Numerical Analysis of Quasiholes of the Moore-Read Wavefunction

TL;DR

This paper numerically investigates non-Abelian quasiholes in the Moore-Read state, demonstrating their properties relevant for quantum computation, including exponential convergence of braiding transformations and degeneracy of wavefunctions.

Contribution

It provides numerical evidence of key properties of quasiholes in the Moore-Read state, including decay lengths and fusion channel energies, supporting their potential for topological quantum computing.

Findings

Braiding transformations converge exponentially with quasihole separation.

Degeneracy of quasihole wavefunctions becomes exponential with distance.

Lower energy fusion channel identified when quasiholes are close.

Abstract

We demonstrate numerically that non-Abelian quasihole excitations of the $\nu = 5/2$ fractional quantum Hall state have some of the key properties necessary to support quantum computation. We find that as the quasihole spacing is increased, the unitary transformation which describes winding two quasiholes around each other converges exponentially to its asymptotic limit and that the two orthogonal wavefunctions describing a system with four quasiholes become exponentially degenerate. We calculate the length scales for these two decays to be $\xi_{U} \approx 2.7 \ell_0$ and $\xi_{E} \approx 2.3 \ell_0$ respectively. Additionally we determine which fusion channel is lower in energy when two quasiholes are brought close together.