Theoretical aspects of quantum information and computation

现代计算机是经典的计算设备：经典计算机的每个组件在计算的每个步骤之前和之后都有一个确定的逻辑状态，并且计算根据确定性（有时是随机的）逻辑运算序列所规定的规则进行。然而，根据量子信息理论，它提供了量子力学系统的信息理论方面的抽象，经典计算只代表了可以通过物理设备实现的计算的有限子集。就像原子中的电子存在于叠加态中，无法在牛顿物理学的背景下明确描述，量子计算机可以存在于逻辑状态的叠加态中，并且它们的计算可以同时沿着沿着多个计算路径进行，这些计算路径可能会建设性地或破坏性地相互干扰。量子计算理论研究了这种计算范式的力量和局限性。我研究的主要目的是更好地理解量子计算机可能实现的计算的性质，以及多个量子计算机在合作和竞争环境中相互作用的性质。我也对量子信息的基本方面感兴趣，以及对量子信息和计算推理有用的数学技术的发展。我工作的一个主要课题是量子计算复杂性理论。本课程的主要目标是识别和理解由量子计算模型定义的计算问题类别之间的关系，并将这些模型和类别与经典计算模型定义的模型和类别联系起来。这包括对各种模型和计算问题类别的研究，包括抽象单个量子计算机编程以尽快解决计算问题的概念的模型;描述分布式和加密设置中多个量子计算机之间交互的模型;以及通过对量子模型设置资源约束和其他限制来定义的问题类别。量子交互式证明系统模型是量子计算模型的一个例子，它已经在量子计算复杂性理论中得到了研究--这个模型是我以前大部分工作的主题，也是我打算与这个研究提案相关的研究的核心。来自不同数学领域的强大数学技术已被应用于量子信息和计算中的问题。我特别感兴趣的技术，从组合优化，凸分析，矩阵分析和研究算子代数。半定规划和矩阵乘性权重更新方法是我最近工作中的两个重要例子。我打算继续研究这些方法和其他方法在量子信息和计算研究中的应用。量子信息有可能给我们构建和使用计算机的方式带来革命性的变化，私密通信和实现加密协议，并研究量子物理系统的性质。如果成功的话，这个研究项目将导致对量子信息和计算的更好的理论理解，导致对其研究有用的新数学方法，并可能导致其可以使用的新方法。

Present-day computers are classical computing devices: each component of a classical computer has a definite logical state before and after each step of a computation, and computations proceed according to rules dictated by sequences of deterministic (or sometimes randomized) logical operations. According to the theory of quantum information, however, which offers an abstraction of the information-theoretic aspects of quantum mechanical systems, classical computations represent only a limited subset of the computations that can potentially be implemented by physical devices. Much like electrons in atoms exist in superpositions that cannot be described definitively within the context of Newtonian physics, quantum computers can exist in superpositions of logical states, and their computations can proceed along multiple computation paths simultaneously that may constructively or destructively interfere with one another. The theory of quantum computation studies the powers and limitations of this computational paradigm. The main objective of my research is to better understand the nature of the computations that can potentially be implemented by quantum computers, as well as the nature of interactions among multiple quantum computers in cooperative and competitive settings. I am also interested in fundamental aspects of quantum information, and in the development of mathematical techniques that are useful for reasoning about quantum information and computation. A primary subject of my work is quantum computational complexity theory. Principal goals of this subject are to identify and understand relationships among classes of computational problems defined by quantum models of computation, and to relate these models and classes to ones defined by classical computational models. This includes the study of a variety of models and classes of computational problems, including models that abstract the notion of a single quantum computer programmed to solve computational problems as rapidly as possible; of models that describe interactions among multiple quantum computers in both distributed and cryptographic settings; and classes of problems defined by placing resource constraints and other limitations on quantum models. The quantum interactive proof system model is one example of a quantum computational model that has been studied within quantum computational complexity theory -- this model has been the subject of much of my previous work, and is central to the research I intend to pursue in association with this research proposal. Powerful mathematical techniques from different areas of mathematics have been applied to problems in quantum information and computation. I am particularly interested in techniques from combinatorial optimization, convex analysis, matrix analysis, and the study of operator algebras. Semidefinite programming and the matrix multiplicative weights update method represent two examples that have been important in some of my recent work. I intend to continue to investigate the uses of these and other methods within the study of quantum information and computation. Quantum information has the potential to bring a transformative change to the way we build and use computers, communicate privately and implement cryptographic protocols, and study the nature of quantum physical systems. If it is successful, this research project will lead to a better theoretical understanding of quantum information and computation, to new mathematical methods that are useful in its study, and possibly to new ways that it can be used.