Quantum polynomials are derived from a specific tribracket structure.
problem Quantum enhancement polynomials for oriented links.
method Defined using a canonical two-element tribracket, proving polynomials can be derived from five specific ones.
result Universal quantum enhancement polynomials are strictly stronger than the Jones polynomial.
Enhances quantum computing for symmetrical systems, proving a new class of problems.
problem Proving the efficiency of a new quantum computing model for symmetrical systems.
method Introducing equivariant convolutional quantum algorithms tailored for SU(d) symmetries.
result Demonstrates a problem that can be solved efficiently on a new quantum model, suggesting it's not classically simulatable.
Study uses big data to analyze quantum invariants.
problem Investigate structural properties of Jones polynomial.
method Exploratory and topological data analysis, including coloring, rank increase, categorification.
result Contrasts behavior of Jones polynomial under various enhancements.
We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY…
In this paper we give a quantum statistical interpretation for the bracket polynomial state sum <K> and for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to al…
We show that the link invariants derived from 3-dimensional quantum hyperbolic geometry can be defined by means of planar state sums based on link diagrams and a new family of enhanced Yang-Baxteroperators (YBO) that we compute explicitly. By a local comparison of the respective YBO's we show that these invariants coin…
Quantization of the Teichmüller space of a punctured Riemann surface S is an approach to 3-dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop γ in S gives rise to a natural trace-of-monodromy function I(γ) on the Teichmüller space. For any…
New algebraic setup defines quantum link invariants.
problem Defining and controlling quantum link invariants.
method Quantum Schur--Weyl duality and variants.
result Global definitions of quantum polynomials.
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.
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.
The paper constructs quantum invariants for knotoid diagrams.
problem Quantum invariants for knotoid diagrams in R2. method Decompose Morse knotoid diagrams into basic elementary diagrams, each associated with a matrix solving the quantum Yang-Baxter equation. Define quantum state sum models to recover various polynomials.
result Recover and define new polynomials for Morse knotoids.
Quantum machine learning models can approximate any continuous function.
problem Theoretical understanding of quantum feature maps in machine learning.
method Proving universal approximation property of quantum machine learning models in quantum-enhanced feature spaces.
result Quantum machine learning models are universal approximators of continuous functions.
Quantum machine learning boosts financial forecasting accuracy.
problem Churn prediction and credit risk assessment in finance.
method Used quantum and classical Determinantal Point Processes for churn prediction, and quantum neural networks for credit risk assessment.
result Significant improvement in precision for churn prediction (6% increase). Quantum models match classical performance with fewer parameters.
Quantum algorithms can enhance machine learning in different aspects. Here, we study quantum-enhanced least-square support vector machine (LS-SVM). Firstly, a novel quantum algorithm that uses continuous variable to assist matrix inversion is introduced to simplify the algorithm for quantum LS-SVM, while retaining expo…
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.
Quantum Kerr learning shows enhancements in convergence and generalization for kernel-based methods.
problem Improving convergence and generalization in kernel-based methods for quantum computing.
method Combining quantum mechanics with neural tangent kernel theory and first-order perturbation theory.
result Quantum enhancements in terms of convergence time and generalization error.
Enhances quantum sensing by eliminating multiple oscillations in field amplitude estimation.
problem Multiple oscillations in field amplitude estimation due to inter-qubit interactions at high qubit densities.
method Adopting a quantum circuit learning framework to approximate a target function by optimizing gate parameters.
result Elimination of multiple oscillations, leading to enhanced dynamic range of quantum sensing.
In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed PGL2-local systems on a punctured surface S. The moduli space is birational to a cluster X-variety, whose positive real points recover the enhanced Teichmüller space of S. Their b…
New methods reveal colored Jones polynomials from quantum R-matrices and knot invariants.
problem Understanding colored Jones polynomials of knots.
method Two realizations: quantum R-matrices and refined quantum modularity conjecture.
result New insights into knot invariants from quantum R-matrices and matrix conjectures.
Kashaev limits of quantum A-polynomials reveal classical action vanishing and hyperbolic volume deformation.
problem Exploring the Kashaev limits of quantum A-polynomials. method Analyzing the double scaling quasiclassical limit.
result Identifying two phases in the Kashaev limit.
Simpler equations derived for knot polynomials coefficients, forming a ring.
problem Complexity of knot polynomials colored with symmetric representations.
method Deriving two difference equations for quantum C-polynomials coefficients.
result Quantum C-polynomials form a ring and are much simpler than colored polynomials.
Study shows quantum modularity in figure-eight knot's colored Jones polynomial.
problem Asymptotic behavior of colored Jones polynomial of figure-eight knot.
method Analyzing polynomial evaluated at specific points and showing asymptotic equivalence.
result Quantum modularity demonstrated in the figure-eight knot's colored Jones polynomial.
