94 research outputs found
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
On the distance between non-isomorphic groups
A result of Ben-Or, Coppersmith, Luby and Rubinfeld on testing whether a map
be two groups is close to a homomorphism implies a tight lower bound on the
distance between the multiplication tables of two non-isomorphic groups.Comment: 2 pages; corrected referenc
On Solving Systems of Diagonal Polynomial Equations Over Finite Fields
We present an algorithm to solve a system of diagonal polynomial equations
over finite fields when the number of variables is greater than some fixed
polynomial of the number of equations whose degree depends only on the degree
of the polynomial equations. Our algorithm works in time polynomial in the
number of equations and the logarithm of the size of the field, whenever the
degree of the polynomial equations is constant. As a consequence we design
polynomial time quantum algorithms for two algebraic hidden structure problems:
for the hidden subgroup problem in certain semidirect product p-groups of
constant nilpotency class, and for the multi-dimensional univariate hidden
polynomial graph problem when the degree of the polynomials is constant.Comment: A preliminary extended abstract of this paper has appeared in
Proceedings of FAW 2015, Springer LNCS vol. 9130, pp. 125-137 (2015
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
We present new deterministic algorithms for several cases of the maximum rank
matrix completion problem (for short matrix completion), i.e. the problem of
assigning values to the variables in a given symbolic matrix as to maximize the
resulting matrix rank. Matrix completion belongs to the fundamental problems in
computational complexity with numerous important algorithmic applications,
among others, in computing dynamic transitive closures or multicast network
codings (Harvey et al SODA 2005, Harvey et al SODA 2006).
We design efficient deterministic algorithms for common generalizations of
the results of Lovasz and Geelen on this problem by allowing linear functions
in the entries of the input matrix such that the submatrices corresponding to
each variable have rank one. We present also a deterministic polynomial time
algorithm for finding the minimal number of generators of a given module
structure given by matrices. We establish further several hardness results
related to matrix algebras and modules. As a result we connect the classical
problem of polynomial identity testing with checking surjectivity (or
injectivity) between two given modules. One of the elements of our algorithm is
a construction of a greedy algorithm for finding a maximum rank element in the
more general setting of the problem. The proof methods used in this paper could
be also of independent interest.Comment: 14 pages, preliminar
Quantum computation of discrete logarithms in semigroups
We describe an efficient quantum algorithm for computing discrete logarithms in semigroups using Shor's algorithms for period finding and the discrete logarithm problem as subroutines. Thus proposed cryptosystems based on the presumed hardness of discrete logarithms in semigroups are insecure against quantum attacks.
In contrast, we show that some generalizations of the discrete logarithm problem are hard in semigroups despite being easy in groups. We relate a shifted version of the discrete logarithm problem in semigroups to the dihedral hidden subgroup problem, and we show that the constructive membership problem with respect to k ≥ 2 generators in a black-box abelian semigroup of order N requires
Θ˜(N12-12k) quantum queries
Deciding universality of quantum gates
We say that collection of -qudit gates is universal if there exists
such that for every every -qudit unitary operation
can be approximated with arbitrary precision by a circuit built from gates of
the collection. Our main result is an upper bound on the smallest with
the above property. The bound is roughly , where is the number of
levels of the base system (the '' in the term quit.) The proof is based
on a recent result on invariants of (finite) linear groups.Comment: 8 pages, minor correction
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