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.
