Periodic Reporting for period 1 - SYMOPTIC (Symmetry and Optimization at the Frontiers of Computation)
Reporting period: 2022-05-01 to 2024-10-31
Noncommutative group optimization is a powerful emerging paradigm. It brings together optimization in curved spaces, which has recently received much attention but is much less developed than the traditional setting, with symmetries, which play a central role across a wide range of disciplines. This combination promises to be key to fast algorithms and deep structural insight. It also provides a unifying perspective on a broad range of important problems across several disciplines that at first glance appear unrelated - from estimation problems in statistics, to program testing in computer science, to optimization on quantum computers. This project aims to develop the theoretical and algorithmic foundations of the new paradigm and apply it to longstanding theoretical problems and practical applications. To achieve this, it aims to bring together fields that are often treated separately. This has the potential for significant impact at several frontiers of computation: optimization, algebra, complexity, and quantum computing.
We have made significant progress on the above objectives.
In optimization, interior point methods offer a highly versatile framework that is effective in theory and practice. We have generalized this framework to curved spaces, along with general constructions of barriers, resulting in broadly applicable algorithms that offer state-of-the-art complexity guarantees for many applications.
In computational complexity, orbit problems connect to a wide range of natural algorithmic problems and applications. We have found new algorithms, complexity theoretic obstructions, and surprising connections, among others linking computational complexity to the famous abc-conjecture in mathematics.
In quantum information, we have discovered new insights into the structure of tensor networks, which play a fundamental role in quantum information and many-body physics, and connected their entanglement to complexity. This gives new structural understanding but also has potential implications on practical algorithms. We have also obtained new insights into the power and limitations of quantum computers.
In optimization, interior point methods offer a highly versatile framework that is effective in theory and practice. We have generalized this framework to curved spaces, along with general constructions of barriers, resulting in broadly applicable algorithms that offer state-of-the-art complexity guarantees for many applications.
In computational complexity, orbit problems connect to a wide range of natural algorithmic problems and applications. We have found new algorithms, complexity theoretic obstructions, and surprising connections, among others linking computational complexity to the famous abc-conjecture in mathematics.
In quantum information, we have discovered new insights into the structure of tensor networks, which play a fundamental role in quantum information and many-body physics, and connected their entanglement to complexity. This gives new structural understanding but also has potential implications on practical algorithms. We have also obtained new insights into the power and limitations of quantum computers.
All results mentioned above go significantly beyond the state of the art. For example, interior point methods revolutionized traditional convex optimization in the past century. Their generalization to curved spaces can be seen as a major advance that opens new opportunities. The same is true for the results on tensor networks, which arguably came as a surprise to the community as they overcome known "no go" results. Overall, the project results have the potential for long-lasting impact at the four frontiers of computation mentioned earlier, which are also being brought together in new ways - in addition to contributing widely applicable methods to optimization, by giving efficient algorithmic solutions to difficult problems in algebra, gaining new insights into the limits of efficient computation, and unlocking the potential of quantum computers and information.