Topology of quantum resources: Homotopical methods in resource theories for quantum information and quantum computing
Abstract
Quantum theory, currently the most fundamental theory of nature, is changing our perspective on information processing by making possible new procedures that bring computational and information-theoretic advantages otherwise unavailable in the classical realm. Understanding the basic reasons behind this quantum advantage is going to help us to build new quantum algorithms with novel applications in emerging quantum technologies and on the foundational level will shed light upon the unexpected behavior of the quantum world. In this proposal we pursue a recently established connection, discovered by the PI and his collaborators including Robert Raussendorf, between algebraic topology, the study of spaces and their algebraic invariants under continuous deformation, and quantum information theory, the quantum way of processing information. This connection is established through a topological approach to quantum contextuality, a notion generalizing Bell's non-locality, that emerged in the study of quantum foundations. Like entanglement, contextuality can be used as a resource for computational speed-up in certain prominent schemes of quantum computation such as quantum computation with magic states and measurement-based quantum computation. In addition, contextuality is a valuable resource for secure cryptography. Our topological approach offers a new perspective on contextuality tailored for resource theoretic applications in computation. On the other hand, it unifies other approaches to contextuality, such as graph-theoretic and sheaf-theoretic. Techniques from simplicial homotopy theory are used to construct classifying spaces, central objects in algebraic topology with applications to bundle theory, whose cohomological and homotopical invariants inform us about contextuality. Such invariants can be translated into results that quantify the fundamental ingredients for computational advantage in quantum computers.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jan 21, 2022
- Source ID
- FA95502110002XX0
Entities
People
- Cihan Okay
Organizations
- Air Force Office of Scientific Research
- Bilkent University
- United States Air Force