Shor’s groundbreaking algorithms for integer factoring and discrete logarithm demonstrated a unique ability of quantum computers to solve problems defined on abelian groups. In this thesis, we study ways in which that ability can be leveraged in order to solve problems on more complex structures such as non-abelian groups, hypergroups, and simplicial complexes.
This leads to new quantum algorithms for the hidden subgroup problem on nilpotent groups whose order contains a large prime factor, the hidden subhypergroup problem on both strongly integral hypergroups and ultragroups, testing equivalence of group extensions, and computing the component parts of the cohomology groups of both group extensions and a generalization of simplicial complexes, amongst other problems. For each of those listed, we also show that no classical algorithm can achieve similar efficiency under standard cryptographic assumptions.
Thesis Supervisor: Aram Harrow, CTP