Searcharxiv⌕ Search

arXiv subjects

Yuan Xu

Publications and source records attributed to Yuan Xu.

At least 145 records · Page 8Linked to original sources
arXiv2020

Non-homogeneous wave equation on a cone

The wave equation $\left(\partial_{tt} - c^2 Δ_x\right) u(x,t) = e^{-t} f(x,t)$ is shown to have a unique solution if $u$ and its partial derivatives in $x$ are in $L^2(e^{-t})$ on the cone, and the solution can be explicit given in the Fourier series of orthogonal polynomials on the cone. This provides a particular solution for the boundary value problems of the non-homogeneous wave equation on the cone, which can be combined with a solution to the homogeneous wave equation in the cone to obtain the full solution.

math.CA↗
arXiv2020

Hahn polynomials on polyhedra and quantum integrability

Orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in ${\mathbb R}^d$, which include hexagons in ${\mathbb R}^2$ and truncated tetrahedrons in ${\mathbb R}^3$, are defined and studied. The polynomials are given explicitly in terms of the classical one-dimensional Hahn polynomials. They are also characterized as common eigenfunctions of a family of commuting partial difference operators. These operators provide symmetries for a system that can be regarded as a discrete extension of the generic quantum superintegrable system on the $d$-sphere. Moreover, the discrete system is proved to possess all essential properties of the continuous system. In particular, the symmetry operators for the discrete Hamiltonian define a representation of the Kohno-Drinfeld Lie algebra on the space of orthogonal polynomials, and an explicit set of $2d-1$ generators for the symmetry algebra is constructed. Furthermore, other discrete quantum superintegrable systems, which extend the quantum harmonic oscillator, are obtained by considering appropriate limits of the parameters.

math.CA↗
arXiv2020

Orthogonal structure on a quadratic curve

Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. As an application, we see that the resulting bases can be used to interpolate functions on the real line with singularities of the form $|x|$, $\sqrt{x^2+ ε^2}$, or $1/x$, with exponential convergence.

math.NA↗
arXiv2019

Orthogonal structure and orthogonal series in and on a double cone or a hyperboloid

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by hyperplanes at the two ends. On each domain a family of orthogonal polynomials, related to the Gegebauer polynomials, is study and shown to share two characteristic properties of spherical harmonics: they are eigenfunctions of a second order linear differential operator with eigenvalues depending only on the polynomial degree, and they satisfy an addition formula that provides a closed form formula for the reproducing kernel of the orthogonal projection operator. The addition formula leads to a convolution structure, which provides a powerful tool for studying the Fourier orthogonal series on these domains. Furthermore, another family of orthogonal polynomials, related to the Hermite polynomials, is defined and shown to be the limit of the first family, and their properties are derived accordingly.

math.CA↗
arXiv2019

Orthogonal polynomials and Fourier orthogonal series on a cone

Orthogonal polynomials and the Fourier orthogonal series on a cone of revolution in $\mathbb{R}^{d+1}$ are studied. It is shown that orthogonal polynomials with respect to the weight function $(1-t)^γ(t^2-\|x\|^2)^{μ-\frac12}$ on the cone $\mathbb{V}^{d+1} = \{(x,t): \|x\| \le t \le 1\}$ are eigenfunctions of a second order differential operator, with eigenvalues depending only on the degree of the polynomials, and the reproducing kernels of these polynomials satisfy a closed formula that has a one-dimensional chacteristic. The latter leads to a convolution structure on the cone, which is then utilized to study the Fourier orthogonal series. This narrative also holds, in part, for more general classes of weight functions. Furthermore, analogous results are also established for orthogonal structure on the surface of the cone.

