Related papers: Simulating hyperbolic space on a circuit board
Exploring the relationship between geometry and the resonant frequencies of a shape is of interest to pure and applied mathematicians. These resonant frequencies are related to the spectrum of the Laplacian, a partial differential operator.…
Hyperbolic embeddings are a class of representation learning methods that offer competitive performances when data can be abstracted as a tree-like graph. However, in practice, learning hyperbolic embeddings of hierarchical data is…
Many high-dimensional practical data sets have hierarchical structures induced by graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform…
Interacting many-body quantum systems show a rich array of physical phenomena and dynamical properties, but are notoriously difficult to study: they are challenging analytically and exponentially difficult to simulate on classical…
The Haldane model on the honeycomb lattice is a paradigmatic example of a Hamiltonian featuring topologically distinct phases of matter. It describes a mechanism through which a quantum Hall effect can appear as an intrinsic property of a…
We investigate the open dynamics of an atomic impurity embedded in a one-dimensional Bose-Hubbard lattice. We derive the reduced evolution equation for the impurity and show that the Bose-Hubbard lattice behaves as a tunable engineered…
We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator, associated with a position-dependent effective mass, by means of the supersymmetric formalism. From the supersymmetric partner…
In this work, we explore a numerical approach for performing the inverse Laplace transformation, with an emphasis on achieving stability and robustness under noisy conditions. Our quadrature-based method integrates reparameterization, data…
We study relationships between asymptotic geometry of submanifolds in the hyperbolic space and their regularity properties near the ideal boundary, revisiting some of the related results in the literature. In particular, we discuss…
Representing data in hyperbolic space can effectively capture latent hierarchical relationships. With the goal of enabling accurate classification of points in hyperbolic space while respecting their hyperbolic geometry, we introduce…
Non-Euclidean geometry, discovered by negating Euclid's parallel postulate, has been of considerable interest in mathematics and related fields for the description of geographical coordinates, Internet infrastructures, and the general…
We start to develop the quantization formalism in a hyperbolic Hilbert space. Generalizing Born's probability interpretation, we found that unitary transformations in such a Hilbert space represent a new class of transformations of…
Photonic implementations of unitary processes on lattice structures, such as quantum walks, have been demonstrated across various architectures. However, few platforms offer the combined advantages of scalability, reconfigurability, and the…
Learning good image representations that are beneficial to downstream tasks is a challenging task in computer vision. As such, a wide variety of self-supervised learning approaches have been proposed. Among them, contrastive learning has…
Momentum space of a gapped quantum system is a metric space: it admits a notion of distance reflecting properties of its quantum ground state. By using this quantum metric, we investigate geometric properties of momentum space. In…
This article presents some methods to control the bottom of the spectrum of the Laplacian $\lambda_0$ on hyperbolic surfaces with infinite volume. Our first result bounds the $\lambda_0$ of a geometrically finite surface in terms of the…
Negatively curved, or hyperbolic, regions of space in an FRW universe are a realistic possibility. These regions might occur in voids where there is no dark matter with only dark energy present. Hyperbolic space is strange and various…
Chemical hypergraphs and their associated normalized Laplace operators are generalized and studied in the case where each vertex--hyperedge incidence has a real coefficient. We systematically study the effect of symmetries of a hypergraph…
This work introduces novel numerical algorithms for computational quantum mechanics, grounded in a representation of the Laplace operator -- frequently used to model kinetic energy in quantum systems -- via the heat semigroup. The key…
We provide a framework to build periodic boundary conditions on the pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian space of constant negative curvature. Starting from the common case of periodic boundary…