Non-Abelian Anyons and Topological Quantum Computation (0707.1889v2)

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 \nu=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 \nu=5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

• The paper introduces non-Abelian anyons as carriers of unique braiding statistics essential for fault-tolerant quantum computation.
• It employs Chern-Simons theory and numerical studies to validate the topological basis and experimental feasibility of non-Abelian states in quantum Hall systems.
• The authors outline experimental approaches and material challenges that are pivotal for advancing robust, topologically protected quantum computing.

Non-Abelian Anyons and Topological Quantum Computation: An Academic Overview

The paper by Nayak et al. provides an extensive overview of the theoretical underpinnings and potential applications of non-Abelian anyons in the context of topological quantum computation. By interpreting these exotic particles' unique statistical properties, the paper explores their role in constructing fault-tolerant quantum computers. The authors discuss how quantum information can be encoded in and manipulated using the non-Abelian braiding statistics of such anyons, offering a pathway to quantum computation that is inherently resistant to error by virtue of the topological nature of the information.

The starting point of the discussion is the revolutionary concept of anyons—particles that can exist in two-dimensional systems with properties interpolating between bosons and fermions. Non-Abelian anyons, in particular, are not just characterized by a phase factor upon interchange, but involve a unitary transformation on the system's state, with multiple particles contributing to a degenerate ground state. This degeneracy, and the unitary transformations that emerge when particles are adiabatically exchanged, is the essence of non-Abelian statistics.

The authors meticulously articulate how topological quantum states, described by Chern-Simons field theories, are prototypical systems for hosting anyons. These theories are fundamentally topological, meaning they lack local degrees of freedom, and their observable properties depend only on the global configuration of the fields. Special emphasis is placed on the SU(2)$_k$ Chern-Simons theory which describes a particular class of non-Abelian anyons related to fractional quantum Hall (FQH) states, such as the Moore-Read Pfaffian state believed to be realized at filling fraction $\nu=5/2$.

Numerical studies support the hypothesis that the $\nu=5/2$ FQH state is non-Abelian and in the universality class of the Moore-Read state. The authors discuss the broader theoretical framework, including the Read-Rezayi series of states, some of which are hypothesized to be realized at fractions like $\nu=12/5$. These states are of interest because they potentially support different types of anyons, including Fibonacci anyons, which are suitable for universal quantum computation due to their richer braiding statistics.

The practical aspects of constructing a topologically protected quantum computer are also addressed. The authors examine both the conceptual framework—how to initialize, manipulate, and measure quantum states using braiding of anyons—and the specific proposal to use quantum Hall states for a realization. They propose experiments, especially involving interferometric measurements, to confirm the non-Abelian nature of excitations in quantum Hall systems. Such experiments link theoretical predictions with measurable quantities, including the tunneling and interference of quasiparticles in constrained geometries like Fabry-Perot interferometers.

The authors acknowledge both the theoretical challenges in constructing Hamiltonians with desired non-Abelian modes and the experimental difficulties, such as maintaining high sample purity at low temperatures to achieve the requisite quantum Hall states. They point out the connection between these challenges and technological improvements needed in material science, particularly focusing on the development of two-dimensional electron systems with high mobility.

Looking ahead, the paper speculates on the potential advancements in quantum computation enabled by non-Abelian anyons. Future work could leverage developments in materials technology and low temperature physics to bring such systems closer to experimental realization, potentially revolutionizing the field of quantum computation through robust, error-free information processing.

In essence, the paper is a vital contribution to the theoretical foundations of quantum computation, underscoring the importance of topological order and offering valuable insights for both theorists and experimentalists endeavoring to utilize non-Abelian anyons for developing topologically robust quantum computers.