Abelian Hidden Subgroup Problem

Years and Authors of Summarized Original Work

1995, Kitaev;

2008, Mosca

Problem Definition

The Abelian hidden subgroup problem is the problem of finding generators for a subgroup K of an Abelian group G, where this subgroup is defined implicitly by a function $f : G \rightarrow X$, for some finite set X. In particular, f has the property that f(v) = f(w) if and only if the cosets (we are assuming additive notation for the group operation here.) v + K and w + K are equal. In other words, f is constant on the cosets of the subgroup K and distinct on each coset.

It is assumed that the group G is finitely generated and that the elements of G and X have unique binary encodings. The binary assumption is only for convenience, but it is important to have unique encodings (e.g., in [22] Watrous uses a quantum state as the unique encoding of group elements). When using variables g and h (possibly with subscripts), multiplicative notation is used for the group operations. Variables x and y...