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The electric dipole polarizabilities and the tune-out wavelengths for the n3S1 (n = 2, 3) states of Be2+ are determined through the application of the relativistic full-configuration-interaction approach. Our calculations directly integrate the mass shift operator into the Dirac–Coulomb–Breit Hamiltonian and further assess the quantum electrodynamics (QED) correction to the dynamic dipole polarizabilities using perturbation theory. The results reveal that the static electric dipole polarizability of the 23S1 and 33S1 states, as well as the 93 nm tune-out wavelength of the 23S1 state and the 238 nm tune-out wavelength of the 33S1 state, exhibits a high sensitivity to QED correction, which exceeds 80 ppm, providing a sensitive test for atomic structure theory. [ABSTRACT FROM AUTHOR]

The electric dipole polarizabilities and the tune-out wavelengths for the n3S1 (n = 2, 3) states of Be2+ are determined through the application of the relativistic full-configuration-interaction approach. Our calculations directly integrate the mass shift operator into the Dirac–Coulomb–Breit Hamiltonian and further assess the quantum electrodynamics (QED) correction to the dynamic dipole polarizabilities using perturbation theory. The results reveal that the static electric dipole polarizability of the 23S1 and 33S1 states, as well as the 93 nm tune-out wavelength of the 23S1 state and the 238 nm tune-out wavelength of the 33S1 state, exhibits a high sensitivity to QED correction, which exceeds 80 ppm, providing a sensitive test for atomic structure theory. [ABSTRACT FROM AUTHOR]

For open-shell atoms and molecules, Kohn–Sham (KS) methods typically resort to spin-polarized approaches that exhibit spin-contamination and often break spatial symmetries. As a result, the KS Hamiltonian operator and the KS orbitals do not exhibit the space and spin symmetry of the physical electron system. The KS formalism can be symmetrized in a rigorous way only in real space, only in spin space, or both in real and spin space. Within such symmetrized KS frameworks, we present exact-exchange-only optimized-effective-potential (OEP) methods that are free of spin contamination and/or spatial symmetry breaking. The effect of symmetrizations on the total energy and its parts and on the exchange potential is analyzed. The presented exact-exchange-only OEP methods may serve as a starting point for high-level symmetrized KS methods based, e.g., on the adiabatic-connection fluctuation-dissipation theorem. [ABSTRACT FROM AUTHOR]

We examine the properties of oblique coordinates. The coordinates, introduced by Zúñiga et al. [J. Phys. B: At., Mol. Opt. Phys. 52, 055101, (2019)], reduce vibrational mode-mixing and enhance the quality of vibrational assignments in quantum mechanical investigations of two-dimensional model Hamiltonians. Oblique coordinates are obtained by making non-orthogonal rotations of the original coordinates that convert the matrix representation of the quadratic Hamiltonian operator into a block-diagonal matrix where the blocks are distinguished by the total quanta of vibrational excitation. Using techniques for the polar decomposition of matrices, we present a scheme for finding these coordinates for systems of arbitrary dimensions. Several molecular examples are presented that highlight the advantages of these coordinates. [ABSTRACT FROM AUTHOR]

In this note, we introduce a notion of the $ \mathcal {J} $ J -kernel of a bounded linear operator on a Krein space and study the $ \mathcal {J} $ J -Fredholm theory for Krein space operators. Using $ \mathcal {J} $ J -Fredholm theory, we discuss and study the $ \mathcal {J} $ J -essential pseudospectra of bounded linear operators on Krein spaces, and some of their properties are studied. Furthermore, we discuss the structure of the $ \mathcal {J} $ J -essential pseudospectra of infinite-dimensional Hamiltonian operators. [ABSTRACT FROM AUTHOR]

For a super Korteweg-de Vries (KdV) equation introduced by Geng and Wu, nonlocal infinitesimal symmetries depending on eigenfunctions of its (adjoint) linear spectral problem are constructed from gradient of the spectral parameter, and one of such symmetries is shown to be related to a nonlocal infinitesimal symmetry of Kupershmidt's super modified KdV equation via a Miura-type transformation. On this basis, a finite symmetry transformation is established for an enlarged system, and leads to a non-trivial exact solution and a Bäcklund transformation of Geng-Wu's super KdV equation. A procedure is explained to generate infinitely many conservation laws. Moreover, these results could be reduced to classical situations, and their bosonic limits are briefly summarized. [ABSTRACT FROM AUTHOR]

Material characterization using high-throughput testing (HTT) techniques has shown great promise in expediting the advancement of high-performance materials. A key step in HTT is the adoption of functionally graded specimens, whose material inhomogeneity, however, imposes a great challenge for analytically evaluating its responsive behaviour under external loads. This work establishes the first symplectic framework for contact analysis of a finite-sized magneto-electro-elastic plane with an exponential material gradient along the horizontal direction. The governing equations are first represented in matrix form in a state space, with the full state vector defined and the operator matrix derived. The operator matrix is found to be distinct from the Hamiltonian operator matrix in the case of a homogeneous medium. With all the eigen-solutions obtained, the Hamiltonian mixed energy variational principle is then introduced to derive the coefficients in the symplectic expansion. The developed analytical solution not only exhibits asymmetry properties but also converges rapidly. The analytical results are validated through comparison with the finite element simulations. The analytical platform established here would work as an effective theoretical basis for the subsequent development of quantitative HTT techniques. [ABSTRACT FROM AUTHOR]

