臺灣博碩士論文加值系統

現實的量子電腦（quantum computer）皆會受到退相干（decoherence）以及雜訊（noise）的影響而產生誤差。因此，為了建造實用的量子電腦，我們必須找尋一組方法，例如：最佳化控制方法（optimal control method），用以降低誤差。
在這篇論文裡，我們以矽基電子施體自旋量子位元（donor electon spin qubits in silicon）之架構為基礎來來進行我們的研究。矽基電子施體自旋量子位元為現今量子電腦的候選系統之一，具有相對長的弛緩時間（ $T_{1}^{*}$）褪相時間（$T_{2}^{*}$）。然而，在建構二位元（two-qubit）量子邏輯閘時，此系統面臨量子位元間距（separation）之不準確性（uncertainty）所造成的誤差，由於量子位元間的交互作用（interaction）對其位置的準確度相當敏感，因此，在此篇論文中，我們透過最佳化方法找到一系列系統參數（system parameter）以架構一組可以容忍不準確性所造成的誤差的脈衝波。在論文的開始，我們將介紹矽基電子施體自旋量子位元，接著是考量實驗上的限制後所選擇的脈衝波波型，與取樣法之最佳化控制。綜合以上考量，我們找到一組能建構受控反閘（CNOT gate）之波形參數，能在保真度99\%的限制下，容忍10\%的量子位元間距誤差。

Every realistic quantum computer inevitably suffers from decoherence and noise, resulting in errors. Hence, for quantum computing to be viable, delving into the errors for quantum systems and looking for a method, such as the optimal control method for error suppression, has become a critical task. This thesis investigates quantum gate operations for quantum computing based on the electron spins of donors in silicon, which is one of the most promising candidates for spin-based quantum computation as the electron spin qubits have relatively long $T_{1}^{*}$ and $T^*_{2}$ times. However, this system faces a stiff challenge on implementing two-qubit gates owing to the strength of the interaction between qubits depending sensitively on the exact positioning of qubits, which is not precisely known. Therefore, in this thesis, we focus on the discussion of the errors coming from this system parameter uncertainty and use an optimal control method to try to find error-suppression pulses for two-qubit gates. The thesis is organized as follows: First, we introduce the electron spins of an exchange-coupled pair of donors in silicon and use the ability to set the donor nuclear spins in arbitrary states to enlarge the effective magnetic detuning. Then, we present an optimal control method, which combines the sampling-based learning control and the gradient-free Nelder-Mead algorithm, to search for error-suppression pulses. Lastly, we present our result of a smooth pulse set for a two-qubit CNOT gate that can tolerate the parameter uncertainty error of about 10% of exchange interaction, while achieving a gate fidelity of 99%.

Contents
誌謝 i
摘要 ii
Abstract iii
1 Introduction 1
1.1 Review...................................1
1.2 Motivation..................................2
2 Introduction of Quantum Computing 4
2.1 Qubit Systems................................5
2.1.1 Qubits and Quantum Gates.....................5
2.1.2 The Quantification of Errors: Infidelity..............12
2.2 Introduction of Electron Spins of Donors in Silicon............13
2.2.1 Introduction of Kane’s Quantum Computer............13
2.2.2 The Architecture of Kane’s Quantum Computer..........14
2.2.3 Errors: Noise and Uncertainty...................24
3 Model, Control Scheme, and Optimal Control Method 26
3.1 Pulsing Scheme of the Control Pulses...................26
3.1.1 The AC Microwave Magnetic Field: Bac(t) ...........27
3.1.2 The Voltage Applied to A-gates: A(t) ..............28
3.1.3 The Voltage Applied to J-gates: J(t) ...............29
3.2 Pulsing Setup ................................30
3.3 Modelingthe1/fnoise...........................31
3.4 ConditionandApproximationtoSpeedupCalculation..........32
3.5 SamplingBasedOptimalControlMethod.................33
3.5.1 SamplingandWeighting......................34
3.5.2 MinimizingtheCostFunction...................35
4 ResultsandDiscussion 37
4.1 Results....................................38
4.2 Discussion..................................43
4.2.1 ComparisonbetweentheSine-CubedandtheSine-ShapedPulses
for CNOTGates..........................43
4.2.2 ErrorCompensationPerformanceofTrainedPulses........43
4.2.3 ImprovementofInfidelitywithIncreasedACMWMagneticField44
5 Conclusion 46
5.1 OptimalControlMethod..........................46
5.2 ResultofConstructingaCNOTGate....................47
Bibliography 48

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