68 research outputs found
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
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 of a group must be determined from a quantum state over that is uniformly supported on a left coset of . These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of 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 of the -hedral groups, i.e., semidirect products , where , and in particular the affine groups , can be information-theoretically reconstructed using the strong standard method. Moreover, if , 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
Quantum algorithm for a generalized hidden shift problem
Consider the following generalized hidden shift problem:
given a function f on {0,...,M − 1} × ZN promised to be
injective for fixed b and satisfying f(b, x) = f(b + 1, x + s)
for b = 0, 1,...,M − 2, find the unknown shift s ∈ ZN.
For M = N, this problem is an instance of the abelian
hidden subgroup problem, which can be solved efficiently on
a quantum computer, whereas for M = 2, it is equivalent
to the dihedral hidden subgroup problem, for which no
efficient algorithm is known. For any fixed positive �, we give
an efficient (i.e., poly(logN)) quantum algorithm for this
problem provided M ≥ N^∈. The algorithm is based on the
“pretty good measurement” and uses H. Lenstra’s (classical)
algorithm for integer programming as a subroutine
Efficient Quantum Algorithm for Identifying Hidden Polynomials
We consider a natural generalization of an abelian Hidden Subgroup Problem
where the subgroups and their cosets correspond to graphs of linear functions
over a finite field F with d elements. The hidden functions of the generalized
problem are not restricted to be linear but can also be m-variate polynomial
functions of total degree n>=2.
The problem of identifying hidden m-variate polynomials of degree less or
equal to n for fixed n and m is hard on a classical computer since
Omega(sqrt{d}) black-box queries are required to guarantee a constant success
probability. In contrast, we present a quantum algorithm that correctly
identifies such hidden polynomials for all but a finite number of values of d
with constant probability and that has a running time that is only
polylogarithmic in d.Comment: 17 page
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