Advancing quantum information through categorification
Abstract
Categorification is a mathematical concept that seeks to enhance existing structure by replacing the basic mathematical objects with more sophisticated (higher categorical) ones. A (higher) category is a mathematical structure that combines many levels of interacting layers. This enhancement usually allows for greater descriptive power. This proposal centers around applying this framework to quantum information science to create machines with higher fundamental capabilities. This project will provide the mathematical framework facilitating the next generation of quantum protocols. By utilizing hierarchies of iteratively constructed enhanced logical comparisons formulated within the context of higher categories, we will lay the groundwork for certifiably correct automated reasoning. The advantages imparted by this higher categorical framework are numerous, but a critical advantage of this approach is that it naturally affords a wealth of tools and techniques for design, development, and generalization. By exploiting the relational power of higher category theory, we will leverage deep connections between quantum field theories, logic, and quantum information to forge the new algorithms and improve DoD fundamental capabilities. In this proposal we argue that to fully realize machines with higher fundamental capabilities we have to reexamine the science of processes and understand the fundamental role of processes between processes, processes between those, and the iterative hierarchies thereby obtained. What is required is categorification, that is, the mathematical framework for uncovering a hidden layer in mathematical structures, revealing a richer theory, capable of describing more complex phenomena. Built into this philosophy is the idea that utilizing higher categories one can develop iteratively more sophisticated structures, enhancing the descriptive power at each stage. We will bring the most current tools from higher category theory, with its enhanced descriptive power, to enable the design of more robust and certifiably correct quantum protocols. We will leverage recent developments in the field of higher representation theory, or categorified representation theory, along with techniques from (topological) quantum field theory to categorify the mathematics of quantum information and unlock its greater descriptive power. Specifically, we address the following: 1) we explain how categorification can be used to develop a topological quantum field theory-based approach to certifiably correct quantum information; 2) we will develop a quantum stochastic differential equation description of `homotopy quantum fields to describe analog quantum computation in the context of adiabatic quantum computation; 3) using higher categorical analogs of Categorical Quantum Mechanics in the context of resource theories, we will create a language to describe and verify the correctness of quantum protocols. While this may at first appear to be a disparate set of objectives, the unifying theme is that of categorification. The core of all of these approaches are built on the mathematical infrastructure of higher categories and the language of categorification. Organizing around these essential mathematical tools, we will establish a unified blueprint that can be implemented in this diverse set of physically realizable systems.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jul 09, 2020
- Source ID
- W911NF2010075
Entities
People
- Aaron Lauda
Organizations
- Army Contracting Command
- United States Army
- University of Southern California