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## "Unpaired Majorana fermions in quantum wires", Alexei Kitaev, 2001

*Reviewed July 25, 2023*

*Citation:* Kitaev, A. Yu. "Unpaired Majorana fermions in quantum wires." Physics-uspekhi 44.10S (2001): 131.

*Web:* https://arxiv.org/abs/cond-mat/0010440

*Tags:* Foundational, Physical, Majorana-fermions, Hardware

This is the paper which introduces the idea of quantum computing with Majorana fermions.
It is based on one-dimensional systems, with quasiparticles created at endpoints.
The method proposed doesn't have a natural way to construct gates, so its more of a "quantum memory" system than a "quantum computing" one.
By cleverly braiding one can implement all Clifford operations. T-gates can then
be obtained by non-topologically-protected methods.

The model itself is a one dimensional wire with n sites. The Hamiltonian has two terms:
a nearest neighbor hopping term, and a nearest number pairing term which corresponds to
giving neighbors some probability of forming Cooper pairs. This is what makes the
Hamiltonian a toy model for (p-wave) superconductors.

This is not the first paper which considers quantum computation with Fermionic systems. Kitaev has an earlier work on this topic with Sergey Bravyi:

*> Bravyi, Sergey B., and Alexei Yu Kitaev. "Fermionic quantum computation." Annals of Physics 298.1 (2002): 210-226.*

These Majorana quasiparticles are bound to topological defects (in this case, boundary components) with zero energy,
and hence they are known as Majorana bound states, or Majorana zero modes.
Even though this work does not consider the braiding of these anyons, this is certainly a topic that has been considered in depth:

*> Kauffman, Louis H., and Samuel J. Lomonaco Jr. "Braiding with majorana fermions." Quantum Information and Computation IX. Vol. 9873. SPIE, 2016.*

Microsoft's approach to building a topological quantum computer is primarily via (non-abelian) Majorana zero modes:

*> Sarma, Sankar Das, Michael Freedman, and Chetan Nayak. "Majorana zero modes and topological quantum computation." npj Quantum Information 1.1 (2015): 1-13.*

The major takeaway from this paper is that special types of superconductors will naturally
have Majorana fermions associated to them. Those Majorana fermions are non-abelian anyons,
and they can be used for topological quantum computation. A more modern/pedagogical introduction
to the Kitaev wire model and its applications is found in

> Leijnse, Martin, and Karsten Flensberg. "Introduction to topological superconductivity and Majorana fermions." Semiconductor Science and Technology 27.12 (2012): 124003.