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Recently the Rydberg blockade effect has been utilized to realize quantum spin liquid (QSL) on a kagome lattice. Evidence of QSL has been obtained experimentally by directly measuring non-local string order. In this paper, we report a Bardeen–Cooper–Schrieffer (BCS)-type variational wave function study of the spin liquid state in this model. This wave function is motivated by mapping the Rydberg blockade model to a lattice gauge theory, where the local gauge conservations replace the role of constraints from the Rydberg blockade. We determine the variational parameter from the experimental measurement of the Rydberg atom population. Then we compare the predictions of this deterministic wave function with the experimental measurements of non-local string order. Combining the measurements on both open and closed strings, we extract the fluctuations only associated with the closed-loop as an indicator of the topological order. The prediction from our wave function agrees reasonably well with the experimental data, with only one fitting parameter determined by measurement of Rydberg atom population. Our variational wave function provides a simple and intuitive picture of the QSL in this system that can be generalized to similar spin liquid phases in other lattice geometry.

AbstractIn this colloquium, we review the research on excitons in van der Waals heterostructures from the point of view of variational calculations. We first make a presentation of the current and past literature, followed by a discussion on the connections between experimental and theoretical results. In particular, we focus our review of the literature on the absorption spectrum and polarizability, as well as the Stark shift and the dissociation rate. Afterwards, we begin the discussion of the use of variational methods in the study of excitons. We initially model the electron–hole interaction as a soft-Coulomb potential, which can be used to describe interlayer excitons. Using an ansatz, based on the solution for the two-dimensional quantum harmonic oscillator, we study the Rytova–Keldysh potential, which is appropriate to describe intralayer excitons in two-dimensional (2D) materials. These variational energies are then recalculated with a different ansatz, based on the exact wavefunction of the 2D hydrogen atom, and the obtained energy curves are compared. Afterwards, we discuss the Wannier–Mott exciton model, reviewing it briefly before focusing on an application of this model to obtain both the exciton absorption spectrum and the binding energies for certain values of the physical parameters of the materials. Finally, we briefly discuss an approximation of the electron–hole interaction in interlayer excitons as an harmonic potential and the comparison of the obtained results with the existing values from both first-principles calculations and experimental measurements.Graphical abstract

Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. This strategy doubles the size of the state preparation circuits but not their depth, while adding a small number of additional two-qubit gates relative to standard variational quantum eigensolver. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H4 and the π-system of hexatriene (C6H8).

We introduce a systematically improvable family of variational wave functions for the simulation of strongly correlated fermionic systems. This family consists of Slater determinants in an augmented Hilbert space involving \"hidden\" additional fermionic degrees of freedom. These determinants are projected onto the physical Hilbert space through a constraint that is optimized, together with the single-particle orbitals, using a neural network parameterization. This construction draws inspiration from the success of hidden-particle representations but overcomes the limitations associated with the mean-field treatment of the constraint often used in this context. Our construction provides an extremely expressive family of wave functions, which is proved to be universal. We apply this construction to the ground-state properties of the Hubbard model on the square lattice, achieving levels of accuracy that are competitive with those of state-of-the-art variational methods.

We formulate variational material modeling in a space-time context. The starting point is the description of the space-time cylinder and the definition of a thermodynamically consistent Hamilton functional which accounts for all boundary conditions on the cylinder surface. From the mechanical perspective, the Hamilton principle then yields thermo-mechanically coupled models by evaluation of the stationarity conditions for all thermodynamic state variables which are displacements, internal variables, and temperature. Exemplary, we investigate in this contribution elastic wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Therein, one key novel aspect are initial and end time velocity conditions for the wave equation, replacing classical initial conditions for the displacements and the velocities. The motivation is intensively discussed and illustrated with the help of a prototype numerical simulation. From the mathematical perspective, the space-time formulations are formulated within suitable function spaces and convex sets. The unified presentation merges engineering and applied mathematics due to their mutual interactions. Specifically, the chosen models are of high interest in many state-of-the art developments in modeling and we show the impact of this holistic physical description on space-time Galerkin finite element discretization schemes. Finally, we study a specific discrete realization and show that the resulting system using initial and end time conditions is well-posed.