This work shows how to efficiently simulate parts of quantum landscapes using classical computers.
problem Identifying where quantum computers are advantageous and offloading computations.
method Developed a quantum-enhanced classical algorithm to simulate sub-regions of quantum landscapes.
result It is possible to generate a classical surrogate of a sub-region of a quantum landscape.
Improved fraud detection in finance with quantum-enhanced federated learning.
problem Challenges in detecting financial fraud with traditional methods.
method Hybrid quantum-enhanced federated learning framework combining quantum LSTM with privacy-preserving techniques.
result Approximately 5% improvement in performance metrics compared to conventional models.
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.
Unified quantum invariants via intersections of embedded Lagrangians.
problem Unified quantum invariants for Uq(sl(2)). method State sum of Lagrangian intersections in configuration spaces.
result Recovery of coloured Jones and Alexander polynomials.
We introduce an infinite family of quantum enhancements of the biquandle counting invariant we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring R using virtual crossings as smoothings, these invariants take the…
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for appro…
Quantum model for knotted graphs from knot theory.
problem Constructing an isotopy invariant polynomial for knotted bipartite ribbon graphs.
method Applying quantum topology to construct an isotopy invariant polynomial.
result Computed the expected number of loops in the double dimer model.
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as Aq polynomial. Quantum invariants are explained as intersections in configuration spaces.
problem Quantum invariants of knots and links.
method Topological intersections in configuration spaces.
result Coloured Jones and Alexander polynomials are special cases of intersection pairings.
Quantum reservoir computing improves volatility forecasting.
problem Forecasting realized volatility in finance.
method Quantum reservoir computing with Ising Hamiltonian and feature selection.
result Quantum reservoir computing outperforms benchmarks in volatility forecasting.
Enhances quantum circuit synthesis using deep learning and geometric methods.
problem Optimizing quantum circuits for time efficiency.
method Combining deep learning with geometric control techniques.
result Improved time-optimal control in quantum circuit synthesis.
A new invariant for links generalizes Alexander polynomial for sl_3.
problem Defining a non-abelian generalization of the Alexander polynomial.
method Using quantum sl3 representations and Laurent polynomials. result Established a direct relation between Δsl3 and the Alexander polynomial. In this short survey article we collect the current state of the art in the nascent field of \textit{quantum enhancements}, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes c…
Simplified geometric derivation of quantum A-polynomials for knots.
problem Deriving quantum A-polynomials for knots in a simple geometric way.
method Geometric derivation using Ward identities in Chern-Simons theory, contact geometry, and Kauffman calculus.
result Simplified presentation of quantum A-polynomials, making them accessible to a broader audience.
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.
Paper proves polynomial equivalence of quantum complexity metrics.
problem Quantum complexity metrics equivalence.
method Study of right-invariant metrics on unitary group.
result All metrics in the equivalence class have polynomial slowdown in approximation.
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…
The colored HOMFLY polynomial is the quantum invariant of oriented links in S3 associated with irreducible representations of the quantum group Uq(slN). In this paper, using an approach to calculate quantum invariants of links via cabling-projection rule, we derive a formula for the colored HOMFLY polyn…
Unified model for knot polynomials using quantum Heegaard diagrams.
problem Categorify knot polynomials using Floer homology.
method Construct quantum Heegaard diagrams, identify gradings, and define a two-variable graded intersection.
result Unified intersection model recovers Alexander and Jones polynomials.
New spin on Khovanov-Rozansky homology categorifies spin link polynomial.
problem Categorify spin link polynomial using colored Khovanov-Rozansky homology.
method Equip Λ^n-colored sl_{2n} Khovanov-Rozansky homology with an involution.
result Categorifies spin-colored so_{2n+1} quantum link polynomial for n=1,2,3.
Quantum-enhanced metrology aims to estimate an unknown parameter such that the precision scales better than the shot-noise bound. Single-shot adaptive quantum-enhanced metrology (AQEM) is a promising approach that uses feedback to tweak the quantum process according to previous measurement outcomes. Techniques and form…
In this paper, we study the quantum sl(n) representation category using the web space. Specially, we extend sl(n) web space for n≥4 as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial Pn(q) specialized to a one variable polynomial …
Quantum correlations enhance generative models, providing a new resource for machine learning.
problem Capturing complex probability distributions in unsupervised learning.
method Theoretical and numerical analysis of quantum correlations in generative models.
result Quantum nonlocality and contextuality provide an expressivity advantage over classical models.
Paper categorifies a polynomial related to ribbon graphs.
problem Enumerating partial duals of ribbon graphs.
method Using an extended Frobenius algebra in unoriented topological quantum field theory.
result A categorification of the partial-dual genus polynomial.
Quantum modularity proven for specific theta series.
problem Proving quantum modularity for partial theta series with periodic coefficients.
method Explicit proof using Kontsevich-Zagier series and colored Jones polynomials.
result Kontsevich-Zagier series is a weight 3/2 quantum modular form.
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.