Author: dabacon

The most famous problem in quantum computing is the hidden subgroup problem. In the hidden subgroup problem you are given access to an oracle which computers a function f. This function is a function from a group G to some set with a promise on the function. The promise is that f is constant and distinct on left cosets of some subgroup H of G. The goal of the hidden subgroup problem is, by querring the function, to determine a generating set for the subgroup H. We say that an algorithm for the hidden subgroup problem is efficient if it uses resources proportional to a polynomial in the logarithm of the size of the group.
The reason the hidden subgroup problem is famous is threefold. The first reason is that when G is Abelian, efficiently solving the hidden subgroup problem leads to an efficient algorithm for factoring integers and for the discrete logarithm problem, i.e. to Shor’s algorithm(s). The second reason is that efficiently solving the hidden subgroup problem when G is the symmetric group would lead to an efficient algorithm for the graph isomorphism problem. The third reason is that efficiently solving the hidden subgroup problem when G is the dihedral group would lead to an efficient algorithm for certain shortest vector in a lattice problems. The first of these reasons is a concrete reason: a quantum algorithm worked! But for the second and third, no efficient quantum algorithm is known.
Now what I find interesting, is that for the second of these reasons, for graph isomorphism, one can get by with solving a much more restricted version of the hidden subgroup problem than the full hidden subgroup problem over the symmetric group. The way to do this is to modify the problem to a problem I call the hidden subgroup conjugacy problem. Call two subgroups, H and K conjugate to each other if there exists an element g of the group G such that gHg^{-1}=K. In the hidden subgroup conjugacy problem, instead of identifying the subgroup (by returning a set of generators, for example) we require that you only identify which conjugacy the subgroup belongs to, i.e. the setup is the same: you are given an f which hides a subgroup H, but now instead of identifying the subgroup you only want to know which conjugacy class the subgroup belongs to. It is easy to see that for the Ableian case, this reduces to the hidden subgroup problem: gHg^{-1}=H for all g in this case. For graph isomorphism, the standard reduction to the hidden subgroup problem reduces to distinguishing between the subgroup being a trivial subgroup and the subgroup being a order two subgroup. So efficiently solving the hidden subgroup conjugacy problem would lead to an efficient algorithm for graph isomorphism. Interesting, the same does not seem to be true for the reduction from the hidden subgroup to certain shortest vector in a lattice problems, although I haven’t thought hard about this.
So the question I ask is, did we overgeneralize the hidden subgroup problem? Did we generalize ourselves into a problem which is just too hard, while efficiently solving a smaller variant would lead to interesting new quantum algorithms? I leave history to judge.