1,863 research outputs found
Robust Entanglement in Anti-ferromagnetic Heisenberg Chains by Single-spin Optimal Control
We demonstrate how near-perfect entanglement (in fact arbitrarily close to
maximal entanglement) can be generated between the end spins of an
anti-ferromagnetic isotropic Heisenberg chain of length , starting from the
ground state in the excitation subspace, by applying a magnetic field
along a given direction, acting on a single spin only. Temporally optimal
magnetic fields to generate a singlet pair between the two end spins of the
chain are calculated for chains up to length 20 using optimal control theory.
The optimal fields are shown to remain effective in various non-ideal
situations including thermal fluctuations, magnetic field leakage, random
system couplings and decoherence. Furthermore, the quality of the entanglement
generated can be substantially improved by taking these imperfections into
account in the optimization. In particular, the optimal pulse of a given
thermal initial state is also optimal for any other initial thermal state with
lower temperature.Comment: 10 pages, revte
Scaling Sparse Constrained Nonlinear Problems for Iterative Solvers
We look at scaling a nonlinear optimization problem for iterative solvers that use at least first derivatives. These derivatives are either computed analytically or by differncing. We ignore iterative methods that are based on function evaluations only and that do not use any derivative information. We also exclude methods where the full problem structure is unknown like variants of delayed column generation. We look at related work in section (1). Despite its importance as evidenced in widely used implementations of nonlinear programming algorithms, scaling has not received enough attention from a theoretical point of view. What do we mean by scaling a nonlinear problem itself is not very clear. In this paper we attempt a scaling framework definition. We start with a description of a nonlinear problem in section (2). Various authors prefer different forms, but all forms can be converted to the form we show. We then describe our scaling framework in section (3). We show the equivalence between the original problem and the scaled problem. The correctness results of section (3.3) play an important role in the dynamic scaling scheme suggested. In section (4), we develop a prototypical algorithm that can be used to represent a variety of iterative solution methods. Using this we examine the impact of scaling in section (5). In the last section (6), we look at what the goal should be for an ideal scaling scheme and make some implementation suggestions for nonlinear solvers.
Order reduction approaches for the algebraic Riccati equation and the LQR problem
We explore order reduction techniques for solving the algebraic Riccati
equation (ARE), and investigating the numerical solution of the
linear-quadratic regulator problem (LQR). A classical approach is to build a
surrogate low dimensional model of the dynamical system, for instance by means
of balanced truncation, and then solve the corresponding ARE. Alternatively,
iterative methods can be used to directly solve the ARE and use its approximate
solution to estimate quantities associated with the LQR. We propose a class of
Petrov-Galerkin strategies that simultaneously reduce the dynamical system
while approximately solving the ARE by projection. This methodology
significantly generalizes a recently developed Galerkin method by using a pair
of projection spaces, as it is often done in model order reduction of dynamical
systems. Numerical experiments illustrate the advantages of the new class of
methods over classical approaches when dealing with large matrices
Low-rank approximate inverse for preconditioning tensor-structured linear systems
In this paper, we propose an algorithm for the construction of low-rank
approximations of the inverse of an operator given in low-rank tensor format.
The construction relies on an updated greedy algorithm for the minimization of
a suitable distance to the inverse operator. It provides a sequence of
approximations that are defined as the projections of the inverse operator in
an increasing sequence of linear subspaces of operators. These subspaces are
obtained by the tensorization of bases of operators that are constructed from
successive rank-one corrections. In order to handle high-order tensors,
approximate projections are computed in low-rank Hierarchical Tucker subsets of
the successive subspaces of operators. Some desired properties such as symmetry
or sparsity can be imposed on the approximate inverse operator during the
correction step, where an optimal rank-one correction is searched as the tensor
product of operators with the desired properties. Numerical examples illustrate
the ability of this algorithm to provide efficient preconditioners for linear
systems in tensor format that improve the convergence of iterative solvers and
also the quality of the resulting low-rank approximations of the solution
Global Control Methods for GHZ State Generation on 1-D Ising Chain
We discuss how to prepare an Ising chain in a GHZ state using a single global
control field only. This model does not require the spins to be individually
addressable and is applicable to quantum systems such as cold atoms in optical
lattices, some liquid- or solid-state NMR experiments, and many nano-scale
quantum structures. We show that GHZ states can always be reached
asymptotically from certain easy-to-prepare initial states using adiabatic
passage, and under certain conditions finite-time reachability can be ensured.
To provide a reference useful for future experimental implementations three
different control strategies to achieve the objective, adiabatic passage,
Lyapunov control and optimal control are compared, and their advantages and
disadvantages discussed, in particular in the presence of realistic
imperfections such as imperfect initial state preparation, system inhomogeneity
and dephasing.Comment: 13 pages, 11 figure
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