math.CA↗
arXiv2019

Observation of topological magnon insulator states in a superconducting circuit

Searching topological states in artificial systems has recently become a rapidly growing field of research. Meanwhile, significant experimental progresses on observing topological phenomena have been made in superconducting circuits. However, topological insulator states have not yet been reported in this system. Here, for the first time, we experimentally realize a tunable dimerized spin chain model and observe the topological magnon insulator states in a superconducting qubit chain. Via parametric modulations of the qubit frequencies, we show that the qubit chain can be flexibly tuned into topologically trivial or nontrivial magnon insulator states. Based on monitoring the quantum dynamics of a single-qubit excitation in the chain, we not only measure the topological winding numbers, but also observe the topological magnon edge and defect states. Our experiment exhibits the great potential of tunable superconducting qubit chain as a versatile platform for exploring non-interacting and interacting symmetry-protected topological states.

quant-ph↗
arXiv2019

Orthogonal polynomials in and on a quadratic surface of revolution

We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation methods.

math.CA↗
arXiv2018

Intertwining operators associated to dihedral groups

The Dunkl operators associated to a dihedral group are a pair of differential-difference operators that generate a commutative algebra acting on differentiable functions in $\mathbb{R}^2$. The intertwining operator intertwines between this algebra and the algebra of differential operators. The main result of this paper is an integral representation of the intertwining operator on a class of functions. As an application, closed formulas for the Poisson kernels of $h$-harmonics and sieved Gegenbauer polynomials are deduced when one of the variables is at vertices of a regular polygon, and similar formulas are also derived for several other related families of orthogonal polynomials.

math.CA↗
arXiv2018

Quantum generative adversarial learning in a superconducting quantum circuit

Generative adversarial learning is one of the most exciting recent breakthroughs in machine learning---a subfield of artificial intelligence that is currently driving a revolution in many aspects of modern society. It has shown splendid performance in a variety of challenging tasks such as image and video generations. More recently, a quantum version of generative adversarial learning has been theoretically proposed and shown to possess the potential of exhibiting an exponential advantage over its classical counterpart. Here, we report the first proof-of-principle experimental demonstration of quantum generative adversarial learning in a superconducting quantum circuit. We demonstrate that, after several rounds of adversarial learning, a quantum state generator can be trained to replicate the statistics of the quantum data output from a digital qubit channel simulator, with a high fidelity ($98.8\%$ on average) that the discriminator cannot distinguish between the true and the generated data. Our results pave the way for experimentally exploring the intriguing long-sought-after quantum advantages in machine learning tasks with noisy intermediate-scale quantum devices.

quant-ph↗
arXiv2018

Experimental repetitive quantum channel simulation

Universal control of quantum systems is a major goal to be achieved for quantum information processing, which demands thorough understanding of fundamental quantum mechanics and promises applications of quantum technologies. So far, most studies concentrate on ideally isolated quantum systems governed by unitary evolutions, while practical quantum systems are open and described by quantum channels due to their inevitable coupling to environment. Here, we experimentally simulate arbitrary quantum channels for an open quantum system, i.e. a single photonic qubit in a superconducting quantum circuit. The arbitrary channel simulation is achieved with minimum resource of only one ancilla qubit and measurement-based adaptive control. By repetitively implementing the quantum channel simulation, we realize an arbitrary Liouvillian for a continuous evolution of an open quantum system for the first time. Our experiment provides not only a testbed for understanding quantum noise and decoherence, but also a powerful tool for full control of practical open quantum systems.

quant-ph↗
arXiv2018

Orthogonal structure on a wedge and on the boundary of a square

Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier orthogonal expansions is studied. These are used to establish analogous results for orthogonal polynomials on the boundary of the square. As an application, we study the statistics of the associated determinantal point process and use the basis to calculate Stieltjes transforms.

math.CA↗
arXiv2018

Demonstration of quantum error correction and universal gate set on a binomial bosonic logical qubit

