Undecidability in Tensor Network States

TL;DR

This paper introduces three new undecidable problems related to tensor networks in quantum information science, highlighting how small changes in constraints can alter problem decidability.

Contribution

It presents novel undecidable problems in tensor network states, connecting physical concepts with computational complexity.

Findings

Certain tensor network problems are undecidable.

Small modifications can switch problems between decidable and undecidable.

Some problems are always satisfiable regardless of constraints.

Abstract

Recent work has examined how undecidable problems can arise in quantum information science. We augment this by introducing three new undecidable problems stated in terms of tensor networks. These relate to ideas of Penrose about the physicality of a spin-network representing a physical process, closed timelike curves, and Boolean relation theory. Seemingly slight modifications of the constraints on the topology or the tensor families generating the networks leads to problems that transition from decidable, to undecidable to even always satisfiable.