Theoretical aspects of quantum information and computation

当今的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明确的逻辑状态，并且计算根据确定性（有时是随机的）逻辑操作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。就像原子中的电子以叠加态存在一样，在牛顿物理学的背景下无法确定地描述，量子计算机可以存在于逻辑状态的叠加态中，它们的计算可以同时沿着多个计算路径进行，这些计算路径可能会相互干扰，产生建设性或破坏性的影响。量子计算理论研究了这种计算范式的能力和局限性。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，以及在合作和竞争环境中多台量子计算机之间相互作用的性质。我也对量子信息的基本方面，以及对量子信息和计算推理有用的数学技术的发展感兴趣。我工作的一个主要主题是量子计算复杂性理论。本学科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由经典计算模型定义的计算问题联系起来。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概念的模型，该概念被编程为尽可能快地解决计算问题；描述分布式和加密设置中多个量子计算机之间相互作用的模型；以及通过在量子模型上放置资源约束和其他限制来定义的问题类别。量子交互证明系统模型是在量子计算复杂性理论中研究的量子计算模型的一个例子——这个模型一直是我以前工作的主题，也是我打算在本研究计划中进行的研究的核心。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问题。我特别感兴趣的技术从组合优化，凸分析，矩阵分析，和算子代数的研究。半定规划和矩阵乘法权值更新方法是我最近一些工作中很重要的两个例子。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、私下通信和实现加密协议以及研究量子物理系统的本质的方式带来革命性的变化。如果它成功了，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，在研究中有用的新的数学方法，以及可能使用它的新方法。

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.