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Anyon computers with smaller groups

Carlos Mochon

TL;DR

The paper demonstrates that anyons derived from finite groups that are solvable but not nilpotent enable universal quantum computation when both magnetic and electric charges are used. It develops a complete gate-set built from braiding, fusion, and probabilistic projections, and shows how to realize measurements and magic states to achieve universality, including construction of Toffoli-like operations in the solvable non-nilpotent setting. A general group-theoretic decomposition guides the extension from the base semidirect-product case to wider classes, with explicit treatment of leakage and error management via a teleportation-style scheme. The results indicate that practical, small symmetry groups (such as $S_3$) can support universal quantum computation in topological models, guiding experimental exploration of realizable anyon systems and informing the design of fault-tolerant protocols.

Abstract

Anyons obtained from a finite gauge theory have a computational power that depends on the symmetry group. The relationship between group structure and computational power is discussed in this paper. In particular, it is shown that anyons based on finite groups that are solvable but not nilpotent are capable of universal quantum computation. This extends previously published results to groups that are smaller, and therefore more practical. Additionally, a new universal gate-set is built out of an operation called a probabilistic projection, and a quasi-universal leakage correction scheme is discussed.

Anyon computers with smaller groups

TL;DR

The paper demonstrates that anyons derived from finite groups that are solvable but not nilpotent enable universal quantum computation when both magnetic and electric charges are used. It develops a complete gate-set built from braiding, fusion, and probabilistic projections, and shows how to realize measurements and magic states to achieve universality, including construction of Toffoli-like operations in the solvable non-nilpotent setting. A general group-theoretic decomposition guides the extension from the base semidirect-product case to wider classes, with explicit treatment of leakage and error management via a teleportation-style scheme. The results indicate that practical, small symmetry groups (such as ) can support universal quantum computation in topological models, guiding experimental exploration of realizable anyon systems and informing the design of fault-tolerant protocols.

Abstract

Anyons obtained from a finite gauge theory have a computational power that depends on the symmetry group. The relationship between group structure and computational power is discussed in this paper. In particular, it is shown that anyons based on finite groups that are solvable but not nilpotent are capable of universal quantum computation. This extends previously published results to groups that are smaller, and therefore more practical. Additionally, a new universal gate-set is built out of an operation called a probabilistic projection, and a quasi-universal leakage correction scheme is discussed.

Paper Structure

This paper contains 50 sections, 138 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Review and Notation
  3. Anyon model and spectrum
  4. Magnetic charge pairs
  5. Electric charge pairs
  6. Superselection sectors, fusion, and vacuum pairs
  7. Requirements for the physical system
  8. Notation for Qudits
  9. Probabilistic projection onto $\mathcal{K}$
  10. Base Case: $G = \mathbb{Z}_p \times_{\theta} \mathbb{Z}_q$
  11. Algebraic structure
  12. Computational basis
  13. Operations involving braiding fluxes
  14. Operations involving fusion of fluxes
  15. Production of ${|{\tilde{0}}\rangle}$ states
  16. ...and 35 more sections

Figures (1)

  • Figure 1: Leakage correction circuit