The paper calculates fractional quantum numbers on complex orbifolds with strong magnetic fields.
problem Understanding fractional quantum numbers in complex orbifolds with strong magnetic fields.
method The study uses Landau Hamiltonians on complex, compact 2D orbifolds and a nontrivial generalisation of the Nahm transform.
result Fractional quantum numbers are calculated as conductance and charge transport is refined.
The twisted Connes-Moscovici higher index theorem is generalized to the case of good orbifolds. The higher index is shown to be a rational number, and in fact non-integer in specific examples of 2-orbifolds. This results in a non-commutative geometry model that predicts the occurrence of fractional quantum numbers in t…
Quantum SVM uses fewer features for faster training.
problem Training high-dimensional SVMs efficiently.
method Quantum linear programming for sparse SVM training.
result Quantum sparse SVM can be trained in sublinear time.
This paper is the continuation of Part I, expanding previous results of math.DG/9803051. This paper uses techniques in noncommutative geometry as developed by Alain Connes in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant u…
We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through…
Quantum probability theory constructs Martingales for non-Brownian financial models.
problem Constructing Martingales for financial models using fractional Brownian motion.
method Quantum probability theory and Wick product.
result Quantum probability framework allows for Martingale construction without Brownian integrals.
We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman's functionals and derived the corresponding fractional evolution …
Quantum algorithms simulate and exponentiate correlated Gaussian vectors for financial modeling.
problem Efficiently simulate and exponentiate correlated Gaussian vectors for financial applications.
method Proposes quantum algorithms for preparing and exponentiating normalised correlated Gaussian random vectors.
result Achieves subcubic complexity in N for quantum state preparation, providing a quantum advantage over classical methods. Study proves certain algebraic structures are symmetric Frobenius algebras.
problem Understanding algebraic structures in bordered surfaces.
method Analyzing stated skein algebras and their fraction rings.
result Fraction ring of stated skein algebra is a symmetric Frobenius algebra.
Quantum computing optimizes ESG portfolios efficiently.
problem Optimizing investment portfolios with risk, return, and ESG considerations.
method Formulated discrete Markowitz portfolio theory (DMPT) for quantum annealers, incorporating ESG ratings.
result Discrete portfolios converge to continuous solutions as budgets increase, outperforming traditional methods.
The study finds new infinite dilogarithm identities related to number sequences and continued fractions.
problem Finding new infinite dilogarithm identities.
method Demonstrating families of identities associated with specific number sequences and continued fractions.
result New infinite dilogarithm identities related to Fibonacci, Lucas numbers, convergents of even period continued fractions, and recurrence relations.
Study finds no significant difference in neural network weights with quantum random numbers.
problem Effects of biased quantum random numbers on neural network initialization.
method Empirical study using quantum hardware and classical pseudo-random numbers.
result No statistically significant difference found between quantum random numbers and other types.
Quantum-inspired tensor network speeds up financial risk assessment.
problem Efficiently pricing multi-asset derivatives in finance.
method Tensor network algorithms for multi-asset options pricing.
result Tensor network approach yields several orders of magnitude speedup.
We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as gr…
Efficient algorithms for WRT invariants of torus bundles using algebraic structures.
problem Computing topological invariants of 3-manifolds is generally intractable.
method Embedding skein algebra into symmetric subalgebra at roots of unity for polynomial-time classical computation and using quantum algorithms for exponential space advantage.
result Polynomial-time classical computation and quantum algorithms for WRT invariants of torus bundles.
We classify all unitary modular tensor categories (UMTCs) of rank ≤4. There are a total of 70 UMTCs of rank ≤4 (Note that some authors would have counted as 35 MTCs.) In our convention there are two trivial unitary MTCs distinguished by the modular S matrix S=(±1). Each such UMTC can be obtained from …
Quantum RNG improves financial risk metrics estimation.
problem Estimating financial risk metrics with high precision.
method Quantum-Enhanced Monte Carlo using QRNG.
result Improved accuracy in VaR and CVaR estimation.
RILA learns HQMMs robustly against adversarial corruption.
problem Robustness of HQMM learning algorithms under adversarial perturbations.
method Adversarially Corrupted HQMM (AC-HQMM) and Robust Iterative Learning Algorithm (RILA).
result RILA outperforms existing algorithms in convergence stability, corruption resilience, and physical validity.
