Categorical Tensor Network States

TL;DR

This paper introduces a categorical tensor network framework that unifies quantum state representation, enabling efficient sampling and decomposition of complex quantum states using category theory and string diagrams.

Contribution

It develops a novel categorical tensor network approach that solves the quantum decomposition problem and reveals a class of states that can be sampled efficiently and exactly.

Findings

New tensor network framework for quantum states

Efficient sampling of a large class of quantum states

General method for tensor network decomposition

Abstract

We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unification of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition problem which has several appealing features. Specifically, given an n-body quantum state S, we present a new and general method to factor S into a tensor network of clearly defined building blocks. We use the solution to expose a previously unknown and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states.