9,645 research outputs found
Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras
Methods of construction of the composition function, left- and
right-invariant vector fields and differential 1-forms of a Lie group from the
structure constants of the associated Lie algebra are proposed. It is shown
that in the second canonical coordinates these problems are reduced to the
matrix inversions and matrix exponentiations, and the composition function can
be represented in quadratures. Moreover, it is proven that the transition
function from the first canonical coordinates to the second canonical
coordinates can be found by quadratures
Fast computation of power series solutions of systems of differential equations
We propose new algorithms for the computation of the first N terms of a
vector (resp. a basis) of power series solutions of a linear system of
differential equations at an ordinary point, using a number of arithmetic
operations which is quasi-linear with respect to N. Similar results are also
given in the non-linear case. This extends previous results obtained by Brent
and Kung for scalar differential equations of order one and two
Computing the Exponential of Large Block-Triangular Block-Toeplitz Matrices Encountered in Fluid Queues
The Erlangian approximation of Markovian fluid queues leads to the problem of
computing the matrix exponential of a subgenerator having a block-triangular,
block-Toeplitz structure. To this end, we propose some algorithms which exploit
the Toeplitz structure and the properties of generators. Such algorithms allow
to compute the exponential of very large matrices, which would otherwise be
untreatable with standard methods. We also prove interesting decay properties
of the exponential of a generator having a block-triangular, block-Toeplitz
structure
On commuting matrices and exponentials
Let A and B be matrices of M_n(C). We show that if exp(A)^k
exp(B)^l=exp(kA+lB) for all integers k and l, then AB=BA. We also show that if
exp(A)^k exp(B)=exp(B)exp(A)^k=exp(kA+B)$ for every positive integer k, then
the pair (A,B) has property L of Motzkin and Taussky. As a consequence, if G is
a subgroup of (M_n(C),+) and M -> exp(M) is a homomorphism from G to
(GL_n(C),x), then G consists of commuting matrices. If S is a subsemigroup of
(M_n(C),+) and M -> exp(M) is a homomorphism from S to (GL_n(C),x), then the
linear subspace Span(S) of M_n(C) has property L of Motzkin and Taussky.Comment: 17 pages, similar version as the one that will be published in Proc.
Amer. Math. So
On the generation of sequential unitary gates from continuous time Schrodinger equations driven by external fields
In all the various proposals for quantum computers, a common feature is that
the quantum circuits are expected to be made of cascades of unitary
transformations acting on the quantum states. A framework is proposed to
express these elementary quantum gates directly in terms of the control inputs
entering into the continuous time forced Schrodinger equation.Comment: 10 page
QIP = PSPACE
We prove that the complexity class QIP, which consists of all problems having
quantum interactive proof systems, is contained in PSPACE. This containment is
proved by applying a parallelized form of the matrix multiplicative weights
update method to a class of semidefinite programs that captures the
computational power of quantum interactive proofs. As the containment of PSPACE
in QIP follows immediately from the well-known equality IP = PSPACE, the
equality QIP = PSPACE follows.Comment: 21 pages; v2 includes corrections and minor revision
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