On Quantum Algorithms for Noncommutative Hidden Subgroups

Abstract

Quantum algorithms for factoring and discrete logarithm 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.

Permutation groups, minimal degrees and quantum computing

“…Several positive results on the power of QFS for the Hidden Subgroup Problem have been obtained previously for groups that are in some ways "close" toAbelian, like some semidirect products of Abelian groups [6], [33], [20], [32], [28], in particular the Dihedral group; Hamiltonian groups [14], groups with small commutator groups [15] and solvable groups of constant exponent and constant length derived series [7]. Often in these cases the irreducible representations are known and can be analysed.…”

“…Let Irr.G/ be the set of irreducible characters of G. Then P H is a distribution on Irr.G/. The strong standard method sometimes provides substantially more information than its weak counterpart, and is indeed necessary to efficiently solve the HSP in the case of groups like the dihedral group [6], [20], [32] and other semidirect product groups [28]. However (see below), for S n it has been shown [11] that for a random basis the additional information provided by the strong method is exponentially small except possibly for very large subgroups.…”

“…Often in these cases the irreducible representations are known and can be analysed. For instance the Dihedral group D n , the first non-Abelian group to be analysed in this context [6], is "nearly" Abelian in the sense that all of its irreducible representations have degree at most two. Indeed hidden reflections of D n can be distinguished from the identity with only polynomial Quantum Fourier Samplings, similar to the Abelian case (where all irreducible representations are one-dimensional).…”

The quantum query complexity of the hidden subgroup problem is polynomial

“…It is possible to improve the success probability to one for Abelian groups of smooth order [4] (a group is of c-smooth order if all prime factors of |G| are at most (log |G|) c for some constant c). For non-Abelian groups, our knowledge is much more limited [5,6,7,8,9,10].…”

On the Complexity of the Hidden Subgroup Problem

“…For a fixed integer B, we say an integer n is B-smooth if all the prime factors of n are less than or equal to B. For such an n, the prime factorization can be obtained in time polynomial in B + log n. Without loss of generality, we assume that the hidden subgroup is an order-two subgroup of D n [EH00]. Proof.…”