Defines a map from lamination spaces to quantum cluster varieties.
problem Quantum cluster varieties from surfaces.
method Using a quantum trace map, we define a canonical map.
result The map satisfies properties conjectured by Fock and Goncharov.
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Study character varieties of surfaces using cluster algebras and Poisson structures.
problem Character varieties of surfaces and their Poisson structures.
method Use Bonahon-Wong's trace map and cluster algebras associated with ideal triangulations.
result Recover Goldman Poisson algebra from cluster algebra structure and show automorphisms.
Quantum Teichmüller theory constants confirmed for cluster varieties.
problem Verifying constants in quantum Teichmüller theory representations.
method Computation of constants using quantum dilogarithm.
result All constants are confirmed to be 1, confirming genuine representations.
Quantum Teichmüller theory constants are shown to be 1.
problem Verifying genuine representations in quantum Teichmüller theory.
method Quantum dilogarithm function and unitary intertwiners.
result Algebraic relations among quantum mutations are satisfied by intertwiners with constants equal to 1.
Quantizes canonical bases for cluster varieties of type A.
problem Deforming algebra of functions on cluster varieties.
method Natural -deformation of Fock and Goncharov's canonical basis.
result Extension to quantum symplectic double.
This thesis studies cluster varieties from surfaces and their properties.
problem Understanding the geometry of cluster varieties associated to surfaces.
method Developed properties of symplectic double and defined measured laminations.
result Proved Fock and Goncharov's duality conjectures for quantum cluster varieties.
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.
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.
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.
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.
The paper constructs bases for cluster varieties using -webs and laminations.
problem Cluster varieties associated to -local systems on surfaces.
method Introducing -laminations, developing quantum and classical trace maps, and constructing bases.
result Bases of regular functions on cluster varieties constructed from -laminations.
Paper proposes a quantum deep clustering framework with improved performance.
problem Improving clustering performance in quantum machine learning.
method Quantum deep SVM, deep convolutional neural networks, and quantum K-Means clustering.
result The proposed quantum deep clustering framework shows significant performance gains over classical methods.
Quantum basis coefficients are positive integers for framed local systems on surfaces.
problem Positivity of quantum basis coefficients for framed local systems on surfaces.
method Introduced a graph to solve a combinatorial ordering problem about ideal triangulations and closed curves on surfaces.
result Laurent coefficients of quantum basis elements are positive integers.
Quantum SVM clustering speeds up big data analysis.
problem Performance degradation of classical SVM clustering on big data.
method Developed a quantum version of SVM clustering using quantum support vector machine and kernels.
result Significant speed-up gain on run-time complexity.
Quantum theory improves counting overlapping clusters.
problem Counting overlapping clusters in machine learning.
method Applied quantum theory using path integral technique.
result Quantum theory provides a robust statistical method for counting clusters.
Quantum character varietiesmath.QA
Quantum character varieties unify four construction methods.
problem No specific problem stated; unification of approaches.
method Four different approaches to construction.
result Unified understanding of quantum character varieties.
Quantum EM algorithm improves clustering for Gaussian mixtures.
problem Improving clustering efficiency for Gaussian mixture models.
method Quantum expectation-maximization algorithm for Gaussian mixture models.
result Quantum EM algorithm demonstrates robustness and speedup.
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.
Adiabatic quantum computing solves binary clustering problems.
problem Binary clustering problems in machine learning.
method Adiabatic quantum computing applied to binary clustering.
result Numerical simulations show feasibility and qubits evolve towards a solution.
Geometric model of unbounded sl3 laminations with tropical coordinates.
problem Modeling unbounded laminations in cluster varieties.
method Introducing tropical cluster coordinates and geometric gluing procedures.
result Established a geometric gluing procedure for unbounded sl3 laminations.
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 .
Quantum annealing speeds up extreme clustering.
problem Efficiently grouping large datasets into many representative clusters.
method Distributed quantum annealing method.
result Optimal clustering assignments achieved under separability assumption.
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.
Quantum transport maps network clusters on a circle, revealing complex community structures.
problem Identifying community structures in complex networks.
method Proposes using the Laplace transform of quantum transport to map network nodes onto a circle, forming clusters.
result QTC algorithm effectively clusters network nodes, robust to cluster heterogeneity.
