56 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 literatureâthe âforgetfulâ abelian method, and measurement in a random basisâand 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
Hidden Subgroup Quantum Algorithms for a Class of Semi-Direct Product Groups
A quantum algorithm for the Hidden Subgroup Problem over the group Z/p^{r}Z rtimes Z/q^{s}Z is presented. This algorithm, which for certain parameters of the group qualifies as \u27efficient\u27, generalizes prior work on related semi-direct product groups
Finding hidden Borel subgroups of the general linear group
We present a quantum algorithm for solving the hidden subgroup problem in the
general linear group over a finite field where the hidden subgroup is promised
to be a conjugate of the group of the invertible lower triangular matrices. The
complexity of the algorithm is polynomial when size of the base field is not
much smaller than the degree.Comment: 12pt, 10 page
Quantum algorithms for algebraic problems
Quantum computers can execute algorithms that dramatically outperform
classical computation. As the best-known example, Shor discovered an efficient
quantum algorithm for factoring integers, whereas factoring appears to be
difficult for classical computers. Understanding what other computational
problems can be solved significantly faster using quantum algorithms is one of
the major challenges in the theory of quantum computation, and such algorithms
motivate the formidable task of building a large-scale quantum computer. This
article reviews the current state of quantum algorithms, focusing on algorithms
with superpolynomial speedup over classical computation, and in particular, on
problems with an algebraic flavor.Comment: 52 pages, 3 figures, to appear in Reviews of Modern Physic
On group invariants determined by modular group algebras: even versus odd characteristic
Let be a an odd prime and let be a finite -group with cyclic
commutator subgroup . We prove that the exponent and the abelianization of
the centralizer of in are determined by the group algebra of over
any field of characteristic . If, additionally, is -generated then
almost all the numerical invariants determining up to isomorphism are
determined by the same group algebras; as a consequence the isomorphism type of
the centralizer of is determined. These claims are known to be false for
.Comment: 18 page
The power of basis selection in fourier sampling: hidden subgroup problems in affine groups
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which a unknown subgroup H of a group G must be determined from a quantum state Ï 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 Ï 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 as well as its name) occurs. It has remained open whether the strong 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 semidirect products of the form â€q Ă â€p, where q | (p - 1) and q = p/polylog(p), can be efficiently determined by the strong standard method. Furthermore, the weak standard method and the "forgetful" abelian method are insufficient for these groups so that, in fact, it appears that use of the corresponding nonabelian representation theory is crucial. We extend this to an informationtheoretic solution for the hidden subgroup problem over the groups â€q Ă â€p where q | (p - 1) and, in particular, the affine groups Ap. Finally, we prove a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently
- âŠ