Quantum Fourier Transform aids machine learning inference.
problem Generalizing from finite data samples to ground truth.
method Inspired by quantum algorithms, uses Quantum Fourier Transform to expose invariant subspace for data comparison.
result Proposes a concrete implementation for machine learning applications leveraging symmetries.
Quantum algorithm for pricing European call options.
problem Accurate valuation of financial derivatives, especially for complex models and options.
method Transforms classical FFT into quantum QFT for pricing European call options.
result Quantum algorithm outperforms classical Monte Carlo simulation in NISQ era.
The paper connects quantum 6j-symbols to tetrahedra volumes via discrete Fourier transforms.
problem Understanding the asymptotic behavior of quantum 6j-symbols and their relation to 3-manifold invariants. method Proposing and proving a conjecture linking discrete Fourier transforms of quantum 6j-symbols to the volumes of deeply truncated tetrahedra. result Supporting evidence for the conjecture in specific cases, with numerical calculations for larger dihedral angles.
Quantum computers can enhance spectral methods in machine learning.
problem Spectral methods are fundamental but challenging for classical models.
method Utilizing quantum Fourier Transform for spectral manipulations.
result Quantum computing can offer more efficient spectral design.
We construct a certain cross product of two copies of the braided dual H~ of a quasitriangular Hopf algebra H, which we call the elliptic double EH, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attache…
Quantum kernel improves solar irradiance forecasting.
problem Improving short-term solar irradiance forecasting accuracy.
method Quantum Fourier Transform kernel in KRR with feature mixing.
result Consistently improves R2 and nRMSE over classical kernels.
A graphical calculus for microformal morphisms simplifies complex operations in classical and quantum physics.
problem Simplifying operations in classical and quantum microformal morphisms.
method Developed a graphical calculus inspired by Cattaneo-Dherin-Felder's work on formal symplectic groupoids, extended to quantum thick morphisms.
result Infinite series can be written as sums over bipartite trees for both classical and quantum thick morphisms.
New MCMC method speeds up quantum physics simulations by a factor of 100.
problem Simulating quantum many-body systems with high computational complexity.
method FFT-accelerated MCMC with coupled particle and auxiliary variables.
result Achieves O(NlogN) scaling, significantly faster than traditional O(N3) methods. We give an explicit formula, as a formal differential operator, for quantum microformal morphisms of (super)manifolds that we introduced earlier. Such quantum microformal morphisms are essentially oscillatory integral operators or Fourier integral operators of a particular kind. They act on oscillatory wave functions, …
Modified Hennings invariant defined using quantum groups and integrals.
problem Defining a modified Hennings invariant using quantum groups.
method Topological ribbon Hopf algebra, discrete Fourier transforms, symmetrized graded integral, modified trace.
result Modified graded Hennings invariant defined and extended to empty manifolds.
Quantum methods improve option pricing accuracy.
problem Pricing financial derivatives using Monte Carlo integration.
method Hybrid classical-quantum methods using Fourier series and QML.
result Quantum methods achieve remarkable accuracy in option pricing.
Quantum ELMs use a quantum reservoir to learn from data, with limits on expressivity and scalability.
problem Understanding the limits of quantum ELMs for machine learning tasks.
method Decomposed QELM predictions into Fourier series to analyze expressivity and scalability.
result Expressivity of QELMs is limited by the number of Fourier frequencies and observables, and scalability is hindered by hardware noise and entanglement.
Quantum-assisted Gaussian process speeds up data regression.
problem High computational complexity of Gaussian process regression for large datasets.
method Quantum-assisted sparse Gaussian process regression using random Fourier features.
result Achieves polynomial-order computational speedup compared to classical methods.
Quantizes the relationship between Koszul and Schouten brackets in Poisson geometry.
problem Quantizing the relationship between Koszul and Schouten brackets in Poisson geometry.
method Employing Voronov's thick morphism technique and quantum Mackenzie-Xu transformations in the framework of L∞-algebroids. result Quantizes the L∞-morphism into a single linear operator, a formal Fourier integral operator. The paper proposes and proves asymptotic expansions for quantum invariants.
problem Quantum invariants and their expansions under varying metrics.
method Asymptotic expansion conjectures for relative Reshetikhin-Turaev, Turaev-Viro invariants and quantum 6j-symbols.
result Proved asymptotic expansions for special cases, showing geometric dependence on metrics.
