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

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.

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 $|1⟩$.

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 antidots 1 and 2. Front gates (shaded regions) also allow tunneling at A, B which flips the qubit.