arXiv research

Defines unitary setting for quantum mechanics, explaining time evolution.

Characterizes optimal-speed quantum state evolution Hamiltonians.

Mackey showed that for a compact Lie group $K$, the pair $(K,C^{0}(K))$ has a unique non-trivial irreducible covariant pair of representations. We study the relevance of this result to the unitary equivalence of quantizations for an infinite-dimensional family of $K\times K$ invariant polarizations on $T^{\ast}K$. The …

Quantum methods model uncertain volatility in financial markets.

We revisit the computation of the phase of the Dirac fermion scattering operator in external gauge fields. The computation is through a parallel transport along the path of time evolution operators. The novelty of the present paper compared with the earlier geometric approach by Langmann and Mickelsson, [LM], is that w…

New complex structures on jet spaces help explain Fock space dynamics.

The paper studies distributions and controllability in quantum mechanical systems.

For the cotangent bundle $T^{*}K$ of a compact Lie group $K$, we study the complex-time evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space $L^{2}(K)$ under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state tra…

Quantum computer method for pricing lookback options with jumps.

Quantum neural networks generalize better due to flatter parameter space.

Recurrent neural networks (RNNs) are notoriously difficult to train. When the eigenvalues of the hidden to hidden weight matrix deviate from absolute value 1, optimization becomes difficult due to the well studied issue of vanishing and exploding gradients, especially when trying to learn long-term dependencies. To cir…

UKM framework optimizes VQCs, showing QCL performance is bounded.

Study geometric mKdV flows for Legendrian curves in a 3-sphere.

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 …

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…

Quantum theory is used to model secondary financial markets. Contrary to stochastic descriptions, the formalism emphasizes the importance of trading in determining the value of a security. All possible realizations of investors holding securities and cash is taken as the basis of the Hilbert space of market states. The…

RUM improves RNN's long-term memory by using unitary matrices.

A major challenge in the training of recurrent neural networks is the so-called vanishing or exploding gradient problem. The use of a norm-preserving transition operator can address this issue, but parametrization is challenging. In this work we focus on unitary operators and describe a parametrization using the Lie al…

New method solves $KP$ problem using global Cartan decompositions.

The paper proves that a specific manifold is unitary cobordant to S^2 × S^6.

Convexity proven for sums of angles of unitary paths.

The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary p…

This article provides an account of the functorial correspondence between irreducible singular $G$-monopoles on $S^1\times Σ$ and $\vec{t}$-stable meromorphic pairs on $Σ$. The main theorem of [1] is thus generalized here from unitary to arbitrary compact, connected gauge groups. The required distinctions and similarit…

Gravitational interactions of higher spin fields are generically plagued by inconsistencies. We present a simple framework that couples higher spins to a broad class of gravitational backgrounds (including Ricci flat and Einstein) consistently at the classical level. The model is the simplest example of a Yang--Mills d…

Compactifies Minkowski space using unitary matrices.

The paper studies the zeta-regularized determinant of a pseudo-Laplacian on a cuspidal end with flat unitary line bundle.

Paper improves tensor completion by reducing sample entries needed.

Private algorithms approximate matrices with private data.

Contact group retracts to unitary subgroup.

Zeta functions for non-unitary twists are shown to have analytic continuation.

Recurrent neural networks are powerful models for processing sequential data, but they are generally plagued by vanishing and exploding gradient problems. Unitary recurrent neural networks (uRNNs), which use unitary recurrence matrices, have recently been proposed as a means to avoid these issues. However, in previous …

The study finds invariant Einstein metrics on complex Stiefel manifolds and special unitary groups.

High-fidelity quantum simulations demonstrated on short-coherence hardware.

Revises super unitary representations by relaxing Hilbert space definition.

Study on determinants of unitary Brownian motion and their asymptotic laws.

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial gen…

We show that for any positive integer $n$, the maps $x \in \mathbb{C}^n \mapsto \{\left|\langle x, z_i \rangle \right|^2\}_{i=1}^{4n} \in \mathbb{R}^{4n}$, where $z_i$ are the columns of four $n\times n$ unitary matrices, are generically injective modulo multiplication by a global phase factor, yielding a family of emb…

New structure on unitary group of Hilbert space.

Study circle actions on unitary manifolds with discrete fixed points.

Quantum Gaussian processes enable scalable quantum learning.

The study shows ergodicity of unitary frame flows on Kähler manifolds with specific curvature conditions.

UGConvs improve CNN accuracy with unitary transforms.

Study asymptotics of unitary matrix elements in quantum mechanics.

New unitary representations for mapping class groups without almost invariant vectors.

New unitary RNN architecture using complex Cayley transform outperforms existing methods.

Study circumcenters in Finsler unitary groups with optimal convexity bounds.

This paper introduces a submanifold of the moduli space of unitary representations of the fundamental group of a punctured sphere with fixed local monodromy. The submanifold is defined via products of involutions through Lagrangian subspaces. We show that the moduli space of Lagrangian representations is a Lagrangian s…

Researchers describe unitary representations of mixed braid groups.