We study nonlocal symmetries of the Degasperis–Procesi equation, which are shown to be closely related to its integrable structure. First, applying the Hamiltonian operator to the gradients of the spectral parameter, we construct nonlocal symmetries of the Kaup–Kupershmidt equation. Next, we show that the nonlocal symmetries can be prolonged to local symmetries for a prolonged system by introducing new dependent variables. Finally, applying the Liouville transformation relating the Degasperis–Procesi and Kaup–Kupershmidt hierarchies, we obtain the corresponding nonlocal symmetries of the Degasperis–Procesi equation. [ABSTRACT FROM AUTHOR]

In this work, the stability of exact smooth multi-solitons for the Dullin-Gottwald-Holm (DGH) equation is investigated for the first time. Firstly, through the inverse scattering approach, we derive the conservation laws in terms of scattering data and multi-solitons related to the discrete spectrum based on the Lax pair of the DGH equation. Notably, a new series Hamiltonian derived from bi-Hamilton structure are equivalent to the conservation laws, and can also be expressed in terms of scattering data. Thus we successfully connect scattering data (multi-solitons) to the recursion operator between Hamiltonian, and to the Lyapunov functional constructed from Hamiltonian. With this Lyapunov functional, it is identified that these smooth multi-solitons serve as non-isolated constrained minimizers, adhering to a suitable variational nonlocal elliptic equation. Moreover, the integrable properties of the recursion operator are presented, which is the crucial for spectral analysis in this work. Consequently, the investigation of dynamical stability transforms into a problem on the spectrum of explicit linearized systems. It is worth noting that, compared to previous works on Camassa-Holm equation and KdV-type equations, recursion operator derived from existing Hamiltonian for the DGH equation do not possess good integrable properties. We construct a new series of Hamiltonians with both analysable recursion operator and simple initial variation derivative (δ H 1 / δ u = m) to overcome this problem. Furthermore, orbital stability of the smooth double solitons is proved at last of the work. [ABSTRACT FROM AUTHOR]

We derive symmetry-based conditions for tight-binding Hamiltonians with flatbands to have compact localized eigenstates (CLS) occupying a single unit cell. The conditions rely on unitary operators commuting with the Hamiltonian and are associated with local symmetries that ensure the existence of orthogonal compact localized states and a flatband. We also demonstrate the inverse: orthogonal flatbands, whose CLS can be made to occupy a single unit cell, always have associated local symmetries that satisfy our conditions.We illustrate our analytical result with several model Hamiltonians with given local symmetries, including examples with finite-range hopping and higher-dimensional ones. [ABSTRACT FROM AUTHOR]

We demonstrate that a pair consisting of a second-order homogeneous Hamiltonian structure in N components and its associated system of conservation laws is in bijective correspondence with an alternating three-form on a N+2 -dimensional vector space. Additionally, we show that the three-form offers N+2 linear equations in the Plücker coordinates that define the associated line congruence. We utilize these results to characterize systems of conservation laws with second-order structure for N≤4. We finally comment on how to extend this result for N=6. [ABSTRACT FROM AUTHOR]

In this work, we investigate a symmetry reduction of the recently discovered (3 + 1)-dimensional equation of the Monge-Ampère type. This equation forms a bi-Hamiltonian system using Magri's theorem when expressed in the two-component form. We select a particular linear combination of the Lie point symmetries belonging to this system to conduct symmetry reduction, resulting in a new (2 + 1)-dimensional system in two-component form. Lagrangian and first Hamiltonian densities are then calculated. We employ Dirac's theory of constraints to obtain symplectic and first Hamiltonian operators. Subsequently, we transform the symmetry condition of the reduced system into a skew-factorized form to determine the recursion operator. Applying the recursion operator to the first Hamiltonian operator yields the second Hamiltonian operator. We demonstrate that the reduced system is a bi-Hamiltonian integrable system in the sense of Magri. Lie point symmetries of the reduced system are identified. Finally, we calculate integrals of motion using the inverse Noether theorem and prove that they have the total divergence form. [ABSTRACT FROM AUTHOR]

Nuclear reactions are complex, with a large number of possible channels. Understanding how different channels contribute to a given reaction is investigated by perturbing the continuous spectrum. Tools are developed to investigate reaction mechanisms by identifying the contributions from each reaction channel. Cluster decomposition methods, along with the spectral theory of proper subsystem problems, is used to identify the part of the nuclear Hamiltonian responsible for scattering into each channel. The result is an expression of the nuclear Hamiltonian as a sum over all scattering channels of channel Hamiltonians. Each channel Hamiltonian is constructed from solutions of proper subsystem problems. Retaining any subset of channel Hamiltonians results in a truncated Hamiltonian where the scattering wave functions for the retained channels differ from the wave functions of the full Hamiltonian by N-body correlations. The scattering operator for the truncated Hamiltonian satisfies an optical theorem in the retained channels. Because different channel Hamiltonians do not commute, how they interact determines their contribution to the full dynamics. [ABSTRACT FROM AUTHOR]

