The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently

Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.Comment: 10 page

The quantum query complexity of the hidden subgroup problem is polynomial

We present a quantum algorithm which identifies with certainty a hidden subgroup of an arbitrary finite group G in only a polynomial (in log |G|) number of calls to the oracle. This is exponentially better than the best classical algorithm. However our quantum algorithm requires exponential time, as in the classical case. Our algorithm utilizes a new technique for constructing error-free algorithms for non-decision problems on quantum computers.Comment: To appear in Information Processing Letters (IPL

From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups

We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probability of success and unitary implementation are closely related to an average-case algebraic problem. By solving this problem, we find efficient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form (Z_p)^r X| Z_p for fixed r (including the Heisenberg group, r=2). In particular, our results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.Comment: 18 pages; v2: updated references on optimal measuremen