Normal subgroup reconstruction and quantum computation using group representations

Abstract

The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups and this was used in Simon's algorithm and Shor's Factoring and Discrete Log algorithms. The non-Abelian case is open; an efficient solution would give rise to an efficient quantum algorithm for Graph Isomorphism. We fully analyze a natural generalization of the Abelian case solution to the non-Abelian case, and give an efficient solution to the problem for normal subgroups. We show, however, that this immediate generalization of the Abelian algorithm does not efficiently solve Graph Isomorphism.

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.…”

“…They also show that the additional information provided by the strong method in a random basis scales with 3 p jH j 2 k.G/=jGj where k.G/ is the number of conjugacy classes of the group G and jH j the size of the hidden subgroup. Both [14] and [11] show that hidden subgroups of S n of size jH j D 2, generated by involutions with large support, cannot be distinguished from identity; exactly the task that needs to be solved for Graph Automorphism. Recently, [29] have essentially shown that the strong standard method cannot distinguish the subgroup generated by a fixed point free involution from identity.…”

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].…”

A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem

“…Many works on hidden subgroup algorithms describe these steps differently [7,8,9,10,18,22]. Instead of defining U f as an embedding that creates f (g), they define it as a unitary operator that adds f (g) to an ancilla.…”

“…An important predecessor is Simon's algorithm [23] for the case G = (Z/2) n . Shor's algorithm extends to the general abelian case [14], to the case when H is normal [10], and to the case when H has few conjugates [9]. Since the main step in the generalized algorithm is the quantum character transform on the group algebra C[G], we will call it the character algorithm.…”