Algorithms
There is a polynomial time quantum algorithm for solving HSP over finite Abelian groups. (In the case of hidden subgroup problem, "a polynomial time algorithm" means an algorithm whose running time is a polynomial of the logarithm of the size of the group.) Shor's algorithm applies a particular case of this quantum algorithm.
For arbitrary groups, it is known that the hidden subgroup problem is solvable using a polynomial number of evaluations of the oracle. This result, however, allows the quantum algorithm a running time that is exponential in log|G|. To design efficient algorithms for the graph isomorphism and SVP, one needs an algorithm for which both the number of oracle evaluations and the running time are polynomial.
The existence of such algorithm for arbitrary groups is open. Quantum polynomial time algorithms exist for certain subclasses of groups, such as semi-direct products of some Abelian groups.
The 'standard' approach to this problem involves: the creation of the quantum state, a subsequent Quantum Fourier transform to the left register, after which this register gets sampled. This approach has been shown to be insufficient for the hidden subgroup problem for the symmetric group.
Read more about this topic: Hidden Subgroup Problem