Many approximate solutions of the time-dependent Schrödinger equation can be formulated as exact solutions of a nonlinear Schrödinger equation with an effective Hamiltonian operator depending on the state of the system. We show that Heller's thawed Gaussian approximation, Coalson and Karplus's variational Gaussian approximation, and other Gaussian wavepacket dynamics methods fit into this framework if the effective potential is a quadratic polynomial with state-dependent coefficients. We study such a nonlinear Schrödinger equation in full generality: we derive general equations of motion for the Gaussian's parameters, demonstrate time reversibility and norm conservation, and analyze conservation of energy, effective energy, and symplectic structure. We also describe efficient, high-order geometric integrators for the numerical solution of this nonlinear Schrödinger equation. The general theory is illustrated by examples of this family of Gaussian wavepacket dynamics, including the variational and nonvariational thawed and frozen Gaussian approximations and their special limits based on the global harmonic, local harmonic, single-Hessian, local cubic, and local quartic approximations for the potential energy. We also propose a new method by augmenting the local cubic approximation with a single fourth derivative. Without substantially increasing the cost, the proposed "single-quartic" variational Gaussian approximation improves the accuracy over the local cubic approximation and, at the same time, conserves both the effective energy and symplectic structure, unlike the much more expensive local quartic approximation. Most results are presented in both Heller's and Hagedorn's parametrizations of the Gaussian wavepacket. [ABSTRACT FROM AUTHOR]

A central building block of many quantum algorithms is the diagonalization of Pauli operators. Although it is always possible to construct a quantum circuit that simultaneously diagonalizes a given set of commuting Pauli operators, only resource-efficient circuits can be executed reliably on near-term quantum computers. Generic diagonalization circuits, in contrast, often lead to an unaffordable SWAP gate overhead on quantum devices with limited hardware connectivity. A common alternative is to exclude two-qubit gates altogether. However, this comes at the severe cost of restricting the class of diagonalizable sets of Pauli operators to tensor product bases (TPBs). In this article, we introduce a theoretical framework for constructing hardware-tailored (HT) diagonalization circuits. Our framework establishes a systematic and highly flexible procedure for tailoring diagonalization circuits with ultra-low gate counts. We highlight promising use cases of our framework and – as a proof-of-principle application – we devise an efficient algorithm for grouping the Pauli operators of a given Hamiltonian into jointly-HT-diagonalizable sets. For several classes of Hamiltonians, we observe that our approach requires fewer measurements than conventional TPB approaches. Finally, we experimentally demonstrate that HT circuits can improve the efficiency of estimating expectation values with cloud-based quantum computers. [ABSTRACT FROM AUTHOR]

Analytical expressions for the description of the time evolution of spin systems beyond the product-operator formalism (POF) can be obtained if a low-dimensional subspace of the Liouville space has been found. The latter can be established by a procedure that consists of a repeated application of the commutator of the Hamiltonian with the density operator. This iteration continues as long as the result of such a commutator operation contains a term that is linearly independent of all the operators appearing in the previous commutator operations. The coefficients of the resulting system of commutator relations can be immediately inserted into the generic propagation formulae given in this article if the system contains two, three or four equations. In cases not satisfying the validity conditions of any of these propagation formulae, the coefficients are used as intermediate steps to obtain both the Liouvillian and propagator matrices of the system. Several application examples are given where an analytical equation can be obtained for the description of the time evolution of small spin systems under the influence of two or more interactions. [ABSTRACT FROM AUTHOR]

We introduce a new approach for describing nonstationary quantum systems with a discrete energy spectrum. The essence of this approach is that we describe the evolution of a quantum system in a time-dependent basis. In a sense, this approach is similar to the description of the system in the interaction representation. However, the time dependence of the basic states of the representation is determined not by the evolution operator with a time-independent Hamiltonian but by the eigenstates of the time-dependent Hamiltonian defined at the current time. The time dependence of the basic states of the representation leads to the appearance of an additional term in the Schrödinger equation, which in the case of slowly changing parameters of the Hamiltonian can be considered as a small perturbation. The adiabatic representation is suitable in cases where it is impossible to apply the standard interaction representation. The application of the adiabatic representation is illustrated by the example of two spins connected by a magnetic dipole–dipole interaction in a slowly varying external magnetic field. [ABSTRACT FROM AUTHOR]

This work presents a new method called Dimensionless Fluctuation Balance (DFB), which makes it possible to obtain distributions as solutions of Partial Differential Equations (PDEs). In the first case study, DFB was applied to obtain the Boltzmann PDE, whose solution is a distribution for Boltzmann gas. Following, the Planck photon gas in the Radiation Law, Fermi–Dirac, and Bose–Einstein distributions were also verified as solutions to the Boltzmann PDE. The first case study demonstrates the importance of the Boltzmann PDE and the DFB method, both introduced in this paper. In the second case study, DFB is applied to thermal and entropy energies, naturally resulting in a PDE of Boltzmann's entropy law. Finally, in the third case study, quantum effects were considered. So, when applying DFB with Heisenberg uncertainty relations, a Schrödinger case PDE for free particles and its solution were obtained. This allows for the determination of operators linked to Hamiltonian formalism, which is one way to obtain the Schrödinger equation. These results suggest a wide range of applications for this methodology, including Statistical Physics, Schrödinger's Quantum Mechanics, Thin Films, New Materials Modeling, and Theoretical Physics. [ABSTRACT FROM AUTHOR]

Although the studies on quantum algorithms have been progressing, it is still necessary to broaden the investigation of open quantum systems. In this study, we present the use of an open quantum system to implement a quantum classifier algorithm. Zhang et al. propose a one-QuBit system interacting with the environment through a unitary operator from the Hamiltonian. In our proposal, the input data are loaded into the amplitude of the environment instead of being in the unitary operator. This change positively impacts the performance of different databases tested and causes a difference in the system entanglement behavior. For evaluation, the Zhang et al. proposed models were tested in four real-world datasets and seven other toy problems. The results are evaluated according to accuracy and F1 score. A deeper analysis of the Iris dataset is also done, checking the creation of entanglement and an extensive random search for better parameters on the proposed model. The results show that for most real-world dataset configurations, the proposed model, although having a simpler decision area, performed better than the one inspired by the Zhang et al. model, and that there is no pattern for the system entanglement in the Iris dataset. [ABSTRACT FROM AUTHOR]

