利用量子運算技術進行二維地下水流數值模擬

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范阮紅玉(PHAN NGUYEN HONG NGOC) 查詢紙本館藏

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論文名稱

利用量子運算技術進行二維地下水流數值模擬
(NUMERICAL SIMULATION OF 2D GROUNDWATER FLOW BY USING QUANTUM COMPUTING TECHNOLOGY)

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摘要(中)

地下水在供水系統中起著至關重要的作用。為了了解地下水的行為，需要使用地下水模型。以往的研究長期以來傾向於使用傳統的計算方法來構建地下水模型, 但最近幾項研究已經證明了量子計算在各種應用(例如反演或隨機問題)中的效率，探索量子技術在地下水水文學領域的應用應該是很有趣的。本研究旨在開發和測試利用量子計算穩態和暫態地下水方程式的解，而這些地下水方程式透過有限差分法直接推導出地下水水頭的變量。 Python 腳本用於在傳統電腦上計算地下水方程式矩陣，其他腳本則連續構建最小二乘問題的 QUBO（量子無限制二進制最佳化）方程，以求解來自 D-Wave System 的 QPU（量子處理單元）直接求解器和迭代直接求解器中的離散矩陣。傳統計算和量子計算的計算結果用於比較穩態時 2x2、3x3、4x4、9x9 和瞬態時 2x2 的網格大小。研究結果顯示量子退火器計算的穩態模型與傳統計算結果一致，QPU直接求解器中的網格大小為 2x2，迭代求解器中的網格大小為 2x2、3x3、4x4。對於兩種求解器而言，不穩定性在 9x9 並具有100個變量的網格中清楚地顯示出來。網格大小為 2x2 的暫態模型也獲得了與穩態相同的結果。大體而言，量子電腦中的測試結果顯示迭代求解器的表現優於 QPU直接求解器。本研究結論為由於電腦硬體中量子位的限制，量子計算能在小尺度的問題中取得較好的結果，計算過程中迭代求解器的結果亦比 QPU直接求解器更加穩定。本研究顯示D-Wave 系統的特定封包和 API 認證應該要透過 SAPI（D-Wave 的 Solver API）直接在本地環境或 D-Wave 的 Leap 線上平台上去執行程式，並證明了地下水模型可以利用目前的量子計算進行求解，儘管在大尺度問題中仍存在錯誤，但仍可視為是透過量子計算解決地下問題的初步研究之一。
關鍵詞：Groundwater 模型，有限差分法，量子退火法，QUBO，QPU 直接求解器，迭代求解器，Python 腳本。

摘要(英)

Groundwater plays a vital role in the water supply system. To understand the groundwater behavior, groundwater model was necessary to carry out. Previous studies prefer to use traditional computational methods to construct groundwater models for a long time. Recently, several studies have proven the efficiency of quantum computing in various applications including inverse problems, stochastic interpretation. It should be interesting to explore the possible implementation of quantum technology for the field of groundwater hydrology. This study aims to develop and test the solutions of groundwater equations for steady state and transient state in quantum computing. These groundwater equations were directly derived by finite difference method to find groundwater head variables. Python scriptings were built to calculate matrices of groundwater equations on classical computer. The other scriptings were continuously constructed QUBO (Quadratic Unconstrained Binary Optimization) equation of least square problem to solve discretized matrices in QPU (Quantum Processing Unit) direct solver and Iterative direct solver from D-Wave System. The results calculated by classical computing and quantum computing were presented for comparison purposes on grid sizes of 2x2, 3x3, 4x4, 9x9 in the steady-state, and 2x2 in the transient state. Our results indicated that the steady state model calculated by the simulated quantum annealer was consistent with classical computing results with grid sizes of 2x2 in QPU (Quantum Processing Unit) direct solver and grid sizes of 2x2, 3x3, 4x4 in Iterative solver. The instability was clearly shown in grid sizes of 9x9 with 100 variables in the matrices for both solvers. Transient state model with grid size of 2x2 also gained the same status as steady-state case. Generally, Iterative solver has performed better than QPU (Quantum Processing Unit) direct solver in testing examples on quantum computer. Based on our analysis, we conclude that quantum computing certainly achieves quality solutions in small-scale problems due to the limitations of qubits in the hardware machine. The results returned from Iterative solver were more stable than QPU (Quantum Processing Unit) direct solver in the calculation process. The research indicated that specific packages and API token authentication from D-Wave System should be prepared to directly run the problems in the local environment or D-Wave′s Leap online platform through SAPI (D-Wave′s Solver API) . Our research has proven that groundwater model is possible to solve by quantum computing in this period. Despite errors in large-scale problems, this research is considered as one of the initial studies in solving subsurface problems by quantum computing in its early stage.
Keywords: Groudwater model, finite difference method, quantum annealing method, QUBO, QPU direct solver , Iterative solver , Python scriptings.

關鍵字(中)

★ Groundwater 模型
★ 有限差分法
★ 量子退火法
★ QUBO
★ QPU 直接求解器
★ 迭代求解器

關鍵字(英)

★ Groudwater model
★ finite difference method
★ quantum annealing method
★ QUBO
★ QPU direct solver
★ Iterative solver

論文目次

ABSTRACT i
摘要 iii
ACKNOWLEDGEMENTS v
LIST OF CONTENTS vi
LIST OF FIGURES viii
LIST OF TABLES xi
LIST OF ABBREVIATIONS xii
LIST OF NOTATIONS xiii
CHAPTER I. INTRODUCTION 1
1.1 Motivation and objectives 1
2.2 Literature review 5
2.3 Thesis structure 7
CHAPTER II. MATHEMATIC FORMULAR OF NUMERICAL METHOD 9
2.1 Governing equation of Groundwater equations 9
2.1.1 Governing equation 9
2.1.2 Boundary conditions and assumptions in the model 13
2.2 Governing equation of Finite Difference Method 14
2.3 Matrix solver in the model 15
CHAPTER III. QUANTUM ANNEALING METHOD 17
3.1 Quantum computing 17
3.1.1 Introcduction to quantum annealing 17
3.1.2 Dwave system 17
3.1.3 Application problem 19
3.1.4 How quantum annealing work 19
3.1.5 Underlying quantum physic 21
3.1.6 Problem formulation: key concepts 22
3.1.7 Graph 24
3.1.8 D-Wave QPU architecture: Chimera 25
3.1.9 Minor-Embedding a problem onto the Chimera graph 26
3.2 Least-squares approximation 28
3.3 Least-squares approximation in quantum computing 29
CHAPTER IV. NUMERICAL EXAMPLES 35
4.1 Conceptual model 35
4.2 Results based on classical methods 36
4.2.1 Steady-state case 36
4.2.2 Transient case 38
4.3 Results based on quantum annealing method 43
4.3.1 Steady-state case 43
4.3.2 Transient case 50
4.3.3 Comparisons between QPU direct solver and Iterative solver 56
4.4 Limitation of quantum computing results 57
CHAPTER V. CONCLUSIONS 60
5.1 Conclusions 60
5.2 Suggestions 62
REFERENCES 63

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指導教授

倪春發(Chuen-Fa Ni)

審核日期

2021-8-2

推文