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arXiv:2006.11883 · Annals of Physics

Majorana orthogonal transformation and Majorana zero modes in free fermionic systems

Jianlong Fu

Published Jul 16, 2021• 13 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2021.168564

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Abstract Fusing Majorana zero modes (MZMs) leads to multiple outcomes, a property unique to non-Abelian anyons. The successful demonstration of this nontrivial fusion rule would be a hallmark for the development of topological quantum computation. Here, we propose that such fusion can be achieved by simply attaching a fermionic mode to a single Majorana zero mode. By modulation of the energy level of the fermionic mode (FM) as well as its coupling with MZM in different sequences, we show that a zero or integer charge pumping can be realized when different fusion loops are chosen. Such fusion loops are intimately related with the nontrivial fusion rule of Majorana modes and are solely determined by the crossings at zero energy in the parameter space. Finally, we demonstrate our proposal in a nanowire-based topological superconductor coupled to a quantum dot. We show that charge pumping is robust for MZMs in the realistic system irrespective of the initial condition of FM state.
Majorana zero modes (MZMs) promise a platform for topologically protected fermionic quantum computation. However, creating multiple MZMs and generating (directly or via measurements) the requisite transformations (e.g., braids) pose significant challenges. We introduce fermion-parity-based computation (FPBC): a measurement-based scheme, modeled on Pauli-based computation, that uses efficient classical processing to virtually increase the number of available MZMs and which, given magic state inputs, operates without transformations. FPBC requires all MZM parities to be measurable, but this conflicts with constraints in proposed MZM hardware. We thus introduce a design in which all parities are directly measurable and which is hence well suited for FPBC. While developing FPBC, we identify the "logical braid group" as the fermionic analog of the Clifford group.
The collective excitations of a system, particularly at low temperatures, behave as quasi-particles with properties that may differ from their constituents.In particular, there are exotic excitations called Majorana fermions or zero-energy modes modes, which have the special characteristic of being their own antiparticle.In that sense, this work seeks tracing these excitations in an artificial arrangement of a nanowire simulated through a one-dimensional chain.In that direction, the Fourier transform is used with the purpose of bringing the Hamiltonian from position to momentum space.Subsequently, it is proposed to diagonalize the system using the Bogoliubov-de Gennes formalism.In this way, we obtain the phase diagram displaying the set of parameters for which zero energy modes are stabilized.