It is remarkable that Heisenberg's position-momentum uncertainty relation leads to the existence of a maximal acceleration for a physical particle in the context of a geometric reformulation of quantum mechanics. It is also known that the maximal acceleration of a quantum particle is related to the magnitude of the speed of transportation in projective Hilbert space. In this paper, inspired by the study of geometric aspects of quantum evolution by means of the notions of curvature and torsion, we derive an upper bound for the rate of change of the speed of transportation in an arbitrary finite-dimensional projective Hilbert space. The evolution of the physical system being in a pure quantum state is assumed to be governed by an arbitrary time-varying Hermitian Hamiltonian operator. Our derivation, in analogy to the inequalities obtained by L. D. Landau in the theory of fluctuations by means of general commutation relations of quantum-mechanical origin, relies upon a generalization of Heisenberg's uncertainty relation. We show that the acceleration squared of a quantum evolution in projective space is upper bounded by the variance of the temporal rate of change of the Hamiltonian operator. Moreover, focusing for illustrative purposes on the lower-dimensional case of a single spin qubit immersed in an arbitrarily time-varying magnetic field, we discuss the optimal geometric configuration of the magnetic field that yields maximal acceleration along with vanishing curvature and unit geodesic efficiency in projective Hilbert space. Finally, we comment on the consequences that our upper bound imposes on the limit at which one can perform fast manipulations of quantum systems to mitigate dissipative effects and/or obtain a target state in a shorter time. [ABSTRACT FROM AUTHOR]

For a -component Camassa–Holm equation, as well as a -component generalization of the modified Camassa–Holm equation, nonlocal infinitesimal symmetries quadratically dependent on eigenfunctions of linear spectral problems are constructed from functional gradients of spectral parameters. With appropriate pseudopotentials, these nonlocal infinitesimal symmetries are prolonged to enlarged systems, and then explicitly integrated to generate symmetry transformations in finite form for the enlarged systems. As implementations of these finite symmetry transformations, some kinds of nontrivial solutions and Bäcklund transformations are derived for both equations. [ABSTRACT FROM AUTHOR]

In this article, we compute the Feynman propagator for a simple harmonic oscillator coupled to a constant electric field using Schwinger's method, which is based on the solution of the Heisenberg equations for the position and canonical momentum operators. Such solutions are then used to write the ordered Hamiltonian operator of the position operators ... (O) and ... (t). The utilization of proper operator ordering, along with subsidiary and initial conditions, results in the yield of such a propagator. We found that the propagator obtained is consistent with the one obtained using the Feynman path integral in the work of Poon and Muñoz (Poon & Muñoz 1999). We anticipate that this technique will be advantageous and widely recognized for physics students. [ABSTRACT FROM AUTHOR]

WaveTrain is an open-source software for numerical simulations of chain-like quantum systems with nearest-neighbor (NN) interactions only. The Python package is centered around tensor train (TT, or matrix product) format representations of Hamiltonian operators and (stationary or time-evolving) state vectors. It builds on the Python tensor train toolbox Scikit_tt, which provides efficient construction methods and storage schemes for the TT format. Its solvers for eigenvalue problems and linear differential equations are used in WaveTrain for the time-independent and time-dependent Schrödinger equations, respectively. Employing efficient decompositions to construct low-rank representations, the tensor-train ranks of state vectors are often found to depend only marginally on the chain length N. This results in the computational effort growing only slightly more than linearly with N, thus mitigating the curse of dimensionality. As a complement to the classes for full quantum mechanics, WaveTrain also contains classes for fully classical and mixed quantum–classical (Ehrenfest or mean field) dynamics of bipartite systems. The graphical capabilities allow visualization of quantum dynamics "on the fly," with a choice of several different representations based on reduced density matrices. Even though developed for treating quasi-one-dimensional excitonic energy transport in molecular solids or conjugated organic polymers, including coupling to phonons, WaveTrain can be used for any kind of chain-like quantum systems, with or without periodic boundary conditions and with NN interactions only. The present work describes version 1.0 of our WaveTrain software, based on version 1.2 of scikit_tt, both of which are freely available from the GitHub platform where they will also be further developed. Moreover, WaveTrain is mirrored at SourceForge, within the framework of the WavePacket project for numerical quantum dynamics. Worked-out demonstration examples with complete input and output, including animated graphics, are available. [ABSTRACT FROM AUTHOR]

This research study employs a Lie algebraic framework to investigate the second and third overtone vibrational frequencies and their combination bands in cyclobutane-d8 (C4D8). The application of this framework ensures the preservation of the point symmetry group D2d, characterized by the symmetry species A1, A2, B1, B2, and E. The analysis encompasses 23 normal vibrational modes within the molecular structure of cyclobutane-d8. Our study extensively explores the vibrational spectra, elucidating the intricate interactions among these vibrational modes. Preserving molecular symmetry allows for a deeper understanding of vibrational interactions, offering valuable insights into spectroscopy. The research enhances the comprehension of molecular structure and its applications in various fields, providing a detailed view of higher energy levels and complex vibrational transitions in cyclobutane-d8.

