageEnglish
Pages (from-to)1389-1404
Number of pages16
JournalJournal of Statistical Physics
Issue number5
Publication statusPublished - 1 Sep 2015
Externally publishedYes


  • Complexity
  • Computational physics
  • Quantum physics
  • Statistical physics


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