The books Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes and Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles contain various applications of fractional calculus to the fields of classical mechanics.

In this paper, the fourth-order Cahn–Hilliard equation is studied, which plays an important role in the development of the spinodal decomposition, phase separation, and phase ordering dynamics. The -expansion method (TEM), Jacobi elliptic function expansion scheme (JEFES), rational multi wave functions (RMWFs), and decomposition variational iteration method (DVIM) are considered to investigate the exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the domain of applied physics and engineering. This equation can be utilized to explain the contact between the modes in describing the process of phase separation of a binary alloy under the critical temperature, phase separation, phase-ordering dynamics, and spinodal decomposition, fluid mechanics, and fluid flow. The dynamics of the assessed solutions in terms of understanding the real phenomena for such nonlinear model is demonstrated by plotting their 3D, 2D, contour, and density profiles using proper parametric values. Consequently, we obtain distinct types of solutions, containing dark, bright, kink, singular, combo kink singular, and combo dark singular soliton solutions. These results are essential to the explanation of several mesmerizing and intricate physical phenomena. Also, the decomposition variational iteration method of the proposed model is analyzed and conditions are developed accordingly. The soliton solutions demonstrate the competency of the proposed technique in identifying traveling wave solutions, offering a useful tool for tackling a variety of NLEEs. We believe that our results would pave a way for future research generating optical memories based on the nonlinear solitons.

Traditionally, the shallow-water equations have been formulated and developed within the Eulerian framework for studying shallow-water wave problems. In this paper, we present a Lagrangian-based approach based on Hamilton’s variational principle to derive an extended displacement shallow-water equation (EDSWE). Using elliptic functions, we obtain exact traveling wave solutions of the resulting EDSWE. The conditions for the formation of various wave types—including cnoidal waves, looped waves, and peaked waves—are systematically analyzed and summarized. The proposed displacement method, grounded in the Lagrangian description, provides an analytical framework for hydrodynamic problems and can be applied to symplectic formulations in fluid mechanics.

Applying a variational analysis, a minimum-enstrophy vortex in three-dimensional (3-D) fluids with continuous stratification is found, under the quasi-geostrophic hypothesis. The buoyancy frequency is held constant. This vortex is an ideal limiting state in a flow with an enstrophy decay while energy and generalized angular momentum remain fixed. The variational method used to obtain two-dimensional (2-D) minimum-enstrophy vortices is applied here to 3-D integral quantities. The solution from the first-order variation is expanded on a basis of orthogonal spherical Bessel functions. By computing second-order variations, the solution is found to be a true minimum in enstrophy. This solution is weakly unstable when inserted in a numerical code of the quasi-geostrophic equations. After a stage of linear instability, nonlinear wave interaction leads to the reorganization of this vortex into a tripolar vortex. Further work will relate our solution with maximal entropy 3-D vortices.

In this paper, we propose a knowledge distillation inspired variational quantum eigensolver (KD-VQE). Inspired by the virtual distillation process in KD, KD-VQE introduces a virtual annealing mechanism to the VQE framework. In KD-VQE, measurement resources (shots) are dynamically allocated among multiple trial wavefunctions, each weighted according to a Boltzmann distribution with a virtual temperature. As the temperature decreases gradually, the algorithm progressively reallocates resources toward lower-energy candidates, effectively filtering out suboptimal states and steering the system toward the global minimum. We demonstrate the effectiveness of KD-VQE through numerical simulations on the two-site Fermi–Hubbard model and the one-dimensional hydrogen chain. Compared to conventional VQE, KD-VQE shows improved convergence and reduced sensitivity to poor initializations.