Prime number factorization using a spinor Bose–Einstein condensate-inspired topological quantum computer

Abstract

Inspired by non-abelian vortex anyons in spinor Bose–Einstein condensates, we consider the quantum double D(Q8)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}(\mathbb {Q}_8)$$\end{document} anyon model as a platform to carry out a particular instance of Shor’s factorization algorithm. We suggest that the D(Q8)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}(\mathbb {Q}_8)$$\end{document} anyon model could be realized by a specific low-temperature phase of a spin-2 Bose–Einstein condensate. We provide the excitation spectrum for this model, as well as the fusion rules, braid group representations, and a circuit architecture that facilitates the computation. All necessary quantum gates, less one, can be compiled exactly for this hybrid topological quantum computer. The required non-topological gate could be implemented using measurement based protocols. To analyze the effect of decoherence on the non-topological gate, a noise model based on stochastic unitary rotations is considered. The computational potential of this quantum double anyon model is similar to that of the Majorana fermion-based Ising anyon model, thus offering a complementary future platform for topological quantum computation.