# Categorical Tensor Network States

*Stephen R.Clark Jacob D. Biamonte and Dieter Jaksch*

### Abstract

We examine the use of the mathematics of category theory in the description

of quantum states by tensor networks. This approach enables the development of

a categorical framework allowing a solution to the quantum decomposition

problem. Specifically, given an n-body quantum state ψ, we present

a general method to factor ψ into a tensor network. Moreover, this

decomposition of ψ uses building blocks defined mathematically in

terms of purely diagrammatic laws. 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.

Blogs about this paper:

(i) http://golem.ph.utexas.edu/category/2010/09/bimonoids_from_biproducts.html

(ii) http://johncarlosbaez.wordpress.com/2010/09/29/jacob-biamonte-on-tensor-networks/

Talks about this paper:

(i) http://new.iqc.ca/news-events/calendar/generated/jacob-biamonte-2010-12-2 (IQC, Institute for Quantum Computing

University of Waterloo, Canada)

Link to arXiv version:

* http://arxiv.org/abs/1012.0531