In this paper, we have proposed a deep quantum SVM formulation, and further demonstrated a quantum-clustering framework based on the quantum deep SVM formulation, deep convolutional neural networks, and quantum K-Means clustering. We have investigated the run time computational complexity of the proposed quantum deep c…
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48 results for “quantum cluster”
Quantum cluster algebras for surfaces with coefficients defined using skein theory.
problem Defining quantum cluster algebras for surfaces with coefficients.
method Introducing a skein algebra and proving it has a quantum cluster structure.
result The skein algebra of a walled surface naturally generalizes quantum cluster algebras of marked surfaces.
The support vector clustering algorithm is a well-known clustering algorithm based on support vector machines using Gaussian or polynomial kernels. The classical support vector clustering algorithm works well in general, but its performance degrades when applied on big data. In this paper, we have investigated the perf…
New knot invariants derived using quantum cluster algebras.
problem Deriving new knot invariants from quantum cluster algebras.
method Interpreting -matrix of as cluster transformation, introducing auxiliary parameter .
result Derives perturbed-Alexander invariants with higher-order terms in .
Counting the number of clusters, when these clusters overlap significantly is a challenging problem in machine learning. We argue that a purely mathematical quantum theory, formulated using the path integral technique, when applied to non-physics modeling leads to non-physics quantum theories that are statistical in na…
Quantum algorithms reduce clustering input size, achieving near-linear approximation.
problem Efficiently clustering large datasets in quantum computing.
method Quantum coresets for -clustering with sublinear query complexity.
result Achieves near-linear approximation for -clustering with coresets.
New solutions to 3D integrability equations using quantum cluster algebras.
problem Constructing solutions to the tetrahedron and 3D reflection equations.
method Extending quantum cluster algebra approach to Fock-Goncharov quivers and investigating cluster transformations.
result Explicit formulas for matrix elements of solutions derived for typical representations.
Develops quantum cluster algebra approach to solve tetrahedron equation.
problem Investigates a three-dimensional generalization of the Yang-Baxter equation.
method Quantum cluster algebra approach with realization of quantum Y-variables in terms of q-Weyl algebras.
result Obtains a solution with three spectral parameters and reproduces Sergeev's R matrix.
Unified 3D R-matrices from quantum cluster algebra.
problem Constructing new solutions to the tetrahedron equation.
method Symmetric butterfly quiver, quantum cluster algebra, quantum dilogarithms, q-Weyl algebra.
result Unified 3D R-matrices from various sources.
Clustering algorithms are a cornerstone of machine learning applications. Recently, a quantum algorithm for clustering based on the k-means algorithm has been proposed by Kerenidis, Landman, Luongo and Prakash. Based on their work, we propose a quantum expectation-maximization (EM) algorithm for Gaussian mixture models…
Quantum GBS boosts asset clustering for robust statistical arbitrage portfolios.
problem Identifying co-moving assets from correlation matrices for statistical arbitrage.
method Mapping S&P 500 correlation data to GBS-compatible adjacency matrices, benchmarking classical and quantum clustering algorithms.
result Quantum GBS generates superior alpha during high volatility periods, persisting under low-loss conditions.
Quantum trace maps for surfaces are shown to be compatible under triangulations.
problem Constructing and understanding quantum trace maps for surfaces.
method Developed quantum mutation maps between subalgebras of quantum torus algebras for different triangulations.
result Quantum trace maps are natural and independent of triangulation choices.
Quantum computing for machine learning attracts increasing attention and recent technological developments suggest that especially adiabatic quantum computing may soon be of practical interest. In this paper, we therefore consider this paradigm and discuss how to adopt it to the problem of binary clustering. Numerical …
We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the R-operator for which hyperbolic octrahedron is assigned. Also shown is that, by taking q to be a root of unity, our braiding operator reduces to the Kashaev R-mat…
Bracelets and theta bases match in various cluster algebras.
problem Matching bracelet and theta bases in cluster algebras.
method Comparing skein and cluster algebras, defining quantum bracelets, and analyzing cluster scattering diagrams.
result Quantum bracelets coincide with theta functions in various cluster algebras.
Quantum time evolution exhibits rich physics, attributable to the interplay between the density and phase of a wave function. However, unlike classical heat diffusion, the wave nature of quantum mechanics has not yet been extensively explored in modern data analysis. We propose that the Laplace transform of quantum tra…
Clustering, or grouping, dataset elements based on similarity can be used not only to classify a dataset into a few categories, but also to approximate it by a relatively large number of representative elements. In the latter scenario, referred to as extreme clustering, datasets are enormous and the number of represent…
This work constructs a finite-dimensional projective representation for a quantum Teichmüller model.
problem Quantum Teichmüller theory and its finite-dimensional representation.
method Explicit construction using cyclic quantum dilogarithm and mutations of coefficients.
result Reconstruction of quantum Teichmüller space with explicit intertwiners.
Quantum cluster algebra constructed from web skein relations on surfaces.
problem Quantization of cluster structures on moduli spaces of SL3 local systems.
method Constructing a quantum cluster algebra inside the skew-field of a skein algebra of unpunctured surfaces.
result Laurent expressions of webs in clusters have positive coefficients.
New Ising models improve consensus clustering on specialized hardware.
problem Consensus clustering optimization problems.
method Formulated consensus clustering as Ising models and evaluated on specialized hardware.
result Our Ising models outperform existing techniques on consensus clustering.
Quantum machine learning improves satellite image alignment.
problem Align satellite images taken at different times and angles.
method Quantum machine learning techniques for feature extraction and matching.
result Quantum methods show potential for future improvements.
New basis confirms Thurston's conjecture and reveals knot configurations.
problem Understanding cluster algebras and their bases from surfaces.
method Topological construction of band basis and comparison with Kazhdan-Lusztig type basis.
result Common triangular basis matches band basis in quantum cluster algebras.
