An exact tensor network for the 3SAT problem

TL;DR

This paper constructs an exact tensor network representing 3SAT solutions, linking tensor contraction complexity to #P-completeness, and shows that finding solutions is computationally hard both theoretically and physically.

Contribution

It introduces a tensor network that encodes 3SAT solutions and demonstrates the computational hardness of contracting it, connecting quantum tensor networks with classical NP-complete problems.

Findings

Tensor network contraction counts solutions to 3SAT.

Exact contraction is #P-complete, indicating high computational complexity.

Physical realization of the tensor network state is as hard as solving 3SAT.

Abstract

We construct a tensor network that delivers an unnormalized quantum state whose coefficients are the solutions to a given instance of 3SAT, an NP-complete problem. The tensor network contraction that corresponds to the norm of the state counts the number of solutions to the instance. It follows that exact contractions of this tensor network are in the #P-complete computational complexity class, thus believed to be a hard task. Furthermore, we show that for a 3SAT instance with n bits, it is enough to perform a polynomial number of contractions of the tensor network structure associated to the computation of local observables to obtain one of the explicit solutions to the problem, if any. Physical realization of a state described by a generic tensor network is equivalent to finding the satisfying assignment of a 3SAT instance and, consequently, this experimental task is expected to be hard.