arXiv research

Quantum connections replace metrics with operator inner products.

New connections found for quantum flag manifolds modules.

Quantum cohomology connects quantum physics with classical math.

Quantum field theory connects Riemannian geometry to quantum fluctuations.

We build metrized quantum vector bundles, over a generically transcendental quantum torus, from Riemannian metrics, using Rosenberg's Levi-Civita connections for these metrics. We also prove that two metrized quantum vector bundles, corresponding to positive scalar multiples of a Riemannian metric, have distance zero b…

The paper explores quantum statistical manifolds and their autoparallelity, providing estimation-theoretical characterizations.

Khovanov homology helps create quantum error-correcting codes.

Generalizes Hodge correlators using quantum master equation concepts.

Quantum invariants of 3-manifolds and links reviewed, with connections to other topological invariants.

A new quantum gauge model is proposed. From this quantum gauge model we derive a quantum invariant of 3-manifolds. We show that this quantum invariant of 3-manifolds gives a classification of closed (orientable and connected) 3-manifolds. From this classification we then prove the Poincaré conjecture.

Program connects quantum computing and topological field theories.

We present the construction of an infinite dimensional Banach manifold of quantum mechanical states on a Hilbert space H using different types of small perturbations of a given Hamiltonian. We provide the manifold with a flat connection, called the exponential connection, and comment on the possibility of introducing t…

Quantum trace map connects Teichmüller theory and quantum groups.

Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed, including the description of a natural differential calculus over the structure …

Formulates quantum jet bundles over noncommutative algebras with connections and braiding.

Quantum theory of curved tetrahedrons yields quantum group intertwiners.

Paper connects knot invariants and Morse flow loops.

First an `irregular Riemann-Hilbert correspondence' is established for meromorphic connections on principal G-bundles over a disc, where G is any connected complex reductive group. Secondly, in the case of poles of order two, isomonodromic deformations of such connections are considered and it is proved that the classi…

Quantum cohomology gives a finite dimensional integrable system via the Dubrovin connection. Motivated by Givental's work on mirror symmetry, we use gauge theory techniques and the Frobenius Integrability Theorem to find flat sections for the Dubrovin connection. An explicit calculation is given for projective space.

The paper proves a quantum modularity conjecture for 3-manifolds.

We construct a covariant functor from a category of Abelian principal bundles over globally hyperbolic spacetimes to a category of *-algebras that describes quantized principal connections. We work within an appropriate differential geometric setting by using the bundle of connections and we study the full gauge group,…

We show how to define invariants of graphs related to quantum $\mathfrak{sl}(2)$ when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions.

Covariant formulation of Barbero-Immirzi connections for spin manifolds.

We give a direct calculation of the curvature of the Hitchin connection, in geometric quantization on a symplectic manifold, using only differential geometric techniques. In particular, we establish that the curvature acts as a first-order operator on the quantum spaces. Projective flatness follows if the Kähler struct…

We propose a protocol to perform quantum reinforcement learning with quantum technologies. At variance with recent results on quantum reinforcement learning with superconducting circuits, in our current protocol coherent feedback during the learning process is not required, enabling its implementation in a wide variety…

In earlier work, Helen Wong and the author discovered certain "miraculous cancellations" for the quantum trace map connecting the Kauffman bracket skein algebra of a surface to its quantum Teichmueller space, occurring when the quantum parameter $q$ is a root of unity. The current paper is devoted to giving a more repr…

4-manifolds have special topological properties which can be used to get a different view on quantum mechanics. One important property (connected with exotic smoothness) is the natural appearance of 3-manifold wild embeddings (Alexanders horned sphere) which can be interpreted as quantum states. This relation can be co…

The abstract discusses connecting quantum mechanics and algebraic index theories.

Quantum GAN improves volatility modeling in finance.

Bayesian methods in machine learning, such as Gaussian processes, have great advantages com-pared to other techniques. In particular, they provide estimates of the uncertainty associated with a prediction. Extending the Bayesian approach to deep architectures has remained a major challenge. Recent results connected dee…

We introduce a framework for coverings of noncommutative spaces. Moreover, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.

Kauffman and Lomonaco explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In the work of G. Alagic, M. Jarret, and S. Jordan it is shown that entanglement is a necessary condit…

New approach connects quantum phases to VQA trainability, enabling better scaling.

Quantum computing techniques applied to Monte Carlo simulations in finance.

Optimizing quantum graphs yields geodesic nets on surfaces.

Cone structures in quantum field theory linked to information geometry.

We introduce a differential geometric framework for describing families of quantum error-correcting codes and for understanding quantum fault tolerance. This work unifies the notion of topological fault tolerance with fault tolerance in other kinds of quantum error-correcting codes. In particular, we use fibre bundles …

In this paper, we present the idea that the formalism of string theory is connected with the dimension 4 in a new way, not covered by phenomenological or model-building approaches. The main connection is given by structures induced by small exotic smooth R^4's having intrinsic meaning for physics in dimension 4. We ext…

Researchers prove unique connection and curvature for Podleś quantum sphere.

Quantum annealing (QA) is a generic method for solving optimization problems using fictitious quantum fluctuation. The current device performing QA involves controlling the transverse field; it is classically simulatable by using the standard technique for mapping the quantum spin systems to the classical ones. In this…

3D quantum trace map connects 3-manifold quantizations.

We show that the small quantum product of the generalized flag manifold $G/B$ is a product operation on $H^*(G/B)\otimes \bR[q_1,..., q_l]$ uniquely determined by the fact that it is a deformation of the cup product on $H^*(G/B)$, it is commutative, associative, graded with respect to $°(q_i)=4$, it satisfies a certain…

Quantum Finance represents the synthesis of the techniques of quantum theory (quantum mechanics and quantum field theory) to theoretical and applied finance. After a brief overview of the connection between these fields, we illustrate some of the methods of lattice simulations of path integrals for the pricing of optio…

In this contribution we review results on the kinematics of a quantum system localized on a connected configuration manifold and compatible dynamics for the quantum system including external fields and leading to non-linear Schrödinger equations for pure states.

The paper shows compatibility between two quantum maps for surfaces and 3-manifolds.

A new hybrid framework reduces quantum runtime and noise effects.

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space $X$ is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the …

Geometric quantization extended to arbitrary connected spaces using path integration.