Study of quantum decorated character stacks and their quantizations.
problem Quantization of decorated character stacks and their compatibility with cutting and gluing.
method Using stratified factorization homology, extend Fock and Goncharov's construction to include stacky points.
result Construction of categorical charts and flips on quantum decorated character stacks.
We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and Goncharov. Our construction is based on the "quantum trace" map introduced by Bonahon …
Quantum Clustering is a powerful method to detect clusters in data with mixed density. However, it is very sensitive to a length parameter that is inherent to the Schrödinger equation. In addition, linking data points into clusters requires local estimates of covariance that are also controlled by length parameters. Th…
This paper defines several algebras associated to an oriented surface with a finite set of marked points on the boundary. The first is the skein algebra , which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defined relations. The product…
A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized …
A novel density-based approach QC detects outliers in data with high precision.
problem Detecting outliers in data with high precision and sensitivity.
method Quantum Clustering (QC) approach based on the density of data points.
result QC effectively finds hidden outliers and subtle outliers with parameter adjustment.
Study of -webs on surfaces, proving cluster algebra structure.
problem Understanding -webs and their cluster algebra properties.
method Introduced skein algebra and cluster structure, proved positivity.
result Proved is a quantum cluster algebra.
In this paper, we show that Alexander polynomials for any 2-bridge knots are specializations of cluster variables. A key tool is an ancestral triangle which appeared in both quantum topology and hyperbolic geometry in different ways.
We describe a natural -deformation of Fock and Goncharov's canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type . We then describe an extension of this construction involving a cluster variety called the symplectic double.
MPE framework proves universal approximation for quantum data distribution.
problem Challenges in generating quantum data from underlying distributions.
method Many-body Projected Ensemble (MPE) framework for quantum state design.
result MPE can approximate any quantum distribution within 1-Wasserstein distance error.
This paper uses QUBO to train machine learning models on quantum computers.
problem Efficiently training machine learning models on quantum computers.
method Formulated three machine learning models (linear regression, SVM, k-means) as QUBO problems.
result Formulations are more efficient or equivalent in time and space complexity to classical methods.
Quantum dynamics algorithm learns manifold from data.
problem Learning manifolds from high-dimensional datasets.
method Simulation of quantum dynamics on a graph embedding of data.
result Algorithm reveals connections between data sampling and quantization.
A cluster variety of Fock and Goncharov is a scheme constructed from the data related to the cluster algebras of Fomin and Zelevinsky. A seed is a combinatorial data which can be encoded as an matrix with integer entries, or as a quiver in special cases, together with formal variables. A mutation is a c…
Quantizes moduli space of 3D gravity metrics.
problem Quantize moduli space of 3D gravity metrics.
method Develops geometrically natural classes of observables and uses cluster -varieties.
result Obtains projective unitary representations of mapping class group.
Pipeline decomposes portfolio optimization problems into smaller, solvable subproblems.
problem Large-scale portfolio optimization with constraints.
method Decomposition pipeline with preprocessing, clustering, and risk rebalancing.
result Pipeline reduces problem size by 80% and computation time.
The paper studies properties of stated SL(n)-skein algebras and their centers.
problem Properties of stated SL(n)-skein algebras and their centers.
method Quantum trace maps and embeddings into quantum tori.
result Finitely generation and PI-degrees of centers of stated SL(n)-skein algebras.
Variational Bayes (VB) inference is one of the most important algorithms in machine learning and widely used in engineering and industry. However, VB is known to suffer from the problem of local optima. In this Letter, we generalize VB by using quantum mechanics, and propose a new algorithm, which we call quantum annea…
The mixture of Gaussian distributions, a soft version of k-means , is considered a state-of-the-art clustering algorithm. It is widely used in computer vision for selecting classes, e.g., color, texture, and shapes. In this algorithm, each class is described by a Gaussian distribution, defined by its mean and covarianc…
Techniques for data-mining, latent semantic analysis, contextual search of databases, etc. have long ago been developed by computer scientists working on information retrieval (IR). Experimental scientists, from all disciplines, having to analyse large collections of raw experimental data (astronomical, physical, biolo…
We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structu…
Efficiently private clustering algorithms with tight approximation ratios.
problem Differentially private clustering of various types.
method Efficient algorithms achieving tight approximation ratios for clustering problems.
result Achieves approximation ratios similar to non-private algorithms with small additive errors.
Study of cluster and skein algebras for surfaces, showing their connection.
problem Understanding algebraic structures of curve algebras on surfaces.
method Generalization and explicit definition of maps between cluster and skein algebras.
result Explicit maps between cluster and skein algebras, showing their close relationship.
We investigate 17 digital currencies making an analogy with quantum systems and develop the concept of eigenportfolios. We show that the density of states of the correlation matrix of these assets shows a behavior between that of the Wishart ensemble and one whose elements are Cauchy distributed. A metric for the parti…
Diffusion maps help learn complex quantum phase transitions from data.
problem Learning quantum phase transitions from experimental data is challenging.
method Diffusion maps for nonlinear dimensionality reduction and spectral clustering.
result Diffusion maps can learn complex phase transitions unsupervised.
Deep QMC methods use neural networks to solve quantum chemistry problems.
problem Solving the electronic Schrödinger equation from first principles.
method Quantum Monte Carlo with neural network wavefunctions.
result Highly accurate solutions at reduced computational cost.
Quantum traces embed into quantum tori for surface skein algebras.
problem Embedding stated skein algebras into quantum tori.
method Two different embeddings using quantum trace maps and lambda length coordinates.
result Quantum cluster algebra of Muller equals reduced stated skein algebra.