Categorical Tensor Network States

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</p