Motivation
The Hidden Subgroup Problem is especially important in the theory of quantum computing for the following reasons.
- Shor's quantum algorithm for factoring and discrete logarithm (as well as several of its extensions) relies on the ability of quantum computers to solve the HSP for finite Abelian groups.
- The existence of efficient quantum algorithms for HSPs for certain non-Abelian groups would imply efficient quantum algorithms for two major problems: the graph isomorphism problem and certain shortest vector problems (SVPs) in lattices. More precisely, an efficient quantum algorithm for the HSP for the symmetric group would give a quantum algorithm for the graph isomorphism. An efficient quantum algorithm for the HSP for the dihedral group would give a quantum algorithm for the poly(n) unique SVP.
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