19 research outputs found
An efficient quantum algorithm for the hidden subgroup problem in extraspecial groups
Extraspecial groups form a remarkable subclass of p-groups. They are also
present in quantum information theory, in particular in quantum error
correction. We give here a polynomial time quantum algorithm for finding hidden
subgroups in extraspecial groups. Our approach is quite different from the
recent algorithms presented in [17] and [2] for the Heisenberg group, the
extraspecial p-group of size p3 and exponent p. Exploiting certain nice
automorphisms of the extraspecial groups we define specific group actions which
are used to reduce the problem to hidden subgroup instances in abelian groups
that can be dealt with directly.Comment: 10 page
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
Hidden Translation and Translating Coset in Quantum Computing
We give efficient quantum algorithms for the problems of Hidden Translation
and Hidden Subgroup in a large class of non-abelian solvable groups including
solvable groups of constant exponent and of constant length derived series. Our
algorithms are recursive. For the base case, we solve efficiently Hidden
Translation in , whenever is a fixed prime. For the induction
step, we introduce the problem Translating Coset generalizing both Hidden
Translation and Hidden Subgroup, and prove a powerful self-reducibility result:
Translating Coset in a finite solvable group is reducible to instances of
Translating Coset in and , for appropriate normal subgroups of
. Our self-reducibility framework combined with Kuperberg's subexponential
quantum algorithm for solving Hidden Translation in any abelian group, leads to
subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in
any solvable group.Comment: Journal version: change of title and several minor update