Quantum machine learning offers advantages for broader learning tasks.
problem Demonstrate QML advantage over classical methods for general learning tasks.
method Construct a new family of supervised learning tasks and prove their hardness.
result Prove provable advantage of QML based on general quantum computational advantages.
A new penalty-free method optimizes portfolios without quantum annealing penalties.
problem Optimizing portfolios with quantum annealing penalties.
method Removing the penalty term and using a classical feasibility projector.
result Significant reduction in chain-break fractions and post-processed regret.
We construct representations of the braid groups B_n on n strands on free Z[q,q^-1,s,s^-1]-modules W_{n,l} using generic Verma modules for an integral version of quantum sl_2. We prove that the W_{n,2} are isomorphic to the faithful Lawrence Krammer Bigelow representations of B_n after appropriate identification of par…
Zagier's conjecture on knot invariants is proven for irrationals, with applications to quantum modular forms.
problem Proving the continuity of a function related to the figure-eight knot's colored Jones polynomial at irrationals.
method Analyzing the asymptotic behavior of the colored Jones polynomial and using properties of continued fractions.
result The continuity conjecture for the function h(x) holds almost everywhere on the real line, and a smooth approximation is established. Quantum states model sequences, revealing complementary system information.
problem Modeling sequences using classical probability distributions.
method Quantum state with entanglement, DMRG algorithm for organizing reduced densities.
result Estimate of generalization error for tensor network model.
Quantum algorithms improve high-frequency trading efficiency.
problem Reducing calculation time in high-frequency statistical arbitrage trading.
method Variable time condition number estimation and quantum linear regression.
result Quantum advantage in trading algorithm complexity reduction.
We present a construction of complete self-dual Einstein metrics of negative scalar curvature on an uncountable family of manifolds of infinite topological type, which are enumerated by continued fraction expansions of irrational numbers. These manifolds may be regarded as limits of the resolutions of cyclic quotient s…
Existence of unstable shrinking solutions in fractional mean curvature flow.
problem Existence and stability of self-shrinkers in fractional mean curvature flow.
method Existence proof of homothetically shrinking solutions with prescribed boundary conditions.
result Unstable shrinking solutions, except the ball, for fractional mean curvature flow.
We introduce multiscale invariant dictionaries to estimate quantum chemical energies of organic molecules, from training databases. Molecular energies are invariant to isometric atomic displacements, and are Lipschitz continuous to molecular deformations. Similarly to density functional theory (DFT), the molecule is re…
Quantum algorithm reduces qubit usage for Monte Carlo simulations.
problem High qubit requirements for Monte Carlo simulations on quantum computers.
method Use of pseudo-random number generator (PRNG) on a quantum circuit.
result Significant reduction in qubit usage without sacrificing quantum speed.
Neural-Network Quantum States have been recently introduced as an Ansatz for describing the wave function of quantum many-body systems. We show that there are strong connections between Neural-Network Quantum States in the form of Restricted Boltzmann Machines and some classes of Tensor-Network states in arbitrary dime…
Geometric methods prove exponential growth in continued fractions.
problem Exponential growth in partial quotients of continued fractions.
method Geometric representation of continued fractions and orbifold triangulations.
result Eventually periodic continued fractions have exponentially growing partial quotients.
Quantum algorithm finds extremal values without direct function access.
problem Finding extremal values of hidden functions without direct access.
method Parametric quantum circuit trained with a trainable quantum feature map.
result Algorithm successfully finds extremal values even with sparse training data.
Unified geometric framework for quantum states using dual number algebras.
problem Representing quantum states in a geometrically unified way.
method Smooth embeddings into higher-order dual number algebras and algebraic flows.
result Established nilpotent dual algebras as a geometric landscape for quantum kinematics.
The paper converts nonalternating forms of rational links into all-even forms and derives formulas for their braid index and HOMFLY polynomial.
problem Finding formulas for the braid index and HOMFLY polynomial of rational links.
method Algorithm to transform nonalternating forms into all-even forms and derivation of formulas.
result Formulas for the braid index and HOMFLY polynomial of rational links in terms of their reduced alternating form.
Quantum speedup for Monte Carlo integration reduces integrand calls.
problem Reducing the number of calls to the integrand subroutine in high-dimensional Monte Carlo integration.
method Combining nested quantum amplitude estimation with pseudorandom numbers for separable integrands.
result Significant reduction in the number of integrand calls for high-dimensional integration.
