利用量子電腦計算分子能階的想法源自於1982年理查·費曼所說的「“大自然不是古典的，如果你想要模擬大自然，你最好把它變成是量子力學的，而這是一個精彩的好問題，因為它看起來並不是那麼容易。”而在1996年Seth Lloyd利用量子相位估算法(PEA)計算出分子系統的基態能階，理論上PEA是量子電腦計算這類問題最具效率的方式，然而在”嘈雜中型量子”(NISQ)電腦因為誤差、雜訊、退相干時間. . .因素，執行上受到諸多的限制，即便小分子的模擬也相當的困難。2015年IBM在量子運算上取得關鍵性突破，如何發揮NISQ電腦的效能變成重要議題，近年基於NISQ電腦諸多限制下開發出「量子變分特徵解演算法」(VQE)。而我們從組態交互作用為出發點，比較不同的Hamiltonian轉換方法以及不同的最佳化方法下的結果，分析這些方法所帶來的優劣。

The idea of using quantum computer to calculate the energies of molecules is came from Richard Feynman. In 1982, he mentioned “Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.” In 1996, Seth Lloyd used the quantum phase estimation algorithm (PEA) to calculate the ground state energy of a molecule. In theory, PEA is the most efficient method to solve this type of problems. However, the power of Noisy Intermediate Scale Quantum (NISQ) computer available in the next five to ten years is limited by error rate, noise, decoherent time, etc. Even for a small molecule, PEA is hard to implement effectively. Due to the breakthrough improvement on quantum devices made by IBM, how to make NISQ devices perform well becomes very important. Based on the limitation of NISQ devices, variational-quantum-eigensolver (VQE) is developed. In this thsis, we start from configuration interaction, discuss results with different encoding and optimization methods for the VQE to find elecreonic structure of molecules, and analyze pros and cons of the different methods.

摘要 I
Abstract II
List of Figures V
List of Tables VIII
1 Introduction 1
2 Transforming Hamiltonian into Pauli Matrices 4
2.1 Jordan-Wigner transformation (local transformation) . . . . . . . . . 5
2.2 The parity transformation . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 The non-local transformation . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 Parity set(P) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Updadte set(U) . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Flip set(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Quantum eigensolver 12
3.1 Qubit and quantum gate . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Unitary couple cluster to VQE . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Spin-adapted configurations . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Conclusion 32
Bibliography 34
A Tapering off qubits 37
A.1 H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
A.2 H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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