Explicit class field theory from quantum modularity, subspace packings, and convolution identities

This project connects number theory and quantum information theory. Number theory studies properties of integers (whole numbers), such as prime factorizations or the existence of integer solutions to equations, and generalizations of those properties to other number systems. Quantum information theory is the theoretic branch of quantum information science, which studies how quantum physics can be used to perform certain types of computation and communication beyond what is possible with classical computers. The project investigates connections between major open problems in both fields. In number theory, Hilbert's 12th Problem—from a famous list of 23 problems created in 1900—and the related Stark conjectures—formulated in the late 1970s—concern explicit expressions for certain abstract structures involving algebraic numbers. In quantum information theory, Zauner's conjecture posed in 1999 predicts the existence of highly-regular geometric configurations called “symmetric informationally complete positive operator-valued measures,” or SICs, that describe quantum measurements. The goal of the project is to discover a unifying mathematical phenomenon that resolves (or makes progress on) these problems in number theory and quantum information theory. Recent advances, including work of the principal investigator supported by the National Science Foundation, provide compelling numerical evidence for the existence of such a unification and lay groundwork for the methods that will be used. Results obtained will enable faster computation of SICs, which has applications to quantum information processing—especially quantum state tomography—and to classical compressed sensing for radar. The project will develop the Subspace Packings Database, a public database of geometric configurations more general than SICs that have applications in quantum information processing, wireless communications, and neural network design. The project will also include the training of graduate students and K-12 outreach initiatives. More precisely, the research in this project aims to advance the understanding of the order 1 abelian real quadratic Stark conjecture and Zauner's conjecture. Analytic methods will be applied toward proving the Twisted Convolution Conjecture for the Shintani-Faddeev modular cocycle using generalized beta integral relations from the theory of mathematical physics. Algebraic methods will be applied to describe the structure of the ring of projective unitary permutation invariants of an arbitrary equiangular tight frame with a fixed number of vectors. Collaborative research on holomorphic quantum modular forms will situate real quadratic Stark units, which are essentially (squares of) real multiplication values of the Shintani-Faddeev modular cocycle, within the context of a larger theory. New optimal subspace packings will be determined using numerical optimization routines, followed by exactification using lattice basis reduction. These examples will be added to the public database, and examined for generalizable features, connections to other number-theoretic phenomena such as cubic Stark units, and suitability for real-world applications. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.