Non-abelian Quantum Statistics on Graphs

TL;DR

This paper develops a topological framework using homology groups of configuration spaces to analyze non-abelian quantum statistics, especially for particles on graphs, with implications for quantum computing models.

Contribution

It introduces a general method for studying quantum statistics via homology of configuration spaces and applies it to graphs, identifying candidates for non-abelian anyon models.

Findings

Identified graph families suitable for non-abelian anyon models

Solved the universal presentation problem for certain graph configuration spaces

Provided a topological approach to quantum statistics on networks

Abstract

We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space $X$. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of $X$ which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.