Variable Ansatz Research Articles

Relativistic scalar fields canonical quantization in Einstein–Yang–Mills–Higgs’s rotating black hole space-time

The quantum theory of relativistic mechanics to deal with the scalar fields behavior in a curved space-time is represented by the Klein–Gordon equation. In this work, we will investigate the gravitationally bound states of massive and massless scalar fields around a Einstein–Yang–Mills–Higgs’s rotating black hole. After applying the standard separation of variables ansatz, we will show in detail how to obtain the novel exact solutions of the radial part of the governing Klein–Gordon equation and express the radial solution in terms of the Confluent Heun functions. By applying the bound state boundary conditions, the Confluent Heun functions are reduced to be polynomials that lead to energy quantization. We find that the scalar fields have complex-valued energy levels that indicate the decaying/growing bound states known as quasibound states. In the end, using the exact radial solution, we derive the radiation distribution function of the black hole’s outer horizon to obtain the equation of the Hawking temperature.

A semi-agnostic ansatz with variable structure for variational quantum algorithms

Quantum machine learning—and specifically Variational Quantum Algorithms (VQAs)—offers a powerful, flexible paradigm for programming near-term quantum computers, with applications in chemistry, metrology, materials science, data science, and mathematics. Here, one trains an ansatz, in the form of a parameterized quantum circuit, to accomplish a task of interest. However, challenges have recently emerged suggesting that deep ansatzes are difficult to train, due to flat training landscapes caused by randomness or by hardware noise. This motivates our work, where we present a variable structure approach to build ansatzes for VQAs. Our approach, called VAns (Variable Ansatz), applies a set of rules to both grow and (crucially) remove quantum gates in an informed manner during the optimization. Consequently, VAns is ideally suited to mitigate trainability and noise-related issues by keeping the ansatz shallow. We employ VAns in the variational quantum eigensolver for condensed matter and quantum chemistry applications, in the quantum autoencoder for data compression and in unitary compilation problems showing successful results in all cases.

Variable ansatz applied to spectral operator decomposition in a physical superconducting quantum device

Variable ansatz applied to spectral operator decomposition in a physical superconducting quantum device

A Gradient‐Cost Multiobjective Alternate Framework for Variational Quantum Eigensolver with Variable Ansatz

AbstractVariational quantum eigensolver (VQE), aiming at determining the ground state energy of a quantum system described by a Hamiltonian on noisy intermediate scale quantum (NISQ) devices. However, the accuracy and trainability of the current VQE algorithm are significantly influenced due to the barren plateau (BP), the non‐negligible gate error and limited coherence time in NISQ devices. To tackle these issues, a gradient‐cost multiobjective alternate framework with variable ansatz is proposed. A theoretical framework is first proposed for VQE with variable ansatz (VA‐VQE) via alternately solving a multiobjective optimization problem defined by cost function values and gradient magnitudes and the original VQE problem. Then, a novel implementation method based on the double ε‐greedy strategy with the candidate tree and modified multiobjective genetic algorithm is proposed. As a result, the local optima are avoided both in ansatz and parameter perspectives, the BP phenomenon is alleviated, and the stability of output ansatz is enhanced. The experimental results indicate that this framework shows considerably average improvement of the error and stability by 59.6% and 78.8% compared with the state‐of‐the‐art cost‐criterion‐based VA‐VQE implementation, and 54.1% and 73.5% compared with the gradient‐criterion‐based counterpart, respectively, with lower quantum costs.

Teukolsky formalism for nonlinear Kerr perturbations

We develop a formalism to treat higher order (nonlinear) metric perturbations of the Kerr spacetime in a Teukolsky framework. We first show that solutions to the linearized Einstein equation with nonvanishing stress tensor can be decomposed into a pure gauge part plus a zero mode (infinitesimal perturbation of the mass and spin) plus a perturbation arising from a certain scalar (‘Debye–Hertz’) potential, plus a so-called ‘corrector tensor’. The scalar potential is a solution to the spin −2 Teukolsky equation with a source. This source, as well as the tetrad components of the corrector tensor, are obtained by solving certain decoupled ordinary differential equations involving the stress tensor. As we show, solving these ordinary differential equations reduces simply to integrations in the coordinate r in outgoing Kerr–Newman coordinates, so in this sense, the problem is reduced to the Teukolsky equation with source, which can be treated by a separation of variables ansatz. Since higher order perturbations are subject to a linearized Einstein equation with a stress tensor obtained from the lower order perturbations, our method also applies iteratively to the higher order metric perturbations, and could thus be used to analyze the nonlinear coupling of perturbations in the near-extremal Kerr spacetime, where weakly turbulent behavior has been conjectured to occur. Our method could also be applied to the study of perturbations generated by a pointlike body traveling on a timelike geodesic in Kerr, which is relevant to the extreme mass ratio inspiral problem.

