Logic and Geometry in Quantum Computing
Abstract
An ambitious aim of the coming decades is construction of a fault-tolerant quantum computer. This promises a revolution in areas ranging from information security to chemistry to artificial intelligence. In many ways, this is one of the most complex physical systems ever attempted. This complexity exists at the physical level of materials to implement quantum circuits, and also at the level of understanding quantum information flow. The proposed research is centered on the use of logical, order-theoretic, and geometric tools to further the understanding of several critical issues in quantum computing, including the topological phases of matter to be used in topological quantum computing, and the picture languages and string diagrams used in categorical approaches to quantum information flow. Since von Neumann formulated quantum mechanics in rigorous mathematical terms, logical, order-theoretic, and geometric concepts have played a guiding role. This viewpoint was key in BirkhoÀ and von NeumannÕs quantum logic, with von NeumannÕs continuous geometry, and with Murray and von NeumannÕs rings of operators. Groundbreaking work of Jones on rings of operators has found application in areas as diverse as knot theory and topological phases of matter. JonesÕ work primarily uses analytic tools, and not the logical and geometric ones that guided von Neumann. One thrust of the study aims to close the circle and recast the work of Jones in the more general setting of continuous geometry. This will hopefully provide new geometric perspective and also allow extensions of JonesÕ results on topological phases of matter. The second thrust of the study uses the same logical and geometric tools to study quantum information flow. Approaches to quantum information make strong use of various types of categories and their associated picture languages. These picture languages have grown from Feynman diagrams and the string diagrams of Penrose. The decompositions approach to quantum foundations realizes the superposition principle via binary product decompositions of a structure. The collection of binary product decompositions plays the logical role of the events or yes/no questions of the system, and has a strong logical and projective geometric flavor. Products are inherently a categorical notion, and the incorporation of the decompositions approach into the study of quantum information flow will add new logical and projective-geometric tools. The logical viewpoint is well suited to view the nature of classical vs. quantum information.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jun 25, 2021
- Source ID
- W911NF2110247
Entities
People
- John Harding
Organizations
- Army Contracting Command
- New Mexico State University
- United States Army