4 research outputs found
Fast Quantum Fourier Transforms for a Class of Non-abelian Groups
An algorithm is presented allowing the construction of fast Fourier
transforms for any solvable group on a classical computer. The special
structure of the recursion formula being the core of this algorithm makes it a
good starting point to obtain systematically fast Fourier transforms for
solvable groups on a quantum computer. The inherent structure of the Hilbert
space imposed by the qubit architecture suggests to consider groups of order
2^n first (where n is the number of qubits). As an example, fast quantum
Fourier transforms for all 4 classes of non-abelian 2-groups with cyclic normal
subgroup of index 2 are explicitly constructed in terms of quantum circuits.
The (quantum) complexity of the Fourier transform for these groups of size 2^n
is O(n^2) in all cases.Comment: 16 pages, LaTeX2
Polynomial-Time Solution to the Hidden Subgroup Problem for a Class of non-abelian Groups
We present a family of non-abelian groups for which the hidden subgroup
problem can be solved efficiently on a quantum computer.Comment: 16 pages, LaTeX2e, 3 figure
Symmetry-based matrix factorization
AbstractWe present a method for factoring a given matrix M into a short product of sparse matrices, provided that M has a suitable “symmetry”. This sparse factorization represents a fast algorithm for the matrix–vector multiplication with M. The factorization method consists of two essential steps. First, a combinatorial search is used to compute a suitable symmetry of M in the form of a pair of group representations. Second, the group representations are decomposed stepwise, which yields factorized decomposition matrices and determines a sparse factorization of M. The focus of this article is the first step, finding the symmetries. All algorithms described have been implemented in the library AREP. We present examples for automatically generated sparse factorizations—and hence fast algorithms—for a class of matrices corresponding to digital signal processing transforms including the discrete Fourier, cosine, Hartley, and Haar transforms