Braiding of non-Abelian anyons using pairwise interactions

Abstract

The common approach to topological quantum computation is to implement quantum gates by adiabatically moving non-Abelian anyons around each other. Here we present an alternative perspective based on the possibility of realizing the exchange (braiding) operators of anyons by adiabatically varying pairwise interactions between them rather than their positions. We analyze a system composed by four anyons whose couplings define a T-junction and we show that the braiding operator of two of them can be obtained through a particular adiabatic cycle in the space of the coupling parameters. We also discuss how to couple this scheme with anyonic chains in order to recover the topological protection.

Superconducting Analogue of the Parafermion Fractional Quantum Hall States

“…Recently, the problem of perturbing a counterpropagating edge mode of a fractional topological insulator [21] or two nearby FQH states with opposite spin polarizations by either electron pairing or backscattering has attracted a lot of attention. It has been shown that the parafermion (fractionalized Majorana fermion) zero modes can be obtained at the domain wall between regions with these different mass terms [6][7][8][22][23][24][25][26][27]. It has also been shown that parafermion zero modes can be found in the bulk of an FQH state which has acquired superconducting pairing through proximity effect [9] as well as in other 2D systems [28][29][30][31][32][33][34][35].…”

New scheme for braiding Majorana fermions

“…The two MFs pick up opposite signs after exchanging their positions, which can be summarized by Equation (13) can be written as with the unitary matrix . This indicates that the braiding of MFs satisfies non-Abelian statistics [24, 25, 31, 40, 41, 43, 44], as will be shown explicitly below.…”

“…There are other possible schemes for realizing non-Abelian statistics in 1D nanowire system [24, 26, 43, 44]. Sau et al [24] proposed to transport MFs by tuning couplings between MFs, instead of driving MFs all the way along the T-junction by tuning chemical potential largely [41].…”

Braiding Majorana zero modes using quantum dots

“…The topological protection of MZMs gave them a very special status: they lie at the heart of current proposals for hardware-based fault-tolerant quantum computation [13][14][15][16][17][18][19][20][21][22][23] . Their non-Abelian statistics can be used to perform non-trivial operations on the ground states through the adiabatic exchange of two anyon positions, which is described by their braiding group [24][25][26][27][28][29][30][31][32][33][34][35][36] . Since it is not sufficient for performing universal quantum computation, it has been suggested that projective measurements could be used to implement the missing gates [37][38][39] .…”