abelian hidden subgroup problem Pell's equation principal ideal ring factoring graph isomorphism lattice problems symmetric group dihedral group algorithm quantum computer computer P
The abelian hidden subgroup problem is a generalization of many problems that can be solved by a quantum computer, such as Simon's problem, solving Pell's equation, testing the principal ideal of a ring R and factoring. There are efficient quantum algorithms known for the Abelian hidden subgroup problem. The more general hidden subgroup problem, where the group isn't necessarily abelian, is a generalization of the previously mentioned problems and graph isomorphism and certain lattice problems. Efficient quantum algorithms are known for certain non-abelian groups. However, no efficient algorithms are known for the symmetric group, which would give an efficient algorithm for graph isomorphism and the dihedral group, which would solve certain lattice problems.
