Cone structures in quantum field theory linked to information geometry.
problem Understanding geometric structures in quantum field theory.
method Analyzing invariant cones under modular automorphism groups and their relation to Wishart laws.
result Explicit connection between CAH cones and Wishart laws.
Quantum Natural Gradient uses quantum geometry for optimization.
problem Optimizing variational quantum circuits efficiently.
method Quantum generalization of Natural Gradient Descent using Quantum Information Geometry.
result Efficient algorithm for computing metric tensor approximations.
New algorithm improves efficiency of quantum system modeling.
problem Intractable complexities in quantum Hamiltonian learning and Gibbs sampling.
method Generalized quantum natural gradient descent and Quantum-Probabilistic Mirror Descent.
result Data sample efficiency proven using information geometry and quantum metrology.
Jordan algebras in information geometry linked to metrics on probability distributions.
problem Understanding Jordan algebras in information geometry.
method Inspired by Kirillov's coadjoint orbits, a pseudo-Riemannian metric is constructed on Jordan algebra leaves.
result Not all points in the dual space lie on a leaf, and the metric structure depends on the cone of positive functionals.
Quantum connections replace metrics with operator inner products.
problem Quantifying geometric properties in quantum systems.
method Defining quantum connections and duals using operator fields and inner products.
result Holonomy and dual connections are equivalent in quantum geometry.
The paper explores quantum statistical manifolds and their autoparallelity, providing estimation-theoretical characterizations.
problem Quantum statistical manifolds and their geometric properties.
method Study of autoparallelity w.r.t. the e-connection, using quantum estimation theory.
result Characterizations of e-autoparallel submanifolds as statistical models with efficient estimators.
Develops a new geometric framework for quantum metrics.
problem Quantum metric generalization for pure two-qubit states.
method Support-projected Petz monotone geometry for pure two-qubit families.
result Strictly generalizes SLD/Bures case and includes other metrics.
Improved graph convolutional networks using perturbed graph Laplacian.
problem Improving graph convolutional networks' generalization.
method Minimizing loss with perturbed graph Laplacian eigendecomposition.
result Consistent improvement on semi-supervised node classification tasks.
The Yukawa term in statistical mechanics quantifies information generation.
problem Understanding information generation in statistical mechanics.
method Defining a statistical product and Yukawa term to quantify information generation.
result The Yukawa term diverges in the quantum case, indicating Bose-Einstein condensation.
Machine learning approximates phase transitions using Fisher information.
problem Understanding phase transitions from data using machine learning.
method Information geometry and Fisher information.
result Machine learning indicators approximate the square root of Fisher information.
Quantum LTA improves document analysis performance.
problem Improving latent topic analysis in quantum information retrieval.
method Proposed a quantum-motivated LTA method combining geometry and probability.
result Quantum-motivated LTA outperforms LSA on three datasets.
Let (M,g) be a compact, connected and oriented Riemannian manifold. We denote D the space of smooth probability density functions on M. In this paper, we show that the Frechet manifold D is equipped with a Riemannian metric g^{D} and an affine connection \nabla^{D} which are infinite dimensional analogues of the Fisher…
We present the construction of an infinite dimensional Banach manifold of quantum mechanical states on a Hilbert space H using different types of small perturbations of a given Hamiltonian. We provide the manifold with a flat connection, called the exponential connection, and comment on the possibility of introducing t…
Information geometry offers new tools for statistical analysis.
problem Statistical analysis of probability distributions.
method Geometric perspective on statistical manifolds.
result New applications in radar sensing, signal processing, etc.
The geometry of cosets in the subgroups H of the two-generator free group G =\textless{} a, b \textgreater{} nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. Dessins stabilize point-line incidence geometries that reflect the commutation of (generalized) Pauli opera…
Quantum field theory connects Riemannian geometry to quantum fluctuations.
problem Generating Riemannian structures from quantum fluctuations.
method QFT approach to Riemannian Geometry, focusing on Ricci curvature.
result Ricci curvature is crucial in generating Riemannian structures.
