Towards Universal Topological Quantum Computation in the $\nu=5/2$ Fractional Quantum Hall State

TL;DR

This paper explores how the $ u=5/2$ fractional quantum Hall state, modeled by the Pfaffian state, can support universal topological quantum computation through topology-changing operations and interference measurements.

Contribution

It proposes methods for achieving fully topologically-protected universal quantum computation in the $ u=5/2$ state using topology-changing operations and interference measurements.

Findings

Topologically-protected qubits can be realized in the $ u=5/2$ state.

Universal quantum computation requires topology-changing operations.

Interference measurements are essential for implementing these operations.

Abstract

The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction $\nu = 5/2$, 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.