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 abelian character varieties to SL2 character varieties.
problem Classifying irreducible representations of Chekhov-Fock algebras.
method Non-commutative deformation of algebraic morphisms.
result Induces birational morphism between torus and SL2 character variety.
Estimates quantum cohomology complexity for Fano varieties and homogeneous spaces.
problem Quantum cohomology complexity estimation for compact symplectic manifolds.
method Estimates the number of states with finite approximate complexity for Fano complete intersections and (co)minuscule homogeneous varieties.
result Sharp upper bound for the dimension of the space spanned by states with finite complexity for Gr(2, n).
Quantum Teichmüller theory solved by linking Bonahon-Wong trace and Gabella's solution.
problem Quantize the trace-of-monodromy function on Teichmüller space.
method Used Bonahon and Wong's quantum trace for skein algebras and Gabella's Seiberg-Witten curves, spectral networks, and writhe of links.
result Bonahon-Wong quantum trace and Gabella's solution coincide and are a twist of each other.
A probabilistic framework optimizes quantum clustering parameters.
problem Optimizing length parameters for quantum clustering sensitivity.
method Bayesian optimization of control parameters within a probabilistic framework.
result Optimized clustering yields better concordance with known data structure.
Paper connects knot theory with cluster algebra via Alexander polynomials.
problem Understanding Alexander polynomials for 2-bridge knots.
method Use of cluster variables and ancestral triangles.
result Alexander polynomials are specializations of cluster variables.
The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum volume conjecture) are extended to $U_q(\fraksl_2)$ colored quantum invariants of the theta and tetrahedron graph. The $\SL(2,\bC)$ character variety of the fundamental group of the complement of a trivalent graph with edges in is a L…
Artificial neural networks predict quantum entanglement types.
problem Predicting entanglement types of quantum states.
method Supervised learning and deep neural networks on algebraic varieties.
result Trained neural networks can classify entanglement types for up to 5 binary qubits and 3 qutrits.
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…
Commuting Line Defects At hep-th
Explains a property of algebras related to quantum field theories.
problem Explains a property of algebras encoding line defects in quantum field theories.
method Physical explanation of a property of quantized algebras using dualities and field theories.
result Physical explanation of a large center in quantized algebras when the deformation parameter is a root of unity.
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 mechanics improves variational Bayes inference.
problem Local optima in variational Bayes inference.
method Quantum annealing variational Bayes (QAVB) inference.
result QAVB drastically improves VB performance.
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.
Quantum interference improves clustering accuracy.
problem Improving clustering accuracy in Gaussian mixture models.
method Modeling classes as wave functions, then mixing them.
result Quantum method outperforms Gaussian mixture in all aspects.
Study calculates instanton homology for simple braids, linking to Fano variety quantum cohomology.
problem Calculating instanton homology for specific braids.
method Using local coefficients and algebraic curves, calculates homology groups and their module structures.
result Equivalent to computing quantum cohomology of a moduli space of parabolic bundles.
Study of Legendrian links using Floer theory and cluster varieties.
problem Understanding exact Lagrangian fillings of positive braid Legendrian links.
method Floer-theoretic approach and exact Lagrangian cobordisms.
result Proves that positive braid Legendrian links admit infinitely many exact Lagrangian fillings.
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 spaces associated to mixed polarizations and their limiting behavior on toric varietiesmath.SG
Study explores quantum spaces on toric varieties and their limiting behavior.
problem Understanding quantum spaces on toric varieties and their limiting behavior.
method Established quantum spaces for mixed polarizations and examined one-parameter families of Kähler polarizations.
result Quantum spaces converge to as .
Complex abelian varieties and quantum error correction: a mathematical framework for GKP codesmath.AG
Quantum codes linked to abelian varieties, providing mathematical rigor.
problem Quantum error correction through complex abelian varieties.
method Mathematical formulation of Gottesman-Kitaev-Preskill codes using abelian varieties.
result Asymptotic isometry of encoding, precise gate realizations, and failure probability optimization.
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.