Logical qubit encoding and quantum error correction (QEC) have been experimentally demonstrated in various physical systems with multiple physical qubits, however, logical operations are challenging due to the necessary nonlocal operations. Alternatively, logical qubits with bosonic-mode-encoding are of particular interest because their QEC protection is hardware efficient, but gate operations on QEC protected logical qubits remain elusive. Here, we experimentally demonstrate full control on a single logical qubit with a binomial bosonic code, including encoding, decoding, repetitive QEC, and high-fidelity (97.0% process fidelity on average) universal quantum gate set on the logical qubit. The protected logical qubit has shown 2.8 times longer lifetime than the uncorrected one. A Ramsey experiment on a protected logical qubit is demonstrated for the first time with two times longer coherence than the unprotected one. Our experiment represents an important step towards fault-tolerant quantum computation based on bosonic encoding.

quant-ph↗
arXiv2018

Avalon: Building an Operating System for Robotcenter

This paper envisions a scenario that hundreds of heterogeneous robots form a robotcenter which can be shared by multiple users and used like a single powerful robot to perform complex tasks. However, current multi-robot systems are either unable to manage heterogeneous robots or unable to support multiple concurrent users. Inspired by the design of modern datacenter OSes, we propose Avalon, a robot operating system with two-level scheduling scheme which is widely adopted in datacenters for Internet services and cloud computing. Specifically, Avalon integrates three important features together: (1) Instead of allocating a whole robot, Avalon classifies fine-grained robot resources into three categories to distinguish which fine-grained resources can be shared by multi-robot frameworks simultaneously. (2) Avalon adopts a location based resource allocation policy to substantially reduce scheduling overhead. (3) Avalon enables robots to offload computation intensive tasks to the clouds.We have implemented and evaluated Avalon on robots on both simulated environments and real world.

cs.RO↗
arXiv2018

Gaussian bounds for the weighted heat kernels on the interval, ball and simplex

The aim of this article is to establish two-sided Gaussian bounds for the heat kernels on the unit ball and simplex in $\mathbb{R}^n$, and in particular on the interval, generated by classical differential operators whose eigenfunctions are algebraic polynomials. To this end we develop a general method that employs the natural relation of such operators with weighted Laplace operators on suitable subsets of Riemannian manifolds and the existing general results on heat kernels. Our general scheme allows to consider heat kernels in the weighted cases on the interval, ball, and simplex with parameters in the full range.

math.CA↗
arXiv2017

Best polynomial approximation on the triangle

Let $E_n(f)_{α,β,γ}$ denote the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_{α,β,γ})$ on the triangle $\{(x,y): x, y \ge 0, x+y \le 1\}$, where $\varpi_{α,β,γ}(x,y) := x^αy ^β(1-x-y)^γ$ for $α,β,γ> -1$. Our main result gives a sharp estimate of $E_n(f)_{α,β,γ}$ in terms of the error of best approximation for higher order derivatives of $f$ in appropriate Sobolev spaces. The result also leads to a characterization of $E_n(f)_{α,β,γ}$ by a weighted $K$-functional.

math.CA↗
arXiv2017

Approximation by polynomials in Sobolev spaces with Jacobi weight

Polynomial approximation is studied in the Sobolev space $W_p^r(w_{α,β})$ that consists of functions whose $r$-th derivatives are in weighted $L^p$ space with the Jacobi weight function $w_{α,β}$. This requires simultaneous approximation of a function and its consecutive derivatives up to $s$-th order with $s \le r$. We provide sharp error estimates given in terms of $E_n(f^{(r)})_{L^p(w_{α,β})}$, the error of best approximation to $f^{(r)}$ by polynomials in $L^p(w_{α,β})$, and an explicit construction of the polynomials that approximate simultaneously with the sharp error estimates.

math.CA↗
arXiv2017

Optimal Points for Cubature Rules and Polynomial Interpolation on a Square

The nodes of certain minimal cubature rule are real common zeros of a set of orthogonal polynomials of degree $n$. They often consist of a well distributed set of points and interpolation polynomials based on them have desired convergence behavior. We report what is known and the theory behind by explaining the situation when the domain of integrals is a square.

math.NA↗