Universal Topological Gates from Braiding and Fusing Anyons on Quantum Hardware

TL;DR

This paper demonstrates that non-Abelian topological order based on the $S_3$ group can be used for universal quantum computation by combining braiding and fusion operations, realized on a 54-qubit quantum processor.

Contribution

It introduces a method to achieve universal topological quantum gates using fusion as a primitive, expanding the capabilities of non-Abelian topological quantum computing.

Findings

Prepared a 54-qubit $S_3$ topological order state on a quantum processor.

Realized a universal set of topological gates combining braiding and fusion.

Successfully prepared a topological magic state for quantum computation.

Abstract

Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, $S_3$, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes $S_3$ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.