Bi-forms extend contrast functions to handle torsion in information geometry.
problem Insufficient contrast-based approaches for geometric structures with torsion.
method Introducing contrast bi-forms, a generalization of contrast functions.
result Bi-forms provide a unified framework for statistical potentials.
QCML uses quantum geometry to represent data.
problem Data representation and the curse of dimensionality.
method QCML encodes data as Hermitian matrices in Hilbert space.
result Data geometry reveals intrinsic dimension and topological properties.
Examines quantum mechanics equivalence with Newtonian geometry.
problem Equivalence principle in quantum mechanics.
method Newton--Cartan geometry, non--relativistic twistor theory.
result Discusses equivalence in quantum mechanics.
We show that, in finite dimensions, the only monotone metrics for which the (+1) and (-1) affine connections are mutually dual are constant multiples of Bogoliubov-Kubo-Mori metric
Develops theory of homogeneous statistical manifolds and classifies Lie groups.
problem Understanding statistical manifolds and Lie groups.
method Constructs examples and classifies Lie groups using information geometry.
result Explicit examples of homogeneous statistical manifolds of low dimension constructed.
Introduces noncommutative geometry for modeling quantum spacetime.
problem Modeling quantum spacetime.
method Operator algebras, K-theory, spectral geometry, quantum groups, and deformation quantization.
result Framework for quantum spacetime.
In this work, differential geometry of the Z3-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.
Introduces Legendre bundle for dually flat manifolds and quantum field theories.
problem Understanding duality in geometric structures and quantum field theories.
method Introduces Legendre bundle and para-Kähler structure.
result Exponential families and Hessian QFTs are realizations of the Legendre bundle.
Quantum statistical models with singularities are studied for state estimation and model selection.
problem Understanding statistical properties of quantum singular models.
method Classical singular learning theory extended to quantum state estimation and model selection using algebraic geometrical methods.
result Asymptotically unbiased estimator (QWAIC) for quantum generalization loss constructed.
In this work, the Z3-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algeb…
Study on topological order on fractal geometries, proving no-go theorem and fault-tolerant gates.
problem Investigating topological order on fractal geometries embedded in n dimensions.
method Using quantum error-correcting codes and systolic geometry to diagnose topological order.
result Proves no-go theorem for topological order on 2D fractals, survival on higher dimensions, and construction of fault-tolerant gates.
Differential geometry of the quantum Lie superalgebra of the extended quantum superplane and its Z2-graded Hopf algebra structure is obtained. Its Z2-graded dual Hopf algebra is also given.
Researchers propose a non-monotone quantum natural gradient for quantum systems.
problem Applying natural gradient methods to quantum systems without monotonicity.
method Introducing a non-monotone quantum natural gradient (QNG) and demonstrating its superiority over conventional QNG.
result Non-monotone QNG outperforms conventional QNG in terms of convergence speed.
Study examines BTZ black hole using information geometry.
problem Understanding the BTZ black hole mechanism.
method Information geometry, Hessian potential, Legendre transformation.
result Exact BTZ metric and entanglement entropy derived.
We propose a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive th…
Quantum complexity lowerbound proved using differential geometry.
problem Proving lower bounds on quantum complexity.
method Applied the Bishop-Gromov bound to Nielsen's complexity geometry.
result Lower bounds on quantum complexity are exponentially large.
We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator x is invertible and furthermore working polynomials in lnx instead of polynomials in x. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual H…
Quantum machine learning classification depends on mutual informations between state and parameter spaces.
problem Generalization in quantum machine learning models.
method Link between quantum machine learning and quantum hypothesis testing, using mutual informations.
result Quantum classifier accuracy and generalization depend on mutual informations between state and parameter spaces.
