以馬約拉納零能模態為基礎的超導位元被視為實現量子運算的有力競
爭者。其原因為在此位元中，資訊基本上存儲於奈米線兩端的邊界態中，
此邊界態即為前述零能模態。由此二馬約拉納零能模態所組成的費米態具
有極強的非局部性，意味著要影響該位元，噪音的來源必須通時耦合兩個
空間上相隔甚遠的零能模態，因此該位元一般被認為在外界環境因子干
擾下，依然能保有良好的穩定性。本篇論文中，我們透過勃格留波夫變換
(Bogoliubov transformation) 模擬 T-junction 的 B12 順時針交換，並在無外界擾動情況下成功得到了理論上預期的結果，即兩個馬約拉納零能模態的混
成態分別得到 ±π/2 的幾何相位。然而，在透過荷斯坦模型 (Holstein model)
導入了聲子的擾動後，我們發現取決於施加擾動的位置，幾何相位有可能
因為該擾動而有向 0 靠近的趨勢。此結果顯示，馬約拉納零能模態為基礎
的超導位元並非完全不受環境擾動影響。

The Majorana-zero-mode based qubit is considered to be a potential candidate
for realizing full-scale quantum computation. In such a qubit, the information is
usually stored in well-separated edge modes of nanowires. These edge modes are
Majorana-zero-modes, which form highly non-local fermionic states. For an error
to occur, the source has to involve the coupling between both zero-modes, making
the qubit theoretically robust against almost any form of noise. In this thesis, we
numerically simulate the B12 braiding operation on a T-junction by utilising the
Bogoliubov-de Gennes formulation. Starting with the noise-free case, we success-
fully obtained the optimal ±π/2
Berry phase for the hybridised Majorana zero-mode
states, after a clockwise braiding; however, after introducing the phonon noise via
the Holstein model, we noticed that the Berry phase may start to degrade towards 0
depending on how the phonon noise couples to the zero-modes. Our results suggest
that the Majorana-based qubit is not entirely immune to noises.

Acknowledgements
Abstract i
1 Introduction 1
2 Theory 3
2.1 Majorana Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Majorana Fermions in p-wave Superconductor . . . . . . . . . . . . . . . . . . 4
2.3 Braiding and Non-Abelian Statistics . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Majorana Zero-modes in p + ip Superconductor . . . . . . . . . . . . 10
2.3.2 Braiding Operation of Majorana Zero-modes . . . . . . . . . . . . . . 14
3 Procedures 19
3.1 T-junction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 BdG Hamiltonian and Eigen-states . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Time Evolution and Berry Phase . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 The Phonon Noise and Holstein Model . . . . . . . . . . . . . . . . . . . . . . 25
4 Result 27
4.1 Noise-free Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Phonon Noise Applied Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Conclusion 33
References 35

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