633 research outputs found
Riemannian optimization of isometric tensor networks
Several tensor networks are built of isometric tensors, i.e. tensors
satisfying . Prominent examples include matrix
product states (MPS) in canonical form, the multiscale entanglement
renormalization ansatz (MERA), and quantum circuits in general, such as those
needed in state preparation and quantum variational eigensolvers. We show how
gradient-based optimization methods on Riemannian manifolds can be used to
optimize tensor networks of isometries to represent e.g. ground states of 1D
quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel
manifolds, the Riemannian manifolds of isometric tensors, and review how
state-of-the-art optimization methods like nonlinear conjugate gradient and
quasi-Newton algorithms can be implemented in this context. We apply these
methods in the context of infinite MPS and MERA, and show benchmark results in
which they outperform the best previously-known optimization methods, which are
tailor-made for those specific variational classes. We also provide open-source
implementations of our algorithms.Comment: 18 pages + appendices, 3 figures; v3 submission to SciPost; v4 expand
preconditioning discussion and add polish, resubmit to SciPos
Complex cobordism classes of homogeneous spaces
We consider compact homogeneous spaces G/H of positive Euler characteristic
endowed with an invariant almost complex structure J and the canonical action
\theta of the maximal torus T ^{k} on G/H. We obtain explicit formula for the
cobordism class of such manifold through the weights of the action \theta at
the identity fixed point eH by an action of the quotient group W_G/W_H of the
Weyl groups for G and H. In this way we show that the cobordism class for such
manifolds can be computed explicitly without information on their cohomology.
We also show that formula for cobordism class provides an explicit way for
computing the classical Chern numbers for (G/H, J). As a consequence we obtain
that the Chern numbers for (G/H, J) can be computed without information on
cohomology for G/H. As an application we provide an explicit formula for
cobordism classes and characteristic numbers of the flag manifolds U(n)/T^n,
Grassmann manifolds G_{n,k}=U(n)/(U(k)\times U(n-k)) and some particular
interesting examples.Comment: improvements in subsections 7.1 and 7.2; some small comments are
added or revised and some typos correcte
GrassmannOptim: An R Package for Grassmann Manifold Optimization
The optimization of a real-valued objective function f(U), where U is a p X d,p > d, semi-orthogonal matrix such that UTU=Id, and f is invariant under right orthogonal transformation of U, is often referred to as a Grassmann manifold optimization. Manifold optimization appears in a wide variety of computational problems in the applied sciences. In this article, we present GrassmannOptim, an R package for Grassmann manifold optimization. The implementation uses gradient-based algorithms and embeds a stochastic gradient method for global search. We describe the algorithms, provide some illustrative examples on the relevance of manifold optimization and finally, show some practical usages of the package
A Framework for Generalising the Newton Method and Other Iterative Methods from Euclidean Space to Manifolds
The Newton iteration is a popular method for minimising a cost function on
Euclidean space. Various generalisations to cost functions defined on manifolds
appear in the literature. In each case, the convergence rate of the generalised
Newton iteration needed establishing from first principles. The present paper
presents a framework for generalising iterative methods from Euclidean space to
manifolds that ensures local convergence rates are preserved. It applies to any
(memoryless) iterative method computing a coordinate independent property of a
function (such as a zero or a local minimum). All possible Newton methods on
manifolds are believed to come under this framework. Changes of coordinates,
and not any Riemannian structure, are shown to play a natural role in lifting
the Newton method to a manifold. The framework also gives new insight into the
design of Newton methods in general.Comment: 36 page
Chern-Simons Theory and Topological Strings
We review the relation between Chern-Simons gauge theory and topological
string theory on noncompact Calabi-Yau spaces. This relation has made possible
to give an exact solution of topological string theory on these spaces to all
orders in the string coupling constant. We focus on the construction of this
solution, which is encoded in the topological vertex, and we emphasize the
implications of the physics of string/gauge theory duality for knot theory and
for the geometry of Calabi-Yau manifolds.Comment: 46 pages, RMP style, 25 figures, minor corrections, references adde
Sigma models for quantum chaotic dynamics
We review the construction of the supersymmetric sigma model for unitary
maps, using the color- flavor transformation. We then illustrate applications
by three case studies in quantum chaos. In two of these cases, general Floquet
maps and quantum graphs, we show that universal spectral fluctuations arise
provided the pertinent classical dynamics are fully chaotic (ergodic and with
decay rates sufficiently gapped away from zero). In the third case, the kicked
rotor, we show how the existence of arbitrarily long-lived modes of excitation
(diffusion) precludes universal fluctuations and entails quantum localization
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