Quantum Mechanical Algorithms for the Nonabelian Hidden Subgroup Problem

Abstract

We give a short exposition of new and known results on the "standard method" of identifying a hidden subgroup of a nonabelian group using a quantum computer. AbstractWe give a short exposition of new and known results on the "standard method"of identifying a hidden subgroup of a nonabelian group using a quantum computer.

Permutation groups, minimal degrees and quantum computing

“…It is shown in [11] that sampling the row index in the strong standard method provides no additional information. 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.…”

“…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.…”

“…In our applications to the Hidden Subgroup Problem, we focus on the weak form of the standard method, since the strong form with random choices of basis does not provide any non-negligible additional information for the symmetric group and the subgroups we consider [11]. It remains to be seen whether judicious choices of basis for each irreducible matrix representation can give more information in the case where random choices do not help; but to our knowledge no such examples have been found and in fact recent results of [29] show that in the case of fixed point free involutions no such good basis exists.…”

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

“…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.…”