David Sun and Zhe Xu Awarded 2020 Math Undergraduate Research Prizes

The hidden subgroup problem is the problem of efficiently determining generators for a subgroup of a group given evaluations of functions distinctly constant on cosets of the subgroup. Many important quantum algorithms can be recast in terms of this problem, for instance, Shor’s quantum algorithm for integer factorization corresponds to the cyclic group case. A solution in the dihedral group case was shown by Regev to have implications to solving hard lattice problems. The standard quantum approach to solving the hidden subgroup problem involves using the quantum Fourier transform, which brings into play the representations theory of the group.

As part of his NSERC USRA project in Summer 2020, David investigated the case of the dihedral group, one of the simplest class of non-abelian groups, but one for which the standard quantum approach runs into essential obstacles and which has been resistant to definitive results. David quickly learned all the basics of quantum computing and investigated a number of possible ways to modify the standard quantum approach. These contained a number of new insights which formed the basis for his honours thesis and later coauthored preprint, including a new relation to quantum cloning and some no-go results. The preprint is currently available on Arxiv and with plans to submit it to a research journal for publication.

Zhe completed an excellent research project during Summer 2020. His project was to explore the behaviour of symmetric differentials around surface singularities of type A_n, in the hope of finding a pattern by extending results for n=1 to the cases n=2 and n=3. Not only did he succeed, he formulated an answer based on lattice point counts in polytopes, valid for arbitrary n. This is particularly exciting because it offers an avenue to get similar results for the remaining canonical surface singularities D_n and E6, E7, E8. Obtaining these results required Zhe to quickly absorb a large and complicated body of mathematical theory in such a way that he could apply it to the problems at hand. He did so in an admirable fashion, and in a very independent way. His accomplishments far exceeded the expectations of his supervisor.