A Universal Circuit Set Using the $S_3$ Quantum Double

TL;DR

This paper presents a comprehensive framework for implementing fault-tolerant universal quantum computation using the non-Abelian topological code $\

Contribution

It introduces explicit circuits for anyon manipulation, a specialized interferometer for charge measurement, and a novel error-correcting code for fault tolerance in the $\

Findings

Successfully constructs circuits for creating, moving, and measuring anyons.

Designs an interferometer for remote charge measurement, avoiding fusion.

Proposes a quantum error-correcting code enabling fault-tolerant logical gate implementation.

Abstract

One potential route toward fault-tolerant universal quantum computation is to use non-Abelian topological codes. In this work, we investigate how to achieve this goal with the quantum double model $\mathcal{D}(S_3)$ -- a specific non-Abelian topological code. By embedding each on-site Hilbert space into a qubit-qutrit pair, we give an explicit construction of the circuits for creating, moving, and locally measuring all non-trivial anyons. We also design a specialized anyon interferometer to remotely measure the total charge of well-separated anyons; this avoids fusion, which would compromise fault tolerance. These protocols enable the implementation of a universal gate set proposed by Cui et al. and active quantum error correction of the circuit-level noise during the computation process. To further reduce the error rate and facilitate error correction, we encode each physical degree of freedom of $\mathcal{D}(S_3)$ into a novel, quantum, error-correcting code, enabling fault-tolerant realization, at the logical level, of all gates in the anyon manipulation circuits. Our proposal offers a promising path to realize robust universal topological quantum computation in the NISQ era.