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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

This is the paper which introduces the idea of quantum computing with Majorana fermions. It is based on one-dimensional systems, which 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.

The Majorana fermions appear as boundaries in second quantized systems, where we have a collection of sites each of which is either inhabited/is not inhabited by an electron. In general, this sort of "empty/full" dichotomy seems to be the easiest way of getting Z2 symmetries. In this case, it is being enforced by Pauli repulsion - we force all the electrons to have spins in the same direction.

This is not the first paper which considers quantum computation with these sorts of 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 considered:

> 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.
This paper is very physics minded, and doesn't seem to have much mathematical insight. All I really need to remember is: Majorana anyons are quasiparticles which are their own inverses. There are cool ways of creating them in the lab, and they are particularly amenable to TQC.