A universal circuit set using the S3 quantum double

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 D(S3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}({S}_{3})$$\end{document}. By embedding each on-site Hilbert space into a qubit-qutrit pair, we explicitly construct 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 compromises fault tolerance. These protocols enable the implementation of a universal gate set proposed by Cui et al. and active correction of the circuit-level noise during computation. To further suppress errors, we encode each physical degree of freedom of D(S3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}({S}_{3})$$\end{document} into a novel 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.