875 research outputs found
One-site density matrix renormalization group and alternating minimum energy algorithm
Given in the title are two algorithms to compute the extreme eigenstate of a
high-dimensional Hermitian matrix using the tensor train (TT) / matrix product
states (MPS) representation. Both methods empower the traditional alternating
direction scheme with the auxiliary (e.g. gradient) information, which
substantially improves the convergence in many difficult cases. Being
conceptually close, these methods have different derivation, implementation,
theoretical and practical properties. We emphasize the differences, and
reproduce the numerical example to compare the performance of two algorithms.Comment: Submitted to the proceedings of ENUMATH 201
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions
We propose algorithms for the solution of high-dimensional symmetrical positive definite (SPD) linear systems with the matrix and the right-hand side given and the solution sought in a low-rank format. Similarly to density matrix renormalization group (DMRG) algorithms, our methods optimize the components of the tensor product format subsequently. To improve the convergence, we expand the search space by an inexact gradient direction. We prove the geometrical convergence and estimate the convergence rate of the proposed methods utilizing the analysis of the steepest descent algorithm. The complexity of the presented algorithms is linear in the mode size and dimension, and the demonstrated convergence is comparable to or even better than the one of the DMRG algorithm. In the numerical experiment we show that the proposed methods are also efficient for non-SPD systems, for example, those arising from the chemical master equation describing the gene regulatory model at the mesoscopic scale
Gauging tensor networks with belief propagation
Effectively compressing and optimizing tensor networks requires reliable
methods for fixing the latent degrees of freedom of the tensors, known as the
gauge. Here we introduce a new algorithm for gauging tensor networks using
belief propagation, a method that was originally formulated for performing
statistical inference on graphical models and has recently found applications
in tensor network algorithms. We show that this method is closely related to
known tensor network gauging methods. It has the practical advantage, however,
that existing belief propagation implementations can be repurposed for tensor
network gauging, and that belief propagation is a very simple algorithm based
on just tensor contractions so it can be easier to implement, optimize, and
generalize. We present numerical evidence and scaling arguments that this
algorithm is faster than existing gauging algorithms, demonstrating its usage
on structured, unstructured, and infinite tensor networks. Additionally, we
apply this method to improve the accuracy of the widely used simple update gate
evolution algorithm.Comment: 47 Pages. 11 Figure
Magnetism, coherent many-particle dynamics, and relaxation with ultracold bosons in optical superlattices
We study how well magnetic models can be implemented with ultracold bosonic
atoms of two different hyperfine states in an optical superlattice. The system
is captured by a two-species Bose-Hubbard model, but realizes in a certain
parameter regime actually the physics of a spin-1/2 Heisenberg magnet,
describing the second order hopping processes. Tuning of the superlattice
allows for controlling the effect of fast first order processes versus the
slower second order ones.
Using the density-matrix renormalization-group method, we provide the
evolution of typical experimentally available observables. The validity of the
description via the Heisenberg model, depending on the parameters of the
Hubbard model, is studied numerically and analytically. The analysis is also
motivated by recent experiments [S. Foelling et al., Nature 448, 1029 (2007);
S. Trotzky et al., Sience 319, 295 (2008)] where coherent two-particle dynamics
with ultracold bosonic atoms in isolated double wells were realized. We provide
theoretical background for the next step, the observation of coherent
many-particle dynamics after coupling the double wells. Contrary to the case of
isolated double wells, relaxation of local observables can be observed. The
tunability between the Bose-Hubbard model and the Heisenberg model in this
setup could be used to study experimentally the differences in equilibration
processes for nonintegrable and Bethe ansatz integrable models. We show that
the relaxation in the Heisenberg model is connected to a phase averaging
effect, which is in contrast to the typical scattering driven thermalization in
nonintegrable models. We discuss the preparation of magnetic groundstates by
adiabatic tuning of the superlattice parameters.Comment: 20 pages, 24 figures; minor changes, published versio
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