臺灣博碩士論文加值系統

隨著現今科學的日益進步，對於量測的精準度之要求也越來越高，並且由於量子雜訊充斥於物理的世界中，根據海森堡提出的測不準原理，執行精準測量時仍會受到最小測不准值之限制。然而在進年來的研究中，使用非線性光學技術將非古典系統--量子壓縮態引入測量當中，可以有效突破量子雜訊所帶來的限制，使得研究如何產生以及測量量子壓縮態成為一個關鍵的技術

對於現今量子電腦以及量子光學領域，量子斷層掃描技術一直以來都是一個十分具有挑戰的問題。在過去的研究中，使用最大似然統計方法重建測量到的量子壓縮態被視為一個相對可靠且穩健的方法。然而使用最大似然方法實現量子態層析成像技術具有以下技術限制：第一，最大似然法之計算量龐大且耗時，然而在未來的研究發展中，將會難以實現即時之量子態估計。第二，最大似然法以推估出量子態密度矩陣為目標，使用適當有限維度的空間去描述量子態密度矩陣，找出最接近實驗結果的量子態密度矩陣。雖然此方法保證所重建的量子態在理論上有效，但是無法唯一的識別量子態，且會需要面對保真度不足的問題。第三，最大似然法以推估出量子態仰賴龐大的數據點來統計出結果，並且會增加資料擷取的時間增加實驗難度。

在本研究中，我們使用傅立葉神經算子網絡架構（Fourier neural operator, FNO）來重建量子態，期待透過資訊萃取以及抽象化的方式將量子態（特別是量子壓縮態）的資訊抽象化成數個特徵參數來描述，並使用傅立葉神經算子網絡架構來完成高效率的資訊萃取。最後，在此研究中，我們也演示了使用傅立葉神經算子網絡架構可以具有高達 99.99\% 的保真度，同時可使用更少的參數，使得實現實時量子斷層掃描技術的可行性大幅提昇。

With the progress of the science and technology, the required accuracy of measurements increases rapidly.Owing to the fact that quantum noise is flooded everywhere around the physics world,the accuracy of measurements are still limited due to the Heisenberg uncertainty principle.According to the research these years, it is possable to breakthrough the quantum limitation by introdice the non-classical light source -- squeezed state generated by non-linear optical effect into the measurements. Therefore, it becomes more and more popular to understand the properties of squeezed state.

The technique of quantum state tomography plays an important part in the field of quantum computing and quantum optics. To reconstruct quantum state via quantum state tomography is always a challenge over the years. According to the past research, it is proposed to be a robust way to reconstruct squeezed states via maximum likelihood estimation. But, there are some disadvantages of maximum likelihood estimation: First, to reconstruct squeezed state via maximum likelihood estimation is time-consuming, and may cause the difficulty to the future study. Second, maximum likelihood estimation uses density matrix as the representation and represent in truncated dimension spaces. Although it is decleared to be a robust method to reconstruct squeezed state, it may not identify the quantum state uniquely, and also, the fidelity of the reconstructed density matrix is not high enough. Third, maximum likelihood estimation calculates reasonable density matrix based on a huge bunch of data points.
To collect such amount of data points may increase the difficulty as well as the time of doing the measurements.

In this thesis, we use Fourier Neural Operators to reconstruct quantum states. We wish to extract features in high efficiency and describe quantum states using characteristic paramiters via Fourier Neural Operators. Last but not least, we also demostrate that the architecture based on Fourier Neural Operator can reach high fidelity up to 99.99\%, and uses less parameters in the architecture at the same time. This make a huge progress on the implementation of real time quantum state tomography.

I 序論 13
1 論文大綱 15
1.1 研究背景 15
1.2 研究目的 16
2 壓縮態之測量以及重建發展近況 17
2.1 基於最大似然估計法 17
2.2 基於捲積神經網絡 19
II 研究原理 21
3 相空間與Wigner 函數 23
3.1 相空間 23
3.2 Wigner 函數 25
4 量子態 27
4.1 真空態 27
4.2 熱態 28
4.3 壓縮真空態 28
4.4 壓縮熱態 30
5 量子態之測量 33
5.1 平衡零差檢測技術 33
III 實時量子態層析成像技術 37
6 量子態層析成像技術 39
6.1 實時系統架構 39
6.2 量子態計算與繪圖系統 42
6.3 資料擷取系統 42
6.4 機器學習模型 43
7 機器學習模型45
7.1 先前的研究 45
7.2 Convolution Neural Network 46
7.3 Neural Operator 47
7.4 Fourier Neural Operator 49
7.5 量子態模型基於FNO 50
IV 問題描述與實驗 53
8 問題描述 55
8.1 實驗產生之量子態 55
8.2 問題設定與模型設計 56
9 實驗 59
9.1 實驗結果 59
9.2 模型比較62
V 總結與未來工作67
10 總結69
11 未來工作71

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