CAREER: Algebraic and Semiclassical Methods for Quantum Computing

该项目探索量子计算机的能力和局限性，使用代数几何和半经典系统物理学的新分析技术。我们是否决定花费大量资源来构建可扩展的量子计算机将在很大程度上取决于这种机器的预期好处，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通过提供量子电路的性质和几何中一些更直观的想法之间的字典，让我们更好地了解以量子力学方式处理信息的有用性。外联部分包括设计和实施一个权威的、带注释的量子计算在线信息源书目，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描述为经典计算的加权和，其中每个权重是复值概率幅度。因此，量子计算的总概率可以使用相应的振幅指数和来计算，这是一个在代数几何中被广泛研究的概念。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量子电路的性质。从物理学的角度来看，振幅的相位与其“作用”成正比，通过最小作用原理，它在连接量子力学与经典力学方面发挥着至关重要的作用。 我们用这个观点来把量子计算与它的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.