Non-Abelian Anyons and Topological Quantum Computation

TL;DR

The paper surveys how non-Abelian anyons in two-dimensional topological phases provide a fault-tolerant platform for quantum computation. It weaves together Chern-Simons theory, conformal field theory, and knot invariants to characterize braiding, fusion, and quantum dimensions, and anchors these ideas in concrete quantum Hall states such as the ν=5/2 Moore–Read Pfaffian and the ν=12/5 Read–Rezayi states. It analyzes interferometric readouts and QC architectures, highlighting that Fibonacci anyons offer universal topological quantum computation while the ν=5/2 state requires supplementary non-topological operations or alternative non-Abelian states for universality. The work also discusses practical challenges—gap sizes, material quality, and error mechanisms—alongside broader non-Abelian platforms beyond quantum Hall systems, painting a roadmap for realizing robust, scalable topological quantum computation.

Abstract

Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as {\it Non-Abelian anyons}, meaning that they obey {\it non-Abelian braiding statistics}. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the ν=5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the ν=5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Figures (23)

• Figure 1: Top: The two elementary braid operations $\sigma_1$ and $\sigma_2$ on three particles. Middle: Here we show $\sigma_2 \sigma_1 \neq \sigma_1 \sigma_2$, hence the braid group is Non-Abelian. Bottom: The braid relation (Eq. \ref{['eq:braidrelation1']}) $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$.
• Figure 2: A quantum Hall analog of a Fabry-Perot interferometer. Quasiparticles can tunnel from one edge to the other at either of two point contacts. To lowest order in the tunneling amplitudes, the backscattering probability, and hence the conductance, is determined by the interference between these two processes. The area in the cell can be varied by means of a side gate $S$ in order to observe an interference pattern.
• Figure 3: If a third constriction is added between the other two, the cell is broken into two halves. We suppose that there is one quasiparticle (or any odd number) in each half. These two quasiparticles (labeled $1$ and $2$) form a qubit which can be read by measuring the conductance of the interferometer if there is no backscattering at the middle constriction. When a single quasiparticle tunnels from one edge to the other at the middle constriction, a $\sigma_x$ or NOT gate is applied to the qubit.
• Figure 4: The functional integrals which give (a) $\langle\chi|\rho\!\left({\sigma_2^2}\right)|\chi\rangle$ (b) $\langle\chi|\chi\rangle$, (c) $\langle\chi|\rho\!\left({\sigma_2}\right)|\chi\rangle$, (d) $\langle\chi|\rho\!\left({\sigma_2^{-1}}\right)|\chi\rangle$.
• Figure 5: The Kauffman bracket is invariant under continuous motions of the arcs and, therefore, independent of the particular projection of a link to the plane.

Theorems & Definitions (1)

• Definition 4.1