Paper proves Fourier transform for valuations, simplifying previous work.
problem Existence of isomorphism for translation-invariant smooth valuations.
method Directly describes Alesker's isomorphism in terms of Fourier transform on functions.
result Simple proofs of Alesker's Fourier transform properties, including a previously conjectured result.
Complex analysis techniques link Gaussian RBF kernels to quantum mechanics.
problem Understanding the Gaussian RBF kernel in machine learning and SVMs.
method Using Fock space and Segal-Bargmann theories in complex analysis.
result Proves connections between Gaussian RBF kernels and quantum mechanics operators.
We present a finite-dimensional version of the quantum model for the stock market proposed in [C. Zhang and L. Huang, A quantum model for the stock market, Physica A 389(2010) 5769]. Our approach is an attempt to make this model consistent with the discrete nature of the stock price and is based on the mathematical for…
Quantum models can approximate any function if data encoding allows for a rich enough frequency spectrum.
problem Theoretical properties of quantum machine learning models, particularly their expressive power.
method Investigated how data encoding affects the expressive power of parametrized quantum circuits.
result Quantum models can access increasingly rich frequency spectra by repeating data encoding gates, potentially making them universal function approximators.
We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid grou…
A new algorithm computes Fourier coefficients for a specified range efficiently.
problem Inefficiency in FFT due to fixed output size for all applications.
method Fast Partial Fourier Transform (PFT) that allows specifying the range of Fourier coefficients to compute.
result PFT achieves significant speedup over state-of-the-art FFT algorithms for small output sizes.
QAOA matches classical tensor power iteration in spiked tensor model recovery.
problem Statistical estimation in spiked tensor model with computational gap.
method Analysis of QAOA performance on spiked tensor model.
result QAOA weak recovery threshold matches tensor power iteration.
The price of a given stock is exactly known only at the time of sale when the stock is between the traders. If we know the price (owner) then we have no information on the owner (price). A more general description including cases when we have partial information on both price and ownership is obtained by using the quan…
Structured CNN designed using the prior information of problems potentially improves efficiency over conventional CNNs in various tasks in solving PDEs and inverse problems in signal processing. This paper introduces BNet2, a simplified Butterfly-Net and inline with the conventional CNN. Moreover, a Fourier transform i…
Quantum kernels can be efficiently embedded into classical feature spaces.
problem Can all quantum kernels be efficiently embedded into classical feature spaces?
method Invoking computational universality and using techniques like random Fourier features, the authors show that certain classes of quantum kernels can be efficiently embedded.
result For shift-invariant and composition kernels, embedding quantum kernels are universal and efficient.
Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
The Fourier transform of Heegaard Floer d-invariants helps classify 3-manifolds.
problem Classifying 3-manifolds up to integer homology cobordism.
method Exploring the Fourier transform of Heegaard Floer d-invariants.
result Lens spaces are cancellable in the monoid of 3-manifolds up to integer homology cobordism.
X-ray transform on H-type groups solved, revealing function injectivity.
problem Injectivity in sub-Riemannian geometry.
method Fourier Slice Theorem adapted to H-type groups.
result Integrable functions on H-type groups are uniquely determined by their integrals over geodesics.
Proves conditions for Fourier transforms in rank 1 symmetric spaces.
problem Understanding Fourier transform bounds in symmetric spaces.
method Proves sufficient and necessary conditions using Lipschitz and Fourier type integral conditions.
result Establishes bounds for Fourier transforms in rank 1 symmetric spaces with specific moduli of continuity.
This work improves Fourier pricing for multi-asset options using RQMC with domain transformation.
problem Efficiently pricing multi-asset options in high dimensions with Fourier methods.
method Randomized quasi-Monte Carlo (RQMC) with domain transformation to handle singularities.
result RQMC with domain transformation provides accurate and scalable Fourier pricing for multi-asset options.
Quantum method improves neural density estimation in high dimensions.
problem High-dimensional density estimation with poor performance and high computational complexity.
method Adaptive Fourier features based on quantum density matrices, integrated with neural networks.
result Competitive performance compared to state-of-the-art methods in various datasets.
