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arXiv:1701.05052

Topological Quantum Computing

Ananda Roy, David Peter Divincenzo

Published Jan 18, 2017• 1 citation
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Abstract

This set of lecture notes forms the basis of a series of lectures delivered at the 48th IFF Spring School 2017 on Topological Matter: Topological Insulators, Skyrmions and Majoranas at Forschungszentrum Juelich, Germany. The first part of the lecture notes covers the basics of abelian and non-abelian anyons and their realization in the Kitaev's honeycomb model. The second part discusses how to perform universal quantum computation using Majorana fermions.

DOI
10.48550/arxiv.1701.05052

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This set of lecture notes forms the basis of a series of lectures delivered at the 48th IFF Spring School 2017 on Topological Matter: Topological Insulators, Skyrmions and Majoranas at Forschungszentrum Juelich, Germany. The first part of the lecture notes covers the basics of abelian and non-abelian anyons and their realization in the Kitaev's honeycomb model. The second part discusses how to perform universal quantum computation using Majorana fermions.
The non-Abelian Berry phase is an essential feature of non-Abelian anyons for the realization of topological quantum computation. This thesis is primarily a study about the numerical calculation of the Berry phase of non-Abelian anyons in the Kitaev honeycomb lattice model. It is also a guide for experimental the realizations of the actual brading process. We give an introduction to the theory of non-Abelian anyons, briefly discussing in what kind of systems they are realized, and their possible use in topological quantum computation. Non-Abelian anyons are studied within the Kitaev honeycomb model where they are realized on the plaquettes of the honeycomb lattice. The Kitaev honeycomb model can be solved exactly by using various fermionization methods. In this thesis, we review a solution based on Jordan-Wigner types of fermions which transform Hamiltonian to a fermionic quadratic form. This kind of fermionization procedure is quite general and can be applied to any trivalent spin lattice models. Moreover, we introduce Hartree-Fock-Bogoliubov method to solve general quadratic fermionic Hamiltonian and employ Bloch-Messiah theorem in order to write ground state wave function explicitly. Later, we apply these methods to honeycomb model and study the eigenstates of the model so that we can do the Berry phase calculation. The final chapter explains the details of the numerical calculation of the non-Abelian Berry phase. First, we show how to create and adiabatically move vortices in the honeycomb model. A brief review of the Berry phase is given including some discussion about a numerical approach. Later on Thouless’ representation of the ground state is introduced to calculate the Berry phase. All these theoretical tools are applied to a 4 vortex configuration of the model to calculate the non-Abelian Berry phase of the system on a particular path in the parameter space.
I will discuss the revolutionary new concept of topological quantum computation, which is fault-tolerant at the hardware level with no need, in principle, of any quantum error correction protocols. Errors simply do not occur since the physical qubits and the computation steps are protected against decoherence by non-local topological correlations in the underlying physical system. The key idea is non-Abelian statistics of the quasiparticles (called 'anyons' as opposed to fermions or bosons), where the space-time braiding of the anyons around each other, i.e. quantum 'knots', form topologically protected quantum gate operations. I will describe in detail the theoretical principles guiding the experimental search for the appropriate topological phases of matter where such non-Abelian anyons, which are low-dimensional solid state versions of the elusive and exotic Majorana fermions hypothesized seventy-five years ago, may exist. I will critically discuss the recent experimental claims of observing the Majorana modes in semiconductor nanowire structures following earlier theoretical proposals, outlining the future developments which would be necessary to eventually build a topological quantum computer.