Spin–orbit coupling (SOC) is crucial for correct electronic structure analysis in molecules and materials, for example, in large molecular systems such as superatoms, for understanding the role of transition metals in enzymes, and when investigating the energy transfer processes in metal–organic frameworks. We extend the GFN-xTB method, popular to treat extended systems, by including SOC into the hamiltonian operator. We followed the same approach as previously reported for the density–functional tight-binding method and provide and validate the necessary parameters for all elements throughout the Periodic Table. The parameters have been obtained consistently from atomic SOC calculations using the density–functional theory. We tested them for reference structures where SOC is decisive, as in the transition metal containing heme moiety, chromophores in metal–organic frameworks, and in superatoms. Our parameterization paves the path for incorporation of SOC in the GFN-xTB based electronic structure calculations of computationally expensive molecular systems. [ABSTRACT FROM AUTHOR]

This work develops and illustrates a new method of calculating "chemically accurate" electronic wavefunctions (and energies) via a truncated full configuration interaction (CI) procedure, which arguably circumvents the large matrix diagonalization that is the core problem of full CI and is also central to modern selective CI approaches. This is accomplished simply by following the standard/ubiquitous Davidson method in its "direct" form—wherein, in each iteration, the electronic Hamiltonian operator is applied directly in second quantization to the Ritz vector/wavefunction from the prior iteration—except that (in this work) only a small portion of the resultant expansion vector is actually even computed (through the application of only a similarly small portion of the Hamiltonian). Specifically, at each iteration of this truncated Davidson approach, the new expansion vector is taken to be twice as large as that from the prior iteration. In this manner, a small set of highly truncated expansion vectors (say 10–30) of increasing precision is incrementally constructed, forming a small subspace within which diagonalization of the Hamiltonian yields clear, consistent, and monotonically variational convergence to the approximate full CI limit. The good efficiency in which convergence to the level of chemical accuracy (1.6 mhartree) is achieved suggests, at least for the demonstrated problem sizes—Hilbert spaces of 1018 and wavefunctions of 108 determinants—that this truncated Davidson methodology can serve as a replacement of standard CI and complete-active space approaches in circumstances where only a few chemically significant digits of accuracy are required and/or meaningful in view of ever-present basis set limitations. [ABSTRACT FROM AUTHOR]

We have discussed the Dirac equation in Schwarzschild spacetime using pseudo-supersymmetric quantum mechanics and have obtained the partner Hamiltonian of the initial Hamiltonian operator. We demonstrate that the partner metric tensors, corresponding to these Hamiltonians, can be derived using the intertwining relations inherent in pseudo-supersymmetric approaches. We have seen that the Schwarzschild metric may be expanded into Schwarzschild–Tangherlini metric using the aspects of pseudo-supersymmetry. Furthermore, we have derived approximate solutions for the radial component within the framework of N = 2 supersymmetry with the corresponding radial potential graphs presented. Subsequently, our discussion extends to the quasinormal modes, focusing particularly on the asymptotic limits as r approaches ± ∞ . [ABSTRACT FROM AUTHOR]

We consider the Schrödinger operator with regular short range complex-valued potential in dimension d ≥ 1 . We show that, for d ≥ 2 , the unitarity of scattering operator for this Hamiltonian at high energies implies the reality of the potential (that is Hermiticity of Hamiltonian). In contrast, for d = 1 , we present complex-valued exponentially localized soliton potentials with unitary scattering operator for all positive energies and with unbroken PT symmetry. We also present examples of complex-valued regular short range potentials with real spectrum for d = 3 . Some directions for further research are formulated. [ABSTRACT FROM AUTHOR]

Context: Quantum computing is an exciting area, which has grown at an astonishing rate in the last decade. It is especially promising for the computational and theoretical chemistry area. One algorithm has received a lot of attention lately, the variational quantum eigensolver (VQE). It is used to solve electronic structure problems and it is suitable to the noisy intermediate-scale quantum (NISQ) hardware. VQE calculations require ansatze and one of the most known is the unitary coupled cluster (UCC). It uses the chosen basis set to generate a quantum computing circuit which will be iteratively minimized. The present work investigates the circuit depth and the number of gates as a function of basis sets and molecular size. It has been shown that for the current quantum devices, only the smallest molecules and basis sets are tractable. The H 2 molecule with the cc-pVTZ and aug-cc-pVTZ basis sets have circuit depths in the order of 10 6 to 10 7 gates and the C 2 H 6 molecule with 3–21G basis set has a circuit depth of 2.2 × 10 8 gates. At the same time the analysis demonstrates that the H 2 molecule with STO-3G basis set, requires at least 500 shots to reduce the error and that, although error mitigation schemes can diminish the error, they were not able to completely negate it. Methods: The quantum computing and electronic structure calculations were performed using the Qiskit package from IBM and the PySCF package, respectively. The ansatze were generated using the UCCSD method as implemented in Qiskit, using the basis sets STO-3G, 3–21G, 6–311G(d,p), def2-TZVP, cc-pVDZ, aug-cc-pVDZ, cc-pVTZ, and aug-cc-pVTZ. The operators and the Hamiltonian were mapped using the Jordan-Wigner scheme. The classical optimizer chosen was the simultaneous perturbation stochastic approximation (SPSA). The quantum computers used were the Nairobi and Osaka, with 7 and 127 qubits respectively. [ABSTRACT FROM AUTHOR]

A Hamiltonian system is developed for the three-dimensional (3D) problem of two-dimensional (2D) decagonal piezoelectric quasicrystals via the variational principle. Based on the full state vector and the properties of the Hamiltonian operator matrix, the superposition principle of solutions obtains the symplectic analytical solutions of the problem under simply supported boundary conditions. Numerical examples are illustrated to display the effects of the stacking sequences and material constants on the stresses, displacements, electric potential, and electric displacements under the mechanical and electric displacement loadings. The symplectic analytical solutions presented in the article can be used as a reference for further numerical research. [ABSTRACT FROM AUTHOR]