This paper outlines an approach to the non-abelian theta functions of the SU(2)-Chern-Simons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier tran…
Algorithm describes Fourier transform of Stokes data at infinity.
problem Understanding the Fourier transform of Stokes data at infinity.
method Topological description and algorithmic approach using recent results and language of Stokes local systems.
result Explicit isomorphisms between wild character varieties are induced.
The paper studies Fourier-Laplace transforms in polynomial OU volatility models for option pricing.
problem Calibrating and pricing options in polynomial Ornstein-Uhlenbeck volatility models.
method Analyzes Fourier-Laplace transforms, connects to Riccati equations, and develops numerical schemes.
result Establishes existence and solution for Riccati equations and provides efficient numerical methods.
New algorithms learn sparse set functions in non-orthogonal Fourier bases.
problem Learning sparse set functions in non-orthogonal Fourier bases.
method Novel algorithms using non-orthogonal Fourier transforms.
result At most nk−klog2k+k queries for k non-zero Fourier coefficients. In this paper, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by usin…
We prove that the Fourier--Laplace--Nahm transform for connections on the projective line is a hyper-Kähler isometry.
Transformers improve with Fourier integral attentions.
problem Inefficiency of dot-product attention in capturing feature dependencies.
method Interpreted attention as kernel regression, proposed FourierFormer with generalized Fourier integral kernels.
result FourierFormer achieves better accuracy and reduces redundancy.
The paper derives statistics of multi-factor functions from their Fourier transforms.
problem Deriving statistics of multi-factor functions from Fourier transforms.
method Developed an m-Coefficient/Index Annihilation Theorem to analyze the moments of a function from its Fourier transform.
result The mth moment of a function becomes a series of terms, each with precisely m Fourier coefficients, and the indices sum to zero.
Study spherical Fourier transform on hypergeometric type harmonic manifolds.
problem Spherical Fourier transform on harmonic Hadamard manifolds.
method Representation of spherical functions by Gauss hypergeometric functions.
result Inversion formula, convolution rule, and Plancherel theorem are derived.
In this paper we prove a new inversion theorem and a refinement of an old support theorem for two Radon transforms on a symmetric space. Included are some new identities for the Abel transform and some results about the Fourier transform from a joint work with Rawat, Sengupta and Sitaram.
Classical clients can verify quantum learning tasks efficiently.
problem Making quantum learning accessible to classical clients.
method Developed a framework for classical verification of quantum learning.
result Quantum learning tasks can be efficiently verified by classical verifiers.
Given two compact hyperkähler surfaces X and Y and a holomorphic vector bundle Q on X×Y, which is a generalized instanton, one can define a Fourier-Mukai transform, which, under suitable assumptions, maps vector bundles on X to vector bundles on Y. If X and Y are dual complex tori, this transform …
Many signals on Cartesian product graphs appear in the real world, such as digital images, sensor observation time series, and movie ratings on Netflix. These signals are "multi-dimensional" and have directional characteristics along each factor graph. However, the existing graph Fourier transform does not distinguish …
RP-GFRFT unifies fractional order and rotation control for graph signals.
problem Lack of rotation-based spectral control in GFRFT and zero-angle degeneracy in AGFT.
method Rotation-parameterized graph fractional Fourier transform (RP-GFRFT) with degeneracy preserving rotation matrix.
result RP-GFRFT improves spectral filtering performance over existing methods.
The paper derives and proves the Helgason Fourier transform for vector bundle-valued differential forms on homogeneous spaces.
problem Deriving the Helgason Fourier transform for vector bundle-valued differential forms on homogeneous spaces.
method Employing the perspective of the functional equation satisfied by the classical Fourier transform, the paper derives the Helgason Fourier transform map and proves its properties.
result The Fourier transform is explicitly given and proven to be a map from vector bundle-valued differential forms to another vector bundle-valued differential form on the product space.
Study identifies and analyzes three types of errors in learning Fourier operators.
problem Statistical, discretization, and truncation errors in learning Fourier operators.
method Analysis of a Discrete Fourier Transform (DFT) based least squares estimator.
result Established upper and lower bounds on statistical, discretization, and truncation errors.
The study establishes uncertainty principles on harmonic manifolds of rank one.
problem Developing uncertainty principles for harmonic manifolds of rank one.
method Derivation of various uncertainty principles including Heisenberg, Morgen, Schrödinger, and Hömanders principles.
result Generalization of Hausdorff-Young inequality to harmonic manifolds of rank one.