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Mathematical Aspects of Quantum Signal Processing
Full text linkQuantum Signal Processing (QSP) is a powerful mathematical and algorithmic framework that has emerged as a cornerstone in quantum computation and quantum information processing. It provides a systematic methodology for transforming quantum systems in an efficient and precise way, enabling the implementation of a wide range of quantum algorithms. At its core, QSP utilizes parameterized quantum circuits to encode polynomial transformations of eigenvalues, which is applicable for tasks such as Hamiltonian simulation, eigenvalue filtering, and solving linear systems. Originally introduced by Guang Hao Low and Isaac L. Chuang in 2017, finding the parameters in the QSP representation was initially regarded as challenging due to the difficulty of designing scalable algorithms for its explicit implementation. However, recent years have seen significant advancements in addressing and understanding this challenge, paving the way for the practical application of QSP in quantum technologies. The essence of QSP lies in its ability to represent a real scalar polynomial of degree d through the real (or imaginary) part of an entry of a product of SU(2) matrices, parameterized by (d+1) real numbers known as phase factors. From a mathematical perspective, the problem of finding QSP phase factors is a nonlinear inverse problem: given a target polynomial, the objective is to determine a set of phase factors such that the polynomial represented through SU(2) matrices precisely matches the target function. A natural extension of this question arises when the target function is generalized to a non-polynomial function, and the phase factors are allowed to be infinite-dimensional. This dissertation addresses both the theoretical investigation and numerical resolution of these two problems, advancing the applicability of QSP to a broader class of functions and expanding its practical potential. This dissertation is organized as follows: Chapter 1 provides a brief introduction to the Quantum Signal Processing (QSP) framework, providing the motivation, and an overview of the results presented in this dissertation. Chapter 2 introduces symmetric QSP, laying the groundwork for the subsequent results, and formulates the problem as an optimization problem. Chapter 3 examines the energy landscape of the optimization problem in detail, offering insights into the practical success of optimization-based methods. Chapter 4 generalizes QSP to the infinite-dimensional setting, termed infinite Quantum Signal Processing, which enables the representation of a broad class of non-polynomial functions. This chapter also presents a fixed-point iteration algorithm for phase factor evaluation, and uncovers a surprising connection between the regularity of the target function and the decay properties of the phase factors. Chapter 5 introduces a novel Newton’s method that exhibits rapid and robust convergence across all parameter regimes, despite the absence of theoretical guarantee. Finally, Chapter 6 provides a complete solution to the problem of infinite QSP using nonlinear Fourier analysis. This chapter also presents the first provably stable numerical algorithm for computing phase factors for nearly all functions that admit a QSP representation. Please note that Chapter 2 and Chapter 3 are based on [42] (joint work with Yulong Dong and Lin Lin), Chapter 4 is based on [10] (joint work with Yulong Dong, Lin Lin and Hongkang Ni), Chapter 5 is based on [11] (joint work with Yulong Dong, Lin Lin and Hongkang Ni), and finally Chapter 6 is based on [1] (joint work with Michel Alexis, Lin Lin, Gevorg Mnatsakanyan and Christoph Thiele)
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Reinforcement Learning and Variational Quantum Algorithms
Full text linkIn recent years, the realms of deep learning and variational quantum algorithms have undergone significant advancements. These innovative algorithms have proven to be exceptionally efficient and robust in addressing complex problems within quantum chemistry, condensed matter physics, and quantum field theory simulations, surpassing the capabilities of traditional classical algorithms. A key factor driving this progress is the development of hybrid quantum algorithms, which blend quantum and classical computational techniques.Prominent examples of these hybrid algorithms include the Quantum Approximate Optimization Algorithm (QAOA), the Variational Quantum Eigensolver (VQE), and various Variational Quantum Algorithms (VQAs). These methods enable the construction of parameterized quantum circuits (PQCs), which are central to the operation of these algorithms. By employing PQCs, these algorithms leverage the unique properties of quantum computing, such as superposition and entanglement, to explore solution spaces more comprehensively than classical methods.Furthermore, the optimization process in these hybrid algorithms involves a sophisticated interplay between quantum and classical computing resources. The quantum computer is used to evaluate the performance of the quantum circuit for given parameters, and classical optimization techniques are then