Advances in quantum information processing can open a way for numerous applications of the processing in various fields of science and technology: communication, precision measurement, computing, nano-scale detectors, and sensors. Classical and various quantum correlations have been studied in real spin 1/2 systems. The nonlocality measures provide a novel classification scheme for bipartite states, highlighting that nonlocality is a quantum resource distinct from other types of quantum correlations. We first studied the temperature and field dependencies of nonlocality measure in the three fictitious spin 1/2 system, which represents a nuclear spin-7/2 placed in magnetic and inhomogeneous electric fields. The relationship between nonlocality and other quantum correlations (entanglement and geometric discord) was studied. The Hamiltonian and spin operators for a spin 7/2 are represented in the basis formed by the Kronecker products of the Pauli matrices. This transformation allows us to represent a spin 7/2 as a system of three coupled fictitious spins 1/2 and, from the quantum information point of view, as an equivalent system of three coupling qubits. Well-developed methods were used to calculate measures of quantum correlations. For example, we consider 59Co (spin 7/2) in the compounds [Co(NH3)5Cl]Cl2 and Ca3Co2O6. The interaction between the fictitious spins of 59Co nuclei depends on the magnitude and direction of the external magnetic field. Other potential cases of quantum correlations can be realized based on nuclear spins 7/2 in solid-state systems, for instance, such as 51V, 165Ho, or 123Sb. [ABSTRACT FROM AUTHOR]

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This study thoroughly investigates the vibrational frequencies of carbonyl sulphide (12C16O32S) and hydrogen cyanide (HCN) up to the fifth harmonic level. It offers comprehensive insights into vibrational modes by using the Hamiltonian operator formalism and concentrating on invariant operators and algebraic parameters with a one-dimensional Lie algebraic method. The findings are significant for atmospheric chemistry, spectroscopy, and quantum chemistry, contributing to a deeper understanding of molecular dynamics. This research sets the groundwork for future studies in comparable compounds and applications.

Heavy element compounds with high symmetries often feature both spin–orbit coupling and vibronic coupling. This is especially true for systems with tetrahedral and octahedral symmetries, whose electronic states may be threefold degenerate and experience complicated Jahn–Teller and pseudo-Jahn–Teller interactions. To accurately describe these interactions, high quality spin–orbit vibronic Hamiltonian operators are needed. In this study, we present a unified one-electron Hamiltonian formalism for spin–orbit vibronic interactions for systems in all tetrahedral and octahedral symmetries. The formalism covers all spin–orbit Jahn–Teller and pseudo-Jahn–Teller problems in the symmetries with arbitrary types and arbitrary numbers of vibrational modes and generates Hamiltonian expansion formulas of arbitrarily high order. [ABSTRACT FROM AUTHOR]

Zombie states are a recently introduced formalism to describe coupled coherent fermionic states that address the fermionic sign problem in a computationally tractable manner. Previously, it has been shown that Zombie states with fractional occupations of spin orbitals obeyed the correct fermionic creation and annihilation algebra and presented results for real-time evolution [D. V. Shalashilin, J. Chem. Phys. 148, 194109 (2018)]. In this work, we extend and build on this formalism by developing efficient algorithms for evaluating the Hamiltonian and other operators between Zombie states and address their normalization. We also show how imaginary time propagation can be used to find the ground state of a system. We also present a biasing method, for setting up a basis set of random Zombie states, that allows much smaller basis sizes to be used while still accurately describing the electronic structure Hamiltonian and its ground state and describe a technique of wave function "cleaning" that removes the contributions of configurations with the wrong number of electrons, improving the accuracy further. We also show how low-lying excited states can be calculated efficiently using a Gram–Schmidt orthogonalization procedure. The proposed algorithm of imaginary time propagation on biased random grids of Zombie states may present an alternative to the existing quantum Monte Carlo methods. [ABSTRACT FROM AUTHOR]

This study introduces a conceptually novel polynomial encoding algorithm for simulating potential energy operators encoded in diagonal unitary forms in a quantum computing machine. The current trend in quantum computational chemistry is effective experimentation to achieve high-precision quantum computational advantage. However, high computational gate complexity and fidelity loss are some of the impediments to the realization of this advantage in a real quantum hardware. In this study, we address the challenges of building a diagonal Hamiltonian operator having exponential functional form, and its implementation in the context of the time evolution problem (Hamiltonian simulation and encoding). Potential energy operators when represented in the first quantization form is an example of such types of operators. Through systematic decomposition and construction, we demonstrate the efficacy of the proposed polynomial encoding method in reducing gate complexity from O (2 n) to O ∑ i = 1 r n C r (for some r ≪ n ). This offers a solution with lower complexity in comparison to the conventional Hadamard basis encoding approach. The effectiveness of the proposed algorithm was validated with its implementation in the IBM quantum simulator and IBM quantum hardware. This study demonstrates the proposed approach by taking the example of the potential energy operator of the sodium iodide molecule (NaI) in the first quantization form. The numerical results demonstrate the potential applicability of the proposed method in quantum chemistry problems, while the analytical bound for error analysis and computational gate complexity discussed, throw light on issues regarding its implementation. [ABSTRACT FROM AUTHOR]