This paper compares classical shadows and direct quantum measurement for efficient information extraction.
problem Efficiently extracting classical information from quantum states with limited classical post-processing.
method Quantitative resource analysis comparing classical shadows and direct quantum measurement.
result An efficiency frontier between classical shadows and direct quantum measurement is identified.
Quantum machine learning faces 'laziness' and 'barren plateaus', but noise can mitigate the latter.
problem Quantum machine learning's loss function landscape issues.
method Theoretical analysis of quantum variational circuits, neural tangent kernels, and noise effects.
result Noise can mitigate barren plateaus in quantum machine learning.
New method uses fractional posteriors for semiparametric inference with improved uncertainty quantification.
problem Semiparametric inference with nonparametric priors and fractional posteriors.
method Established a general Bernstein--von Mises theorem for fractional posterior distributions, proposed shifted-and-rescaled credible sets.
result Fractional posterior credible sets provide reliable uncertainty quantification but have inflated size; shifted-and-rescaled set is an efficient confidence set.
Quantum computing speeds up risk analysis by efficiently sampling copulas.
problem Efficiently modeling tail dependence and risk measures in financial risk analysis.
method Quantum computing implementation of copula models for risk aggregation.
result The MB11 copula family is suitable for capturing tail dependence structures in risk factors.
The paper sets a lower bound on the crossing number of 2-bridge knots and answers a question about their epimorphism number.
problem Determining the epimorphism number of 2-bridge knots and answering a specific question posed by Suzuki.
method Using techniques related to continued fraction expansions and parsings of 2-bridge knots.
result Established a lower bound on the crossing number of 2-bridge knots and answered Suzuki's question about the epimorphism number.
We study the fractional gravity for spacetimes with non-integer dimensions. Our constructions are based on a geometric formalism with the fractional Caputo derivative and integral calculus adapted to nonolonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical …
Quantum reservoirs risk bounds are analyzed using Rademacher complexity.
problem Bounding generalization errors of quantum reservoirs.
method Using Rademacher complexity, specific bounds are derived for quantum reservoir classes.
result Risk bounds converge with increasing training samples and qubits.
We demonstrate how quantum computation can provide non-trivial improvements in the computational and statistical complexity of the perceptron model. We develop two quantum algorithms for perceptron learning. The first algorithm exploits quantum information processing to determine a separating hyperplane using a number …
Quantum machine learning tackles large datasets with randomized measurements.
problem Efficiently process large, high-dimensional datasets on quantum computers.
method Randomized measurements to scale linearly with dataset size and quadratic for post-processing.
result Substantial speed-up for noisy quantum computers, enabling image classification.
Origami structures are enumerated and shown to be quantum modular.
problem Counting and understanding origami structures with real structures.
method Using combinatorics of zonal polynomials and Schur polynomials, and relating to quantum modular forms and double Hurwitz numbers.
result The generating functions of certain origami structures are quantum modular forms.
The study examines how quantum resources enhance the complexity of quantum circuits.
problem Quantum resource enhancement on circuit complexity.
method Utilizing quantum resource theories, the study analyzes statistical complexities of quantum circuits with limited quantum resources.
result Bounds for statistical complexities of quantum circuits are derived and applied to specific cases.
Unified framework combines trace-induced quantum kernels for improved machine learning models.
problem Improving performance of quantum machine learning models using trace-induced kernels.
method Developed a unified framework combining various trace-induced quantum kernels, including global fidelity and local projected kernels, as Lego kernels.
result Local projected kernels can achieve comparable performance to global fidelity kernels with fewer quantum resources.
This paper explores how rational numbers on the Stern-Brocot diagram map to lines when terms are extended.
problem Understanding the geometry of rational numbers on the Stern-Brocot diagram.
method Analyzing continued fraction expansions and their geometric implications on the diagram.
result Vertices of the Stern-Brocot diagram corresponding to extended rational numbers lie on two Euclidean lines.
Quantum Signal Processing reduces derivative pricing quantum resource requirements.
problem Efficiently pricing financial derivatives on quantum computers.
method Quantum Signal Processing (QSP) to encode payoffs directly into quantum amplitudes.
result Significantly reduces quantum resources (T-gates and qubits) for practical derivative contracts.