An approach for obtaining integrable Hamiltonians from Poisson-commuting polynomial families

We discuss a general approach permitting the identification of a broad class of sets of Poisson-commuting Hamiltonians, which are integrable in the sense of Liouville. It is shown that all such Hamiltonians can be solved explicitly by a separation of variables ansatz. The method leads in particular to a proof that the so-called “goldfish” Hamiltonian is maximally superintegrable and leads to an elementary identification of a full set of integrals of motion. The Hamiltonians in involution with the “goldfish” Hamiltonian are also explicitly integrated. New integrable Hamiltonians are identified, among which some have the property of being isochronous, that is, all their orbits have the same period. Finally, a peculiar structure is identified in the Poisson brackets between the elementary symmetric functions and the set of Hamiltonians commuting with the “goldfish” Hamiltonian: these can be expressed as products between elementary symmetric functions and Hamiltonians. The structure displays an invariance property with respect to one element and has both a symmetry and a closure property. The meaning of this structure is not altogether clear to the author, but it turns out to be a powerful tool.

The scaled boundary finite element method for the analysis of 3D crack interaction

The scaled boundary finite element method (SBFEM) can be applied to solve linear elliptic boundary value problems when a so-called scaling center can be defined such that every point on the boundary is visible from this scaling center. From a more practical point of view, this means that in linear elasticity, a separation of variables ansatz can be used for the displacements in a scaled boundary coordinate system. This approach allows an analytical treatment of the problem in the scaling direction. Only the boundary needs to be discretized with finite elements. Employment of the separation of variables ansatz in the virtual work balance yields a Cauchy–Euler differential equation system of second order which can be transformed into an eigenvalue problem and solved by standard eigenvalue solvers for nonsymmetric matrices. A further obtained linear equation system serves for enforcing the boundary conditions. If the scaling center is located directly at a singular point, elliptic boundary value problems containing singularities can be solved with high accuracy and computational efficiency. The application of the SBFEM to the linear elasticity problem of two meeting inter-fiber cracks in a composite laminate exposed to a simple homogeneous temperature decrease reveals the presence of hypersingular stresses.

Envelope solitons on anharmonic damped atomic chains

We investigate the influence of dissipation on envelope solitons on anharmonic chains. We consider both Stokes and hydrodynamical damping and derive the evolution equations for the envelope in both the continuum and the quasi-continuum approximation of the chain. We introduce an appropriate collective variable ansatz for the envelope in order to describe the effect of damping on the soliton shape. We derive ordinary differential equations for the evolution of the three collective variables amplitude, width, and chirp which describe the spatial modulation of the envelope. The analytical results are in good agreement with the simulations of the discrete system for high-energy excitations on the chain. Our results derived from the quasi-continuum approximation show significant improvements compared to the continuum approximation.

Behaviour of the nonlinear Dirac soliton in an external field of the Lorentz scalar type

The nonlinear Dirac soliton in an external field of the Lorentz scalar type in (1+1) dimensions is examined by solving the nonlinear Dirac equation numerically. A remarkable feature of the behaviour of the soliton is found, namely, for the linear potential the motion of the soliton centroid is independent of the strength of the nonlinear interaction within the accuracy of the calculation. This feature can be understood by means of a version of the so-called collective variable Ansatz, which leads to an approximate equation of motion for the soliton centroid. It is argued that this feature is characteristic of the case with the linear external potential of the Lorentz scalar type. For the quadratic potential, however, the independence of the motion of the soliton centroid on the strength of the nonlinear interaction is broken very slightly.PACS Nos.: 03.50-w; 03.40.Kf; 11.10.Qr

Nonlinear Dirac soliton in an external field

The behaviour of the nonlinear Dirac soliton in an external potential in 1+1 dimensions is examined by means of a collective variable ansatz and also by solving the nonlinear Dirac equation numerically. When the potential is linear with respect to coordinate x, the motion of the soliton centroid is found to be consistent with the classical relativistic equation of motion for a point particle. For a general potential there is a deviation from the behaviour of the corresponding classical point particle. This deviation can be interpreted as being caused by the finite size of the soliton.

Stationary and axisymmetric perfect fluid solutions with conformal motion

Stationary and axisymmetric perfect fluid metrics are studied under the assumption of the existence of a conformal Killing vector field and in the general case of differential rotation. The possible Lie algebras for the conformal group and corresponding canonical line elements are explicitly given. It turns out that only four different cases appear, the Abelian and three others called I, II and III. We explicitly find all the solutions in the Abelian and I cases. For the Abelian case the general solution depends on an arbitrary function of a single variable and the perfect fluid satisfies the equation of state . This class of metrics is the one presented recently by one of us. The general solution for case I is a new Petrov type D metric, with the velocity vector outside the 2-space spanned by the two principal null directions and a barotropic equation of state . For the cases II and III, the general solution has been found only under the further assumption of a natural separation of variables ansatz. The conformal Killing vectors in the solutions that come out here are, in fact, homothetic. No barotropic equation of state exists in these metrics unless for a new Petrov type D solution belonging to case II and with which cannot be interpreted as an axially symmetric solution and such that the velocity vector points in the direction of one of the Killing vectors. This solution has the previously unknown curious property that both commuting Killing vectors are timelike everywhere.