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

TL;DR

The paper addresses whether the observed $ν=5/2$ fractional quantum Hall state is the non-Abelian Pfaffian phase and proposes an interference-based experiment to both measure quasiparticle braiding statistics and, if non-Abelian, realize a topologically protected qubit manipulated by braiding. The proposed device uses anti-dot interferometry in a quantum Hall bar to initialize, flip, and read out the qubit encoded in a neutral fermion in the Pfaffian core, with readout via state-dependent Aharonov-Bohm interference. A key result is a remarkably small projected error rate, $rac{ ext{rate}}{ ext{gap}} o rac{T}{ riangle} e^{- riangle/T}$, giving $<10^{-30}$ for current best estimates of the gap and temperature, and potentially $<10^{-100}$ if the ideal gap of $ riangle o 2$ K is realized; this underscores the robustness of topological protection for quantum computation and provides a concrete path toward fault-tolerant qubits in non-Abelian quantum Hall states. The work also discusses extensions to other non-Abelian states and the role of disorder, emphasizing the practical significance of topological quantum computation in solid-state platforms.

Abstract

The Pfaffian state is an attractive candidate for the observed quantized Hall plateau at Landau level filling fraction $ν=5/2$. 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 $ν=5/2$ 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 $ν=5/2$, we estimate the error rate to be $10^{-30}$ or lower.

Figures (2)

• Figure 1: By evaluating the Jones polynomial at $q=\exp(\pi i/4)$ for these links, we can obtain the desired matrix elements for braiding operations manipulating the qubit. The boxed $1$ is a projector on the pair of quasiparticles which puts them in the state $|1\rangle$.
• Figure 2: A schematic depiction of a Hall bar with front gates which enable tunneling between the two edges at M, N and P, Q, thereby allowing a measurement of the qubit formed by the correlation between anti-dots 1 and 2. Front gates (shaded regions) also allow tunneling at A, B which flips the qubit.