Coding for Storage, Computing, and Quantum Error Correction

Coding theory is a mathematical branch that dates back to the work of Claude Shannon and Richard Hamming in the late 1940s. They sought to ensure reliable communication and computation even in the presence of noisy channels and faulty machines. Over the years, the discipline, techniques, and challenges have evolved. Today, we are at an inflection point where we must control noise and errors along with several conditions for classical and quantum computers. In the classical setting, coding theory should satisfy other restrictions, like protecting data and ensuring tasks are performed quickly. In the quantum setting, coding theory is one of the needed pillars for the development of large-scale quantum computers that will revolutionize and benefit society. Specifically, quantum error-correcting codes are fundamental for the reliability of quantum computers and for applying and implementing quantum algorithms, like Shor’s, among others. In this proposal, we aim to develop more agile erasure recovery and repair for use in distributed storage, faster and secure frameworks for distributed computing, and fault-tolerant quantum error correction. Additional broader impacts include running online seminars in cyber-security, training of students and contributing to publicly available software. To address the proposed coding theory problems and challenges, the investigators harness their expertise in commutative algebra and algebraic geometry and build on the results of their long-standing collaboration as well as new developments in the field. Evaluation codes are generalizations of the ubiquitous Reed-Solomon codes, which are employed in a stunning array of applications to support data transmission and storage. Evaluation codes are a more nimble family based on evaluating rational functions over curves or higher-dimensional varieties or evaluating multivariate polynomials over points in affine space over a finite field, offering enhanced performance in a variety of settings. This project builds on recent advances, suggesting that the full capabilities of evaluation codes have not yet been realized. Algebraic function fields support reduced complexity distributed computing. Folded evaluation codes may provide additional noise control over smaller alphabets than their Reed-Solomon counterparts. Powers of codes play a role in distributed storage and, along with their duals, in fault-tolerant quantum computation. 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.