New method learns quantum states using neural networks, revealing hidden dynamics.
problem High-precision ground state estimation of quantum many-body problems.
method Stochastic reconfiguration method with neural network Ansatz states.
result Learning landscape modes with least entanglement have largest eigenvalues, suggesting correlations are encoded in large flat valleys.
The paper develops a framework for quantum geometry of localized σ-models.
problem Quantum geometry of localized σ-models with small quantum fluctuations. method General framework using Gauss-Manin connection and exact semi-classical approximation.
result Proof of the algebraic index theorem using quantum mechanics.
Holonomy invariants from SL2(C) link complements detect link geometry.
problem Detecting geometric information about links using algebraic quantum invariants.
method Enhanced RT construction with SL2(C) holonomy representations. result Holonomy invariants JN compute Reidemeister torsion for N=2. Quantum learning complexity reviewed using information theory.
problem Learning properties of quantum systems or processing data via quantum computing.
method Information-theoretic techniques focusing on data, copy, and model complexity.
result Copy complexity due to irreversible quantum measurements limits information extraction.
Paper explores Monge-Ampère in deep learning and quantum geometry.
problem Understanding the Monge-Ampère equation in deep learning.
method Review of Boltzmann learning, connection to optimal transport, insights from quantum geometry, renormalization group flow.
result Space of covariance matrices in learning dynamics coincides with the CAH cone.
This paper solves the normalizability crisis in sequential inference by introducing bounded information geometry.
problem Structural failure in standard sequential inference architectures when dealing with extreme outliers.
method Non-parametric field actions and bounded information geometry to truncate infinite tails of spatial distributions.
result Empirical benchmarks across three domains show robust estimation without infinite-tailed distributional assumptions.
4-manifolds have special topological properties which can be used to get a different view on quantum mechanics. One important property (connected with exotic smoothness) is the natural appearance of 3-manifold wild embeddings (Alexanders horned sphere) which can be interpreted as quantum states. This relation can be co…
Study evaluates capacity and trainability of parametrized quantum circuits.
problem Finding the best type of circuits for hybrid quantum-classical algorithms.
method Geometric structure of parameter space, effective quantum dimension, and circuit expressiveness.
result Identifies a transition in quantum geometry leading to decay of quantum natural gradient for deep circuits.
We study possible real structures in the space of solutions to the quantum differential equation. We show that, under mild conditions, a real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry near the large radius limit. We compute an example of P^1 which is pure and polarized over the …
The abstract discusses financial irreversibility using quantum mechanics and projective geometry.
problem Financial irreversibility and its limitations in trading strategies.
method Projective geometry and Taylor expansion of directed distance in quantum systems.
result Fundamental asymmetry under state exchange is a key factor in financial irreversibility.
Physics: Similar long-distance properties can mask vastly different short-distance metrics.
problem Classifying homogeneous metrics on group manifolds by long-distance properties.
method Apply universality concept to geometry, focusing on metrics on Lie groups.
result Many metrics on low-dimensional Lie groups have similar long-distance properties despite differing short-distance properties.
Paper defines compact quantum spaces with Kähler structures.
problem Noncommutative Kähler structures on quantum spaces.
method Introduces compact quantum homogeneous Kähler spaces and studies their properties.
result Analytic properties of Dolbeault-Dirac operators and their indices.
The abstract semiclassicalises quantum group principal bundles to Poisson geometry.
problem Semiclassicalising quantum group principal bundles to Poisson geometry.
method The theory is developed for Poisson manifolds with Poisson-compatible contravariant connections, and for Poisson-Lie groups with bicovariant Poisson-compatible contravariant connections.
result The construction of the Poisson level of the q-Hopf fibration and the spin connection on a principal bundle. This work proposes a new quantum neural network model and training algorithm.
problem Classical neural networks are computationally expensive, especially for tasks like image classification.
method Defines quantum neural networks via quantum time evolution and Hamiltonian, proposing a quantum backpropagation algorithm.
result Validated the proposed quantum backpropagation algorithm on the MNIST dataset using a quantum computer simulation.