Non-abelian statistics from an abelian model

TL;DR

This paper demonstrates that the same highly entangled lattice states supporting abelian anyons can also support non-abelian Ising anyons, by using Clifford group operators and additional structures, revealing a new way to realize non-abelian statistics.

Contribution

It shows that non-abelian Ising anyons can be realized on abelian toric code states using Clifford operators and auxiliary qubits, bridging abelian and non-abelian models.

Findings

Non-abelian Ising anyons are superpositions of abelian toric code anyons.

The model reproduces fusion, braiding, and statistical properties of Ising anyons.

A string framing and ancillary qubits enable implementation of non-trivial chirality.

Abstract

It is well known that the abelian $Z_2$ anyonic model (toric code) can be realized on a highly entangled two-dimensional spin lattice, where the anyons are quasiparticles located at the endpoints of string-like concatenations of Pauli operators. Here we show that the same entangled states of the same lattice are capable of supporting the non-abelian Ising model, where the concatenated operators are elements of the Clifford group. The Ising anyons are shown to be essentially superpositions of the abelian toric code anyons, reproducing the required fusion, braiding and statistical properties. We propose a string framing and ancillary qubits to implement the non-trivial chirality of this model.