On Quantum Algorithms for Noncommutative Hidden Subgroups

Abstract

Quantum algorithms for factoring and finding discrete logarithms have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden subgroups of noncommutative groups. We present a quantum algorithm for the special case of dihedral groups which determines the hidden subgroup in a linear number of calls to the input function. We also explore the difficulties of developing an algorithm to process the data to explicitly calculate a generating set for the subgroup. A general framework for the noncommutative hidden subgroup problem is discussed and we indicate future research directions.

Quantum mechanical algorithms for the nonabelian hidden subgroup problem

“…Ettinger, Høyer and Knill [5] show that for any group there exists a sequence of polynomially many queries, from which, with exponentially many measurements, we can reconstruct the hidden subgroup. For the special case of the dihedral group Dn Ettinger and Høyer [4] showed how to obtain sufficient statistical information about the hidden subgroup using polynomially many queries and polynomially many measurements; leaving open the question of whether there is an efficient reconstruction algorithm using that data. The dihedral group is interesting because by some measures it is not far from abelian, for instance none of its irreps have dimension greater than 2; on the other hand by our measure defined above, it is highly nonabelian, since |M (Dn)| ≤ 2.…”

Quantum computation and lattice problems

“…Although the dihedral group has a much simpler structure than the symmetric group, no solution to the HSP on the dihedral group is known. Ettinger and Høyer [7] showed that one can obtain sufficient statistical information about the hidden subgroup with only a polynomial number of queries. However, the question of finding an efficient algorithm that uses this information to solve the HSP is still open.…”

Quantum Mechanics on Finite Groups

“…(8,17−19) An important unsolved problem in quantum computation theory is the hidden subgroup problem. (5)(6)(7)(11)(12)(13)22) Let H be a subgroup of a finite group G and let X be a nonempty set. A function f : G → X separates cosets of H if for every g 1 , g 2 ∈ G, f (g 1 ) = f (g 2 ) if and only if Hg 1 = Hg 2 .…”