Numerical Analysis of Quasiholes of the Moore-Read Wave Function

Abstract

We demonstrate numerically that non-Abelian quasihole (qh) excitations of the ν = 5/2 fractional quantum Hall state have some of the key properties necessary to support quantum computation. We find that as the qh spacing is increased, the unitary transformation which describes winding two qh's around each other converges exponentially to its asymptotic limit and that the two orthogonal wavefunctions describing a system with four qh's become exponentially degenerate. We calculate the length scales for these two decays to be ξU ≈ 2.7 0 and ξE ≈ 2.3 0 respectively. Additionally we determine which fusion channel is lower in energy when two qh's are brought close together.The proposal to use quantum Hall states as a platform for quantum computation has spurred a great deal of interest 1,2,3 . These quantum Hall systems are believed to have natural "topological" immunity to decoherence and therefore hold particular promise for quantum computation. In so-called non-Abelian quantum Hall systems, the ground state is highly degenerate in the presence of quasiparticles (qp's), and this degenerate space can be used to store quantum information. Operations on this space are then performed by adiabatically dragging qp's around each other, thus "braiding" their world-lines in 2+1 dimensions.Although there is currently no definitive experimental evidence that non-Abelian quantum Hall states even exist, the community now strongly suspects 1 that the quantum Hall plateau observed at Landau level (LL) filling fraction ν = 5/2 is the non-Abelian Moore-Read (MR) phase 4 (or its closely related particle-hole conjugate 5 ). While the MR phase is, strictly speaking, not capable of universal topological quantum computation (computation by braiding qp's around each other at large distances), a scheme has been devised 6 that in principle allows error free quantum computation by supplementing these topological processes with nontopological processes where qp's are moved together and allowed to interact. Furthermore, the MR phase is frequently viewed as the simplest paradigm of a non-Abelian state of matter, and is therefore a logical starting point for detailed analysis 1 . In order for topological (or partially topological) schemes for quantum computation to be scalable (i.e., to allow large scale quantum computation), a number of crucial conditions must hold 1 . Condition (1) As all of the qp's are moved apart from one another, the splitting of the energy levels of the putatively degenerate ground state space must converge to zero at least as fast as e −R/ξ E where R is the minimum distance between qp's. In the literature, there has been numerical work suggesting that condition (1) may not be true 7 for the MR state. One of the goals of our work is to perform more precise numerical calculations to determine whether this numerical conclusion holds up to more careful scrutiny. Condition (2) As qp's are moved apart from each other, the unitary transformation that results from adiabatically dragging one qp around another must converge to its...

Majorana zero modes and topological quantum computation

“…In the Moore-Read Pfaffian state 24,25 and the anti-Pfaffian state, 40,41 proposed as candidate non-Abelian states for the 5/2-FQHS, charge-e/4 quasiparticles are Ising anyons. 26,[42][43][44][45][46][47][48] There is theoretical 28,[49][50][51][52][53][54][55][56] and experimental [29][30][31][32][57][58][59][60][61][62] evidence that the ν = 5/2 fractional quantum Hall state is in one of these two universality classes. However, there are also some experiments [63][64][65][66] that do not agree with this hypothesis.…”

Interacting non-Abelian anyons as Majorana fermions in the honeycomb lattice model

“…By simulating continuous vortex transport [30], we uncovered the characteristic oscillations in the bi-partite degeneracy lifting and characterized their physical parameter dependence. The oscillations have already been discovered for both p-wave superconductors [24] and the Moore-Read state [23], but these calculations involve mean fields and trial wave functions, respectively. Our results demonstrate the oscillations in an actual microscopic model.…”

“…However, these states can 3 become locally distinguishable due to tunneling processes that lead to the degeneracy being lifted [21]. This has been studied in Kitaev's honeycomb lattice model [22], in the Moore-Read state [23] and in p-wave superconductors [24][25][26]. It has also been discovered recently that in many-anyon systems the tunneling processes can not only lift the degeneracy but also even drive transitions between topological phases [27][28][29].…”

Three- and four-body interactions from two-body interactions in spin models: A route to Abelian and non-Abelian fractional Chern insulators

“…If the energy U n is sufficiently large, or if additional flux quanta are inserted into the system, then fractional quasiholes will be pinned at the impurity sites, as they exactly eliminate the energy cost of the impurities. By adiabatically moving these potentials around the lattice, non-abelian quasiholes can be braided to perform quantum logic gates [2,28,[32][33][34][35][36][37].…”