applied to refine these parameters iteratively. This synergistic approach enhances the efficiency and effectiveness of the optimization process, making it particularly suitable for problems that are intractable for classical computers alone.We primarily concentrate on a specific issue: the preparation of ground states. A notable challenge in this process is the noise originating from measurements or the device itself. It's crucial to consider this noise when preparing ground states. To address this, we need to develop algorithms that are robust to noise. Our approach involves the development of variational quantum algorithms, which allow for parameter updates during iterative processes. Effectively preparing the ground state is vital, as it has significant applications in subsequent downstream tasks.In addressing the ground state preparation challenge, our objective is to generate the ground state, defined as the lowest eigenstate of the Hamiltonian H. Our exploration is two-pronged: firstly, we investigate various parametrization methods for the variational circuits, aiming to enhance the flexibility and efficiency of the quantum circuits. Secondly, we scrutinize different optimization strategies. This includes examining policy gradients and incorporating optimization challenges within the framework of reinforcement learning, thereby expanding the scope and capability of our optimization methodologies.In evaluating the optimization process, we utilize two critical metrics: fidelity and ground state energy. Fidelity measures the overlap between the target quantum states and the evolved quantum states from the quantum circuit, serving as an indicator of the precision in achieving the desired quantum state. Ground state energy, conversely, relates to observables that can be measured in experimental settings, offering valuable insights into the physical characteristics of the quantum system under investigation.The algorithms we discuss are specifically engineered to operate effectively in environments where quantum computer measurements are subject to noise. Demonstrating robustness against such measurement noise, these optimization algorithms efficiently identify optimal parameters for the variational quantum circuits. This efficiency and resilience are pivotal in advancing the field of quantum computing, particularly in the context of practical, noisy quantum systems.Chapter 1 introduces the background knowledge and overview of deep learning techniques and optimization algorithms, quantum circuits, and variational quantum algorithms the basic problem setup and provides an overview of the results in this paper. Chapter 2 introduces a policy gradient approach to the Quantum Approximate Optimization Algorithm (QAOA) using methods. Chapter 3 presents reinforcement learning techniques for the preparation of many-body ground states in quantum systems. It specifically leverages counter-diabatic driving, a method that guides the system adiabatically to avoid non-equilibrium excitations, thus ensuring more reliable ground state preparation. Chapter 4 presents a noise-robust, deep autoregressive policy networks based end-to-end quantum control framework as to the challenge of noise in quantum systems. Chapter 5 presents another approach which integrates MCTS with quantum circuit optimization, aiming to enhance the efficiency and effectiveness of the circuit design and operation. Chapter 6 presents a random coordinate descent method as a straightforward yet effective technique for optimizing parameterized quantum circuits.Please note that Part 2 is based on [Yao, J., Bukov, M., & Lin, L. Mathematical and Scientific Machine Learning (pp. 605-634). PMLR.] (joint work with Marin Bukov, Lin Lin), Part 3 is based on [Yao, J., Lin, L., & Bukov, M. (2021). Physical Review X, 11(3), 031070.] (joint work with Marin Bukov, Lin Lin), Part 3 is based on [Yao, J., Kottering, P., Gundlach, H., Lin, L., & Bukov, M. Mathematical and Scientific Machine Learning (pp. 1044-1081). PMLR.] (joint work with Paul Kottering, Hans Gundlach, Lin Lin, Marin Bukov), and Part 5 is based on [Yao, J., Li, H., Bukov, M., Lin, L., & Ying, L. Mathematical and Scientific Machine Learning (pp. 49-64). PMLR.] (joint work with Haoya Li, Marin Bukov, Lin Lin, Lexing Ying). Finally, Part 6 is based on a joint work in preparation with Zhiyan Ding, Taehee Ko, Lin Lin, Xiantao Li)
Developing a Green Bonds Market: The Case of China
No full textEuropean Business Organization Law ReviewGerman
sj-docx-1-tag-10.1177_17562848221137758 – Supplemental material for Adjuvant chemotherapy improves survival in high-risk stage II colon cancer: a retrospective cohort study
No full textSupplemental material, sj-docx-1-tag-10.1177_17562848221137758 for Adjuvant chemotherapy improves survival in high-risk stage II colon cancer: a retrospective cohort study by Lin-Lin Liu and Zuo-Lin Xiang in Therapeutic Advances in Gastroenterology</p
Shi ge za lun.
No full text白話詩與方言詩 ---p.1詩歌與英雄主義 ---p.11論詩的感情 ---p.23叙事詩的寫作問題 ---p.37關於詩腔 ---p.54談詩歌的用詞 ---p.60詩歌與比喻 ---p.70閩南歌謠的藝術性 ---p.77對唱式的民歌 ---p.90魯迅先生與詩歌 ---p.98陶行知詩歌的生活化 ---p.115李煜的教訓 ---p.119關於海湼的諷刺詩 ---p.124論詩的主題 ---p.129後記 ---p.146林林著.Lin Lin zhu
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