• The symplectic method is applied to derive the analytical solution of micropolar plane annular sector in polar coordinate. • The solution to an eigenproblem of the Hamiltonian operator matrix is obtained in form of the Bessel and power functions. • The analytical solutions of the micropolar plane annular sector under the different boundary conditions are obtained. • Size-dependent behaviors in the micropolar plane annular sector are captured. The symplectic approach is utilized to derive the solutions of the plane annular sector in micropolar elasticity. According to the Hellinger-Reissner variational principle, the Hamiltonian canonical equations are obtained for the plane problem in an annular sectorial region. Through the method of separation of variables, we obtain an eigenvalue equation of the Hamiltonian operator matrix which is a system of linear differential equations with variable coefficients. By the method of series solutions, its general solutions can be expressed as the linear combination of Bessel functions and power functions. The corresponding eigensolutions for three types of homogeneous boundary conditions are derived. According to the adjoint symplectic orthogonality of the eigensolutions, the solution of the micropolar plane annular sector is expressed as a series expansion of eigensolutions. Numerical analyses are presented to demonstrate the validity of the proposed method and confirm its ability to accurately capture the size effect. [ABSTRACT FROM AUTHOR]

The phenomenon of laser-assisted alpha decay is studied within the framework of non-Hermitian quantum theory integrated with the (t, t ′)-formalism. The width and the lifetime of the alpha cluster of a specific isotope is extracted from the spectrum of the complex-scaled Hamiltonian operator of the nuclear system and estimated numerically. The laser-field-induced correction to the lifetime is computed by first-order (t, t ′)-perturbation calculation with regard to different polarization states and control parameters of the external laser field, adjusting appropriate intensity and photon energy pairs by examining the limit of the non-relativistic approximation of the problem. [ABSTRACT FROM AUTHOR]

In quantum metrology, achieving optimal simultaneous multiparameter estimation is of great significance but remains highly challenging. The research approach involving evolution on SU(N)$\mathrm{SU}(N)$ dynamics provides a framework to investigate simultaneous multiparameter estimation within graph states. For single‐parameter estimation, it is observed that the precision limit exceeds the Heisenberg limit in higher‐dimensional SU(2)$\mathrm{SU}(2)$ spin systems. For multiparameter estimation, two scenarios are considered: one with commutative Hamiltonian operators and another with non‐commutative Hamiltonian operators. The results demonstrate that the global estimation precision exceeds the local estimation precision. Under the conditions of parameter limit, the precision of parameter estimation for simultaneously estimating each parameter is equal to that of single‐parameter estimation. Furthermore, a precision‐enhancement scheme has been identified that depends on the dynamics of SU(N)$\mathrm{SU}(N)$. The smaller the value of N$N$ in the dynamic evolution, the higher the precision of the parameter estimation. Finally, it is demonstrated that graph states serve as optimal states in quantum metrology. A set of optimal measurement bases is also identified, and it is illustrated that the precision limit of multiparameter estimation can attain the quantum Cramér‐Rao bound. [ABSTRACT FROM AUTHOR]

We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product. [ABSTRACT FROM AUTHOR]

This research study employs a Lie algebraic framework to investigate the second and third overtone vibrational frequencies and their combination bands in cyclobutane-d8 (C4D8). The application of this framework ensures the preservation of the point symmetry group D2d, characterized by the symmetry species A1, A2, B1, B2, and E. The analysis encompasses 23 normal vibrational modes within the molecular structure of cyclobutane-d8. Our study extensively explores the vibrational spectra, elucidating the intricate interactions among these vibrational modes. Preserving molecular symmetry allows for a deeper understanding of vibrational interactions, offering valuable insights into spectroscopy. The research enhances the comprehension of molecular structure and its applications in various fields, providing a detailed view of higher energy levels and complex vibrational transitions in cyclobutane-d8. [ABSTRACT FROM AUTHOR]

A hyperlink is a finite set of non-intersecting simple closed curves in R 4 ≡ R × R 3 , and each curve is either a matter or geometric loop. We consider an equivalence class of such hyperlinks, up to time-like isotopy, preserving time-ordering. Using an equivalence class and after coloring each matter component loop with an irreducible representation of s u (2) × su (2) , we can define its Wilson loop observable using an Einstein–Hilbert action, which is now thought of as a functional acting on the set containing equivalence classes of hyperlink. Construct a vector space using these functionals, which we now term as quantum states. To make it into a Hilbert space, we need to define a counting probability measure on the space containing equivalence classes of hyperlinks. In our previous work, we defined area, volume and curvature operators, corresponding to given geometric objects like surface and a compact solid spatial region. These operators act on the quantum states and, by deliberate construction of the Hilbert space, are self-adjoint and possibly unbounded operators. Using these operators and Einstein's field equations, we can proceed to construct a quantized stress operator and also a Hamiltonian constraint operator for the quantum system. We will also use the area operator to derive the Bekenstein entropy of a black hole. In the concluding section, we will explain how loop quantum gravity predicts the existence of gravitons, implies causality and locality in quantum gravity and formulates the principle of equivalence mathematically in its framework. [ABSTRACT FROM AUTHOR]

