Tensor Network Contractions for #SAT

Abstract

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 P), determining the number of solutions is #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, counting problems can be solved efficiently when their tensor network expression has at most O(log c) 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.

Hand-waving and interpretive dance: an introductory course on tensor networks

“…As determining the existence of a q-colouring is an NP-complete problem [8], contracting this graph is therefore #P-complete [9]. Indeed similar constructions exist for tensor networks corresponding to #SAT and other #P-complete problems [10]. As we will see later in section 7, there also exists a quantum hardness result which shows approximate contraction to be Post−BQP-hard, putting it inside a class of problems not believed to be efficiently solvable on even a quantum computer.…”

Report on scipost_201908_00005v1

“…Instances of (#)SAT problems can be straightforwardly represented as tensor networks [32][33][34]. Each variable and clause is encoded into a tensor T i 1 i 2 ...i d , where d is the rank of the tensor and all indices are boolean.…”

“…CSPs are commonly defined on random graphs. Even though some CSPs -in particular, decision (SAT) and counting (#SAT) problems [32][33][34] -have been formulated as tensor networks, no practical strategies for their efficient contraction exist. This is partly because it is nontrivial to define coarse-graining protocols in arbitrary graphs, but also because there is frequently no obviously advantageous order of contraction.…”

A Complete Set of Invariants for LU-Equivalence of Density Operators

“…which is a polynomial for q a natural number. The Rényi entropies [2,3,4,12,44] are a wellstudied measurement of entanglement. Positive integral (q P Z ě1 ) Rényi entropies can be measured experimentally without computing the density operators explicitly [1,7,9,41,45].…”