Parafermion braiding in fractional quantum Hall edge states with a finite chemical potential

Abstract

Parafermions are non-Abelian anyons which generalize Majorana fermions and hold great promise for topological quantum computation. We study the braiding of Z 2n parafermions which have been predicted to emerge as bound states in fractional quantum Hall systems at filling factor ν = 1/n (n odd). Using a combination of bosonization and refermionization, we calculate the energy splitting as a function of distance and chemical potential for a pair of parafermions separated by a gapped region. Braiding of parafermions in quantum Hall edge states can be implemented by repeated fusion and nucleation of parafermion pairs. We simulate the conventional braiding protocol of parafermions numerically, taking into account the finite separation and finite chemical potential. We show that a nonzero chemical potential poses challenges for the adiabaticity of the braiding process because it leads to accidental crossings in the spectrum. To remedy this, we propose an improved braiding protocol which avoids those degeneracies. arXiv:1907.12799v1 [cond-mat.str-el]

Holonomic implementation of CNOT gate on topological Majorana qubits

“…This holonomic protocol is completely equivalent to the physical clockwise braiding of γ 1 and γ 2 . This can be easily understood by tracking the motion of the zero-energy modes, initially associated with γ 1 and γ 2 , throughout the evolution of the system along Γ [31,51]. It is therefore not surprising that this holonomic protocol inherits the same topological protection featured by the physical braiding of MZMs.…”

“…Indeed, it is possible to tune the coupling strengths c j away from the idle configuration without destroying the qubit, keeping the fixed-parity computational space degenerate and separated from the excited states by a finite energy gap. A sufficient condition for the stability of the qubit is that, at each time t, either one or two of the five different couplings strengths c j must be non-vanishing [31,50,51].…”

“…Faster implementations of the protocol would result in | f (T )〉 = | f (∞)〉. The oscillations featured by the overlap |〈 f (∞)| f (T )〉| 2 for short T can be understood as interference patterns resulting from subsequent transitions between ground and excited states, in analogy with the Landau-Zener-Stückelberg effect [51,62].…”

Report on scipost_202010_00010v1

“…Faster implementations of the protocol would result in |f (T ) = |f (∞) . The oscillations featured by the overlap | f (∞)|f (T ) | 2 for short T can be understood as interference patterns resulting from subsequent transitions between ground and excited states, in analogy with the Landau-Zener-Stückelberg effect [51,62]. Let us define the "adiabatic threshold" T ad as the minimal duration of the holonomic process for which diabatic effects become negligible, say 1 − | f (∞)|f (T ) | 2 < 10 −3 for all T ≥ T ad .…”

Current correlations of Cooper-pair tunneling into a quantum Hall system

“…In the recent past, it has become possible to investigate such transport problems not only between identical ballistic chiral QH states but also between distinct systems, such as QH edge states and superconductors, both theoretically [61][62][63][64][65][66][67][68][69][70] and experimentally [19,20,22,[71][72][73][74]. This line of research is particularly relevant for the creation of parafermion bound states, non-Abelian quasiparticles with potential application in topological quantum computation [75][76][77][78][79][80][81][82][83][84]. Motivated by this progress, we investigate the noise properties of the tunneling current between a superconductor and QH edge states at integer and Laughlin filling factors.…”