In this paper, we study the mass spectra of heavy quarkonia and heavy-light mesons in a curved space-time with conical geometry induced by a topological defect. The interactions between particles are modeled by an extended version of the Cornell potential, namely the singularized harmonic potential. To compute energy levels as well as the corresponding wave functions, the Hamiltonian operator is constructed within a torsion free space-time with a point-like global monopole. Then the full spectrum is obtained by solving the Schrödinger equation using the bi-confluent Heun equation. It is found that the energy spectra and wave functions of each meson differ from their flat Minkowski space-time counterparts. To check the efficiency of our model, we use the results of calculation to reproduce the mass spectra of heavy quarkonia such as b b ¯ , c c ¯ and some heavy-light mesons such as b ¯ c and c s ¯ . The mass spectra are calculated up to 4 S state at different values of the topological defect, then compared with experimental data and some relevant theoretical works. Decay constants are also computed for the selected mesons. The effect of the topological defect is also analyzed in details. It is observed that, for small values of the parameter α , our predictions approach the experimental data, especially when α = 0.39 . Whereas for α close to unity, the mass spectrum of the system gives values quite far from the experiment. [ABSTRACT FROM AUTHOR]

The scaling function g of the scaled Schrödinger equation (SSE) is generalized to obtain accurate solutions of the Schrödinger equation (SE) with the free complement (FC) theory. The electron–nuclear and electron–electron scaling functions, giA and gij, respectively, are generalized. From the relations between SE and SSE at the inter-particle distances being zero and infinity, the scaling function must satisfy the collisional (or coalescent) condition and the asymptotic condition, respectively. Based on these conditions, general scaling functions are classified into "correct" (satisfying both conditions), "reasonable" (satisfying only collisional condition), and "approximate but still useful" (not satisfying collisional condition) classes. Several analytical scaling functions are listed for each class. Popular functions riA and rij belong to the reasonable class. The qualities of many electron–electron scaling functions are examined variationally for the helium atom using the FC theory. Although the complement functions of FC theory are produced generally from both the potential and kinetic operators in the Hamiltonian, those produced from the kinetic operator were shown to be less important than those produced from the potential operator. Hence, we used only the complement functions produced from the potential operator and showed that the correct-class gij functions gave most accurate results and the reasonable-class functions were less accurate. Among the examined correct and reasonable functions, the conventional function rij was worst in accuracy, although it was still very accurate. Thus, we have many potentially accurate "correct" scaling functions for use in FC theory to solve the SEs of atoms and molecules. [ABSTRACT FROM AUTHOR]

An analytical solution for the bending problem of micropolar plates is derived based on the symplectic approach. By applying Legendre's transformation, we obtain the Hamiltonian canonical equation for the bending problem of a micropolar plate. Utilizing the method of separation of variables, the homogeneous Hamiltonian canonical equation can be transformed into an eigenvalue problem of the Hamiltonian operator matrix. We derive the eigensolutions of the eigenvalue problem for the simply supported, free, and clamped boundary conditions at the two opposite sides. Based on the adjoint symplectic orthogonal relation of the eigensolutions, the solution of the bending problem of the micropolar plate is expressed as a series expansion of eigensolutions. Numerical results confirm the validity of the present approach for the bending problem of micropolar plates under various boundary conditions and demonstrate the capability of the proposed approach to capture the size-dependent behavior of micropolar plates. [ABSTRACT FROM AUTHOR]

We introduce a notion of residual diffeomorphism covariance in quantum Kantowski–Sachs (KS) describing the interior of a Schwarzschild black hole. We solve for the family of Hamiltonian constraint operators satisfying the associated covariance condition, as well as parity invariance, preservation of the Bohr Hilbert space of the Loop Quantum KS and a correct (naïve) classical limit. We further explore the imposition of minimality for the number of terms and compare the solution with those of other Hamiltonian constraints proposed for the Loop Quantum KS in the literature. In particular, we discuss a lapse that was recently commonly chosen due to the resulting decoupling of the evolution of the two degrees of freedom and the exact solubility of the model. We show that such a choice of lapse can indeed be quantized as an operator that is densely defined on the Bohr Hilbert space and that any such operator must include an infinite number of shift operators. [ABSTRACT FROM AUTHOR]

We show that the equivalence classes of deformations of localizable semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura-reciprocal group contain a local representative and are in one-to-one correspondence with the equivalence classes of deformations of local semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura group. [ABSTRACT FROM AUTHOR]

In the present paper, the unique solvability of the direct and inverse source problems for the pseudo-parabolic equation involving the bi-ordinal Hilfer fractional derivative and 2D Landau Hamiltonian operator is considered. Applying the Fourier analysis for the operator Landau Hamiltonian, the theorems of uniqueness and existence of solutions to direct and inverse source problems are proved. In the investigation of the inverse source problem, we have used the value of unknown function at the final time in order to find the right-hand side of the equation. It is also presented the stability result of the inverse problem. [ABSTRACT FROM AUTHOR]

In this work, the impact of screened and decimal potential to the energy spectra and eigenfunctions associated with these energies is investigated in the case of specific, triaxial platinum isotope nuclei namely: 1 9 2 Pt, 1 9 4 Pt and 1 9 6 Pt for numerical results and the plotting of the curves of the potential. To better do this work, we rely on the Extended Nikiforov–Uvarov method and general Heun equation using the Bohr Hamiltonian operator in the presence of a modified fractional Kratzer screened potential. Our calculated energies and associated wave functions here take into account the screening factor and decimal power of potential term. We discuss steeply the impact of the screened parameter k on the spectra of the quantum energies. The results obtained and their transition are closer to the experimental values than those of other researchers with whom we made the comparison. We prove that the power of decimal term influences well the energy. At this stage, the nucleons of nucleus remain at the bottom of the potential. Consequently, there follows a great stability of the nucleus atomic by improving or reducing the decimal power term contained in the potential, one obtained with great instability, that can lead to other atomic phenomena like radioactivity. [ABSTRACT FROM AUTHOR]