臺灣博碩士論文加值系統

為了建造一個量子電腦，高精準度的邏輯閘運算是必要的。特別是兩個量子位元(two qubit)所能進行的控制非門(CNOT)的運算，是一個很重要的量子邏輯閘。然而，現實世界中有需多的問題導致實驗上的結果和理論上的預測是不盡相同的。在這篇碩論中，我們探索了在量子最佳化控制理論下，考慮了流失態(leakage)和非馬可夫(non-Markovian)環境下的控制非門的運算。首先，我們給了一個簡短的超導體量子位元介紹，然後決定了單量子位元的漢米爾頓(Hamiltonian)矩陣。然後我們描述了環境的雜訊能譜(noise power spectrum)，接著就用費米黃金定律(Fermi golden rule)去連結馳耗速率(relaxation rate)和關聯函數(correlation function)之間的關係。然後我們描述了一個非馬可夫環境下的控制方程式，接著用我們的模型實行了量子最佳化控制。科羅多夫最佳化控制是我們在這篇碩論中採用的方法，目的是為了最小化邏輯閘運算的誤差，然後找到相對應的控制參數，並且考慮非同時的記憶效應(non-local memory effect)的非馬可夫開放系統。我們也討論了不同控制參數的不同波形，以及控制參數和關聯函數影響最佳化邏輯閘控制的行為。我們發現在環境的參數影響下(此參數參考真實實驗系統結果)，使用超導體電量量子位元是可以完成一個誤差約為1e-4~1e-5的控制非門的高精準度邏輯閘控制的。

When building a quantum computer, high precision gate operations are needed. In particular, the controlled-not (CNOT) operation regarded as a crucial universal two-qubit gate is a very important quantum gate to implement. However, real world contains a lot of problems and causes the difference between experimental results and theoretical simulations. In this thesis, we investigate CNOT gate operation using quantum optimal control theory for superconducting charge qubit system taking into account the effects of leakage states and a non-Markovian environment. First, we give a brief introduction to superconducting qubits and decide the Hamiltonian of a simple one-qubit model. Then we describe the noise power spectrum of environments, which gives the relation between the relaxation rate and the bath correlation function through the Fermi golden rule . After that, we describe the non-Markovian master equation approach and apply it to our model together with the quantum optimal control theory. The Krotov optimal control method that we used in this thesis can minimize the error of gate operations and find the corresponding optimal control pulses to realize the gate operations. Considering the non-local memory effect in non-Markovian open quantum systems. We also discuss the effect of different shapes and behaviors in the bath correlation function on the optimal control gate fidelity. We find that it is possible to implement high-fidelity CNOT gates with error about 1e-4~1e-5 in superconducting charge qubit system with environment parameters extracted from the realistic noise power spectrum of experiments.

口試審定書 i
誌謝 ii
中文摘要 iii
Abstract iv
1 Introduction and Background 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Review of Josephson Charge Qubit . . . . . . . . . . . . . . . . . . . 2
1.3 Hamiltonian of the One Josephson Charge Qubit Model . . . . . . . . 2
1.4 Power Spectrum and Noise Fluctuation . . . . . . . . . . . . . . . . . 5
1.5 Transition Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Superconducting Qubit and Noise 11
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Hamiltonian of Two Coupled Josephson Charge Qubits . . . . . . . . 12
2.3 Spectral Density and Correlation Function . . . . . . . . . . . . . . . 18
2.4 Diagram of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Optimal Control 25
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Error/Fidelity Definition . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 One Qubit Optimal Control . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Two Qubit Optimal Control . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 State-independent Optimal Control . . . . . . . . . . . . . . . . . . . 31
3.5.1 Computational States and Leakage States . . . . . . . . . . . 31
3.5.2 Superoperator and Column Representation . . . . . . . . . . . 37
4 Results 41
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Closed System with No Leakage Levels . . . . . . . . . . . . . . . . . 43
4.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.2 Pulse Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Open System with No Leakage Levels . . . . . . . . . . . . . . . . . . 44
4.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 Closed System with Leakage Levels . . . . . . . . . . . . . . . . . . . 51
4.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4.2 Pulse Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Open System with Leakage Levels . . . . . . . . . . . . . . . . . . . . 55
5 Conclusion 59
A RK4 and expm 61
B Derivatives of Matrix, Traces 65
B.1 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
B.2 Derivatives of Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
B.3 Derivatives of Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
B.3.1 Derivation of Trace REAL Matrix . . . . . . . . . . . . . . . . 66
B.3.1.1 First Order . . . . . . . . . . . . . . . . . . . . . . . 67
B.3.1.2 Second Order . . . . . . . . . . . . . . . . . . . . . . 67
B.3.2 Derivation of Trace COMPLEX Matrix . . . . . . . . . . . . . 67
B.3.2.1 Generalized Complex Derivative: . . . . . . . . . . . 68
B.3.2.2 Useful Formula . . . . . . . . . . . . . . . . . . . . . 68
C Leakage Optimal Calculation 69

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