Quantum Probabilistic Spaces on Graphs for Topological Evolutions

TL;DR

This paper develops a framework combining quantum probability, algebraic structures, and graph theory to model topological evolutions of anyonic particles and their quantum walks, with implications for quantum simulations.

Contribution

It introduces a novel approach using fusion rules, association schemes, and automorphisms to analyze quantum dynamics on graphs for topological quantum computation.

Findings

Fusion rules induce association schemes with Krein parameters.

Automorphisms describe the unitary dynamics of anyons.

Quantum states are defined on Bose-Mesner and von Neumann algebras.

Abstract

We start with the consideration of fusion rules of anyonic particles evolving on a 2D surface and the a hypergroup comes with it to construct entangled quantum Markov chains. The fusion rules induce an association scheme with Krein parameters and their duals the intersection numbers. One useful way to think of the schemes as regular graphs encoding the paths of possible quantum walks (automorphisms). We consider braid B3 that describes the unitary dynamics of the anyons as the automorphism subgroup of the graphs. The dynamics induced by the fusions (and the adjoint splitting operations) may be viewed as the chain evolving on a growing graph and the braiding as automorphisms on a fixed graph. In our quantum probability framework infinite iterations of the unitaries, which can encode algorithmic content for quantum simulations, can describe asymptotics elegantly if the particles are allowed to evolve coherently for a longer period. We will define quantum states on the Bose-Mesner algebra which is also a von Neumann algebra as well as a Frobenius algebra to build the quantum Markov chains providing another perspective to topological computation.