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arXiv:cond-mat/0412343 · Physical Review Letters

Topologically Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State

S. Das Sarma, Michael Hartley Freedman, Chetan Nayak

Published Apr 27, 2005• 670 citations
arXivHTMLPDF

Abstract

The Pfaffian state is an attractive candidate for the observed quantized Hall plateau at a Landau-level filling fraction nu=5/2. This is particularly intriguing because this state has unusual topological properties, including quasiparticle excitations with non-Abelian braiding statistics. In order to determine the nature of the nu=5/2 state, one must measure the quasiparticle braiding statistics. Here, we propose an experiment which can simultaneously determine the braiding statistics of quasiparticle excitations and, if they prove to be non-Abelian, produce a topologically protected qubit on which a logical Not operation is performed by quasiparticle braiding. Using the measured excitation gap at nu=5/2, we estimate the error rate to be 10(-30) or lower.

DOI
10.1103/physrevlett.94.166802

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The Pfaffian state is an attractive candidate for the observed quantized Hall plateau at a Landau-level filling fraction nu=5/2. This is particularly intriguing because this state has unusual topological properties, including quasiparticle excitations with non-Abelian braiding statistics. In order to determine the nature of the nu=5/2 state, one must measure the quasiparticle braiding statistics. Here, we propose an experiment which can simultaneously determine the braiding statistics of quasiparticle excitations and, if they prove to be non-Abelian, produce a topologically protected qubit on which a logical Not operation is performed by quasiparticle braiding. Using the measured excitation gap at nu=5/2, we estimate the error rate to be 10(-30) or lower.
The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction $\ensuremath{\nu}=\frac{5}{2}$, can support topologically-protected qubits with extremely low error rates. Braiding operations also allow perfect implementation of certain unitary transformations of these qubits. However, in the case of the Pfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is not dense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian state supports universal quantum computation, albeit with some operations which require error correction. On the other hand, if certain topology-changing operations can be implemented, then fully topologically-protected universal quantum computation is possible. In order to accomplish this, it is necessary to measure the interference between quasiparticle trajectories which encircle other moving trajectories in a time-dependent Hall droplet geometry [cond-mat/0512072].
Fractional quantum Hall states are a cornerstone of topological physics, hosting fractionally charged quasiparticles with exotic statistics that promise to enable topologically protected quantum information processing. Among these, the Pfaffian state introduced by Moore and Read implements a p-wave pairing structure that supports excitations with non-Abelian exchange statistics. Despite extensive study in electronic systems, direct access to its pairing structure has remained limited. Here we realize a three-particle bosonic Pfaffian state of ultracold 87Rb^{87}\mathrm{Rb}87Rb atoms in an optical lattice subject to a Floquet-engineered synthetic magnetic field. Using a Bayesian-optimized adiabatic protocol, we prepare a state exhibiting Pfaffian pairing correlations. Site-resolved measurements of multi-point density correlations reveal a pronounced suppression of short-range three-body coincidences, reflecting the underlying pairing structure. We further probe the state's transport response through Hall drift measurements. Our results establish a bottom-up approach to engineering non-Abelian topological order and lay the groundwork for future explorations of anyonic braiding in synthetic matter.
We review some theoretical progress towards understanding the origin of the fractional quantum Hall effect at filling factor 5/2 using numerical techniques. The leading candidate thought to describe the physics is the Moore-Read Pfaffian state which supports non-abelian anyonic excitations. Interestingly, these excitations have possible applications in topologically protected quantum information processing. Specifically, we discuss how the finite-thickness of realistic quasi-two-dimensional systems stabilizes the Moore-Read Pfaffian in the fractional quantum Hall effect at 5/2 and the use of bilayer systems in the second Landau level to investigate the possible quantum phase transition between abelian and non-abelian fractional quantum Hall states.
The Pfaffian fractional quantum Hall (FQH) states are incompressible non-Abelian topological fluids present in a half-filled electron Landau level, where there is a balanced population of electrons and holes. They give rise to half-integral quantum Hall plateaus that divide critical transitions between integer quantum Hall (IQH) states. On the other hand, there are Abelian FQH states, such as the Laughlin state, that can be understood using partons, which are fermionic divisions of the electron. In this paper, we propose a family of incompressible paired parton FQH states at filling $\ensuremath{\nu}=\frac{1}{6}$ (modulo 1) that emerge from critical transitions between IQH states and Abelian FQH states at filling $\ensuremath{\nu}=\frac{1}{3}$ (modulo 1). These paired parton states are originated from a half-filled parton Landau level, where there is an equal amount of partons and holes. They generically support Ising-type anyonic quasiparticle excitations and carry non-Abelian Pfaffian topological orders (TO) for partons. We prove the principal existence of these paired parton states using exactly solvable interacting arrays of electronic wires under a magnetic field. Moreover, we establish a notion of particle-hole (PH) symmetry for partons and relate the PH symmetric parton Pfaffian TO with the gapped symmetric surface TO of a fractional topological insulator in three dimensions.
The observed fractional quantum Hall (FQH) plateaus follow a recurring hierarchical structure that allows an understanding of complex states based on simpler ones. Condensing the elementary quasiparticles of an Abelian FQH state results in a new Abelian phase at a different filling factor, and this process can be iterated ad infinitum. We show that condensing clusters of the same quasiparticles into an Abelian state can instead realize non-Abelian FQH states. In particular, condensing quasiparticle pairs in the ν=2/3 Laughlin state yields the anti-Pfaffian phase at half filling. We moreover show that the successive condensation of Laughlin quasiparticles produces quantum Hall states whose fillings coincide with the most prominent plateaus in the first excited Landau level of GaAs. More generally, such condensation can realize any non-Abelian FQH state that admits a parton representation. This surprising result is supported by an exact analysis of explicit wave functions, field theory arguments, conformal-field theory constructions of trial states, and numerical simulations.