Quantum Statistical Mechanics Research Articles

Investigation of the complex dynamical structure of bifurcation and dark soliton solutions to fractional generalized double [formula omitted]-Gordon equation

In the current research work, the classical wave equation is combined with a nonlinear sinh source term in the sinh-Gordon equation. It has been used in several scientific domains such as differential geometry theory, integrable quantum field theory, kink dynamics, and statistical mechanics. It makes more comprehensible the dynamics of strings and multi-strings in the constant curvature space. The current study has three main objectives. Examine the governing model to get novel solutions by employing the modified Kudryashov technique. Then compare it with the numerical technique of modified variational iteration method (MVIM) to calculate the error terms. Furthermore, employing bifurcation theory to produce a dynamical system. Additionally, use the dynamical system’s sensitivity analysis to investigate the model’s sensitivity. At last, for the validation of acquired results, the cryptography technique of novel image encryption and decryption is used. The research is greatly enhanced by the presentation of thorough 2D and 3D phase portraits. The field of mathematics and other sciences will benefit from these discoveries.

A formulation of quantum fluid mechanics and trajectories

A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schrödinger equation, respectively, where the Schrödinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schrödinger equations that are a generalization of the Hartree–Fock equations are also obtained. These equations contain a quantum Coulomb’s law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.

Simulating quantum field theories on continuous-variable quantum computers

We delve into the use of photonic quantum computing to simulate quantum mechanics and extend its application towards quantum field theory. We develop and prove a method that leverages this form of continuous-variable quantum computing (CVQC) to reproduce the time evolution of quantum-mechanical states under arbitrary Hamiltonians, and we demonstrate the method's remarkable efficacy with various potentials. Our method centers on constructing an , a specially prepared quantum state that induces the desired time evolution on the target state. This is achieved by introducing a non-Gaussian operation using a measurement-based quantum computing approach, enhanced by machine learning. Furthermore, we propose a framework in which these methods can be extended to encode field theories in CVQC without discretizing the field values, thus preserving the continuous nature of the fields. This opens new avenues for quantum computing applications in quantum field theory. Published by the American Physical Society 2024

Simulating quantum field theories on continuous-variable quantum computers

We delve into the use of photonic quantum computing to simulate quantum mechanics and extend its application towards quantum field theory. We develop and prove a method that leverages this form of continuous-variable quantum computing (CVQC) to reproduce the time evolution of quantum-mechanical states under arbitrary Hamiltonians, and we demonstrate the method's remarkable efficacy with various potentials. Our method centers on constructing an , a specially prepared quantum state that induces the desired time evolution on the target state. This is achieved by introducing a non-Gaussian operation using a measurement-based quantum computing approach, enhanced by machine learning. Furthermore, we propose a framework in which these methods can be extended to encode field theories in CVQC without discretizing the field values, thus preserving the continuous nature of the fields. This opens new avenues for quantum computing applications in quantum field theory. Published by the American Physical Society 2024

Canonical Typicality for Other Ensembles than Micro-canonical

AbstractWe generalize Lévy’s lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a much more general class of measures, so-called GAP measures. For any given density matrix $$\rho $$ ρ on a separable Hilbert space $${\mathcal {H}}$$ H , $${\textrm{GAP}}(\rho )$$ GAP ( ρ ) is the most spread-out probability measure on the unit sphere of $${\mathcal {H}}$$ H that has density matrix $$\rho $$ ρ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue $$\Vert \rho \Vert $$ ‖ ρ ‖ of $$\rho $$ ρ is small. We use this fact to generalize and improve well-known and important typicality results of quantum statistical mechanics to GAP measures, namely canonical typicality and dynamical typicality. Canonical typicality is the statement that for “most” pure states $$\psi $$ ψ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a $$\psi $$ ψ -independent matrix. Dynamical typicality is the statement that for any observable and any unitary time evolution, for “most” pure states $$\psi $$ ψ from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state $$\psi _t$$ ψ t is very close to a $$\psi $$ ψ -independent distribution. So far, canonical typicality and dynamical typicality were known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble, and for rather special mean-value ensembles. Our result shows that these typicality results hold also for $${\textrm{GAP}}(\rho )$$ GAP ( ρ ) , provided the density matrix $$\rho $$ ρ has small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles.

Analysis of HERA data with a PDF parametrization inspired by quantum statistical mechanics

We present a determination of the parton distribution functions (PDFs) of the proton from HERA data using a PDF parametrization inspired by a quantum statistical model of the proton dynamics. This parametrization is characterised by a very small number of parameters, yet it leads to a reasonably good description of the data, comparable with other parametrizations on the market. It may thus provide an alternative to standard parametrizations, useful for studying parametrization bias and to possibly simplify the fit procedure thanks to the small number of parameters. Interestingly, the model reproduces key physical features, such as a d¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{d}$$\\end{document} distribution larger than u¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{u}$$\\end{document}, that HERA data alone are not able to constrain when using more flexible parametrizations. Moreover, polarized distributions are described in the model by the same parameters of the unpolarized ones, giving us the possibility of extracting both types of distributions within the same fit.

Gauge Field Marginal of an Abelian Higgs Model

We study the gauge field marginal of an Abelian Higgs model with Villain action defined on a 2D lattice in finite volume. Our first main result, which holds for gauge theories on arbitrary finite graphs and does not assume that the structure group is Abelian, is a loop expansion of the Radon–Nikodym derivative of the law of the gauge field marginal with respect to that of the pure gauge theory. This expansion is similar to the one of Seiler (Gauge theories as a problem of constructive quantum field theory and statistical mechanics, volume 159 of lecture notes in physics, Springer, Berlin, p v+192. https://doi.org/10.1007/3-540-11559-5, 1982) but holds in greater generality and uses a different graph theoretic approach. Furthermore, we show ultraviolet stability for the gauge field marginal of the model in a fixed gauge. More specifically, we show that moments of the Hölder–Besov-type norms introduced in Chevyrev (Commun Math Phys 372(3):1027–1058. https://doi.org/10.1007/s00220-019-03567-5, 2019) are bounded uniformly in the lattice spacing. This latter result relies on a quantitative diamagnetic inequality that in turn follows from the loop expansion and elementary properties of Gaussian random variables.

Driven generalized quantum Rayleigh-van der Pol oscillators: Phase localization and spectral response.

Driven classical self-sustained oscillators have been studied extensively in the context of synchronization. Using the master equation, this work considers the classically driven generalized quantum Rayleigh-van der Pol oscillator, which is characterized by linear dissipative gain and loss terms as well as three nonlinear dissipative terms. Since two of the nonlinear terms break the rotational phase space symmetry, the Wigner distribution of the quantum mechanical limit cycle state of the undriven system is, in general, not rotationally symmetric. The impact of the symmetry-breaking dissipators on the long-time dynamics of the driven system are analyzed as functions of the drive strength and detuning, covering the deep quantum to near-classical regimes. Phase localization and frequency entrainment, which are required for synchronization, are discussed in detail. We identify a large parameter space where the oscillators exhibit appreciable phase localization but only weak or no entrainment, indicating the absence of synchronization. Several observables are found to exhibit the analog of the celebrated classical Arnold tongue; in some cases, the Arnold tongue is found to be asymmetric with respect to vanishing detuning between the external drive and the natural oscillator frequency.

A hybrid quantum ensemble learning model for malicious code detection

Quantum computing as a new computing model with parallel computing capability and high information carrying capacity, has attracted a lot of attention from researchers. Ensemble learning is an effective strategy often used in machine learning to improve the performance of weak classifiers. Currently, the classification performance of quantum classifiers is not satisfactory enough due to factors such as the depth of quantum circuit, quantum noise, and quantum coding method, etc. For this reason, this paper combined the ensemble learning idea and quantum classifiers to design a novel hybrid quantum machine learning model. Firstly, we run the Stacking method in classical machine learning to realize the dimensionality reduction of high-latitude data while ensuring the validity of data features. Secondly, we used the Bagging method and Bayesian hyperparameter optimization method applied to quantum support vector machine (QSVM), quantum K nearest neighbors (QKNN), variational quantum classifier (VQC). Thirdly, the voting method is used to ensemble the predict results of QSVM, QKNN, VQC as the final result. We applied the hybrid quantum ensemble machine learning model to malicious code detection. The experimental results show that the classification precision (accuracy, F1-score) of this model has been improved to 98.9% (94.5%, 94.24%). Combined with the acceleration of quantum computing and the higher precision rate, it can effectively deal with the growing trend of malicious codes, which is of great significance to cyberspace security.

Quantifying the quantumness of pure-state ensembles via coherence of Gram matrix

The Gram matrix of a set of pure states plays a crucial role in exploring the information content of quantum states. Since the Gram matrix of a pure-state ensemble can be regarded as a quantum state, we can quantify the quantumness of the corresponding quantum ensemble by means of the coherence of the Gram matrix. In particular, we introduce two natural measures of quantumness of an ensemble by means of the coherence of the Gram matrix based on Hilbert-Schmidt distance and skew information, respectively, and investigate their fundamental properties. In addition, notice that the average coherence and the maximum coherence are independent of the choice of a reference basis, we further propose to use the average coherence and the maximum coherence of the Gram matrix to quantify the generalized quantumness of the corresponding ensemble, and find that the generalized quantumness is not only related to the quantum states in the ensemble, but also to the probability distribution. We discuss the validity of these quantumness measures by calculating some important pure-state ensembles in quantum measurement and quantum cryptography. Finally, we compare these quantifiers with other existing quantifiers in the literatures and illustrate that different quantifiers can yield different features and quantumness orderings for quantum ensembles.

Stability Enhancement by Zero‐Point Spin Fluctuations: A Quantum Perspective on Bloch Point Topological Singularities

AbstractBloch points represent singularities within magnetic materials. From a macroscopic viewpoint, their cores are points where the magnetization vector is undefined, resulting in unique topological characteristics that influence the magnetic behavior of their hosts. The picture is very different at the microscopic level, where quantum effects enter the scene. The spin variables' quantum dynamics effect on the BP's stability is revealed. Zero‐point fluctuations, intrinsic fluctuations within the quantum mechanical ground state originating from the uncertainty principle, play a fundamental role. It is found that quantum fluctuations bloom in the vicinity of the singularity, thereby reducing the effective magnetic moment in its neighborhood. This increases the overall stability of the BP. These methods also allow for a characterization of the magnonic eigenmodes surrounding and bound to the singularity. The latter leads to predict on quite general grounds several features of the magnonic spectra, its degeneration structure, and its splitting response under a magnetic field. The last result is coherent with the association of a magnetic moment to the orbital angular momentum of the magnons. This approach allows integration with multiscale algorithms to provide a realistic description of generic topological singularities.

Quantum statistical mechanics from a Bohmian perspective

We develop a general formulation of quantum statistical mechanics in terms of probability currents that satisfy continuity equations in the multi-particle position space, for closed and open systems with a fixed number of particles. The continuity equation for any closed or open system suggests a natural Bohmian interpretation in terms of microscopic particle trajectories, that make the same measurable predictions as standard quantum theory. The microscopic trajectories are not directly observable, but provide a general, simple and intuitive microscopic interpretation of macroscopic phenomena in quantum statistical mechanics. In particular, we discuss how various notions of entropy, proper and improper mixtures, and thermodynamics are understood from the Bohmian perspective.

Quantum statistical mechanics and the boundary of modular curves

The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of GL2 and an analog for SL2. The structure of equilibrium states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.

Cellular automaton ontology, bits, qubits and the Dirac equation

Cornerstones of the Cellular Automaton Interpretation of Quantum Mechanics are its ontological states that evolve by permutations, in this way never creating would-be quantum mechanical superposition states. We review and illustrate this with a classical Ising spin chain. It is shown that it can be related to the Weyl equation in the continuum limit. Yet, the model of discrete spins or bits unavoidably becomes a model of qubits by generating superpositions, if only slightly deformed. We study modifications of its signal velocity which, however, do not relate to mass terms. To incorporate the latter, we consider the Dirac equation in 1+1 dimensions and sketch an underlying discrete deterministic “necklace of necklaces” automaton that qualifies as ontological.

High-pressure high-temperature EXAFS and Debye–Waller factor of platinum

The temperature and pressure effects on the mean-square relative displacement (MSRD) corresponding to the extended X-ray absorption fine structure (EXAFS) Debye–Waller factor of platinum have been studied based on the statistical moment method in quantum statistical mechanics. We implement numerical calculations for platinum up to pressure of 400 GPa. Examining the temperature effects on the EXAFS Debye–Waller factor at ambient pressure we show that theoretical MSRD curve is in good agreement with the previous measurement and calculations up to nearly 800 K. When pressure increases, our pressure-dependent MSRD curve follows very well with the prediction based on the numerical integration along the Hugoniot. Using the theoretical EXAFS Debye–Waller factor, we obtain the Hugoniot temperatures which are highly consistent with recent experimental results. This research advances the correlated Einstein model on the calculation of MSRDs of materials at high temperature and high pressure, specifically, it includes the zero-point vibrations at low temperature and contains the anharmonicity contributions caused by thermal lattice vibrations at high temperature.

Single-Shot Factorization Approach to Bound States in Quantum Mechanics

Single-Shot Factorization Approach to Bound States in Quantum Mechanics

The set of basis functions generated by Pearson type IV distributions and its application to problems of statistical data analysis and quantum mechanics

На примере распределений Пирсона IV типа предложена процедура дополнения классического распределения вероятностей до квантового состояния. Получена волновая функция, отвечающая распределениям Пирсона IV типа, а также построен соответствующий набор базисных функций. Продемонстрировано применение разработанного метода к задачам статистического анализа данных и квантовой механики. Показана эффективность разработанного подхода при аппроксимации статистических распределений с тяжелыми хвостами.

Underpinnings behind the magnetic order-to-disorder transition and property anomaly of disproportionated insulating samarium nickelate

Perovskite nickelates (RNiO3), as benchmark materials for exploring underlying physics of phase transitions in strong correlated systems, are well-known for their rich structural characteristics and physicochemical properties. Understanding phase transition from the spin-ordered antiferromagnetic (AFM) state to the spin-disordered paramagnetic (PM) state of RNiO3 requires accurate models at finite temperatures. Here, we investigate the underpinning physics behind the property anomaly and AFM-to-PM transition in samarium nickelate (SmNiO3) using a thermodynamic zentropy approach, which depicts the temperature-dependent total entropy for a system of interest via a nested formula through integration of quantum mechanics and statistical mechanics. The SmNiO3 displays the ground-state of AFM insulator as well as the PM metal and PM insulator at elevated temperatures. It turns out that the magnetic configurational entropy induced by competition among different spatially fluctuating polymorphous spin configurations, contributes to the specific-heat anomaly and the phase transition of SmNiO3 from AFM to PM state. The predicted Néel temperature of 241±25 K for SