Hidden Subgroup Zoo

The Problem

Hidden Subgroup Problem (HSP)

Given: A function <math>f</math> from a group <math>G</math> to a set <math>S</math>, <math>f:G \rightarrow S</math> which is promised to be constant and distinct on different left cosets of an unknown subgroup <math>H</math>:

<math>f(g_1)=f(g_2)~{\rm iff}~g_1H = g_2H</math>

Problem: Find the hidden subgroup <math>H</math> by returning a set of generators for <math>H</math>

An algorithm for the hidden subgroup problem is efficient if the algorithm runs in a time polynomial in the logarithm of the size of the group (for infinite groups the definition is more subtle.