CAREER: Algebraic and Semiclassical Methods for Quantum Computing

这个项目使用代数几何和半经典系统物理学的新分析技术来探索量子计算机的能力和局限性。我们是否决定花费大量资源来构建可伸缩的量子计算机，很大程度上将取决于这样一台机器的预期好处，但我们目前对这个问题的理解仍然非常有限。这个项目的研究应该会让我们更好地了解以量子力学的方式处理信息的有用性，因为它提供了一本关于量子电路的性质和一些更直观的几何概念之间的词典。扩展部分包括设计和实施一个关于量子计算的在线信息来源的权威、带注释的书目，该书目将免费提供给学术界以外的人。量子计算机的演变最好被描述为经典计算的加权和，其中每个权重都是复值概率幅度。因此，量子计算的总体概率可以使用相应的幅度指数和来计算，这是一个在代数几何中得到广泛研究的概念。在这个项目中，我们将这些指数和背后的几何和拓扑思想转化为相应的量子电路的性质。从物理的角度看，振幅的相位与它的“作用力”成正比，通过最小作用量原理，它在连接量子力学和经典力学中起着至关重要的作用。我们使用这一观点将量子计算与其经典计算路径之和联系起来。

This project explores the power and limitations of quantum computers using new analytical techniques from algebraic geometry and the physics of semi-classical systems. Whether or not we decide to spend large amounts of resources on the construction of a scalable quantum computer will depend largely on the expected benefits of such a machine, but our current understanding of this issue is still very limited. The research of this project should give us a better picture of the usefulness of processing information in a quantum mechanical way by providing a dictionary between the properties of quantum circuits and some of the more intuitive ideas in geometry. The outreach component consists of the design and implementation of an authoritative, annotated bibliography of online sources of information on quantum computing, which will be made freely available to those outside of academia.The evolution of a quantum computer is best described as a weighted sum of classical computations, where each weight is a complex valued probability amplitude. As a result, the overall probabilities of the quantum computation can be calculated using the corresponding exponential sums of amplitudes, which is a concept that has been studied extensively in algebraic geometry. In this project we translate the geometric and topological ideas behind these exponential sums to the properties of the corresponding quantum circuits. From a physical point of view, the phase of an amplitude is proportional to its `action', which, through the Least Action Principle, plays a crucial role in connecting quantum mechanics with classical mechanics. We use this viewpoint to relate quantum computation to its sum of classical computational paths.