量子動力學主要是描述量子系統隨時間的演化。雖然孤立量子系統的動力學由薛丁格方程式所決定，但模擬開放性量子系統需要更廣泛的理論架構，才能精確地刻劃系統與周遭環境耦合時所產生的複雜交互作用以及潛在的非馬可夫效應。
本論文首先回顧兩種描述開放性量子系統動力學的方法：階層式運動方程以及林德布拉德方程。這兩套方法適用於不同的系統環境耦合強度。由於解析解通常難以獲得，發展高效率的數值工具和實驗方法用以模擬量子動力學便顯得格外重要。
因此，我們開發兩個開源套件：QuantumToolbox.jl 和 HierarchicalEOM.jl。這兩個套件皆以 Julia 程式語言撰寫，並且同時支援高效能的 CPU 和 GPU 運算。我們藉由以下兩個物理模型來展示這些套件的功能：與近藤效應相關的單一雜質安德森模型以及一個電荷–共振腔極強耦合系統同時與玻色子和費米子環境交互作用。除此之外，我們也比較階層式運動方程與林德布拉德方程兩個方法的數值模擬結果，突顯非馬可夫效應在準確描述開放性量子系統動力學上的重要性。
受到費曼 (R. P. Feynman) 於 1982 年所提出構想的啟發，我們進一步透過雲端量子電腦直接模擬量子動力學，並且探討以下兩個模型：量子態傳輸和量子資訊置亂。這兩個模型在量子資訊處理上分別展現不同面向的應用。我們針對這兩個模型，分別提出刻劃其量子關聯性的方法。接著，透過量子電腦進行實驗來驗證其行為，同時也能夠比較不同量子電腦所得到的實驗結果。我們的研究結果指出，如今的嘈雜中等規模量子電腦已具備有模擬小規模量子動力學和產生真正量子關聯性的能力。
本論文旨在提供量子理論的見解與實際應用，推動量子模擬、量子電腦效能評估、量子資訊處理以及量子科技的發展。

Quantum dynamics describes the time evolution of quantum systems. While the dynamics of closed quantum systems are governed by the Schr\"odinger equation, open quantum systems require more general theoretical frameworks to accurately capture complex interactions and potential non-Markovian effects arising from their couplings to the surrounding environments.
In this dissertation, we begin by reviewing two approaches for describing open quantum system dynamics: the hierarchical equations of motion and the Lindblad equation. These approaches are suitable for different system-environment coupling regimes. Because analytical solutions are seldom available, the development of efficient numerical tools and experimental methods for simulating quantum dynamics becomes essential.
Therefore, we develop two open-source software packages: QuantumToolbox.jl and HierarchicalEOM.jl. Both of them are written in the Julia programming language and support high-performance CPUs and GPUs computing. We demonstrate their functionalities using two physical examples: the single impurity Anderson model relevant to Kondo effects, and a ultra-strongly coupled charge–cavity system interacting with bosonic and fermionic reservoirs. Furthermore, a comparison between the hierarchical equations of motion and Lindblad equation highlights the importance of non-Markovian effects in accurately describing open quantum system dynamics.
Inspired by R. P. Feynman's proposal in 1982, we directly simulate quantum dynamics on cloud-based quantum computers and investigate the following two models: quantum state transfer and quantum information scrambling. Each of them demonstrates different application aspects of quantum information processing. For both models, we propose methods to characterize quantum correlations and validate their behavior through experiments on quantum computers. We then compare the results obtained from different quantum computers. Our findings highlight the capability of current noisy intermediate-scale quantum computers to simulate small-scale quantum dynamics and generate genuine quantum correlations.
This dissertation aims to contribute both theoretical insights and practical applications to advance quantum simulations, quantum computer benchmarking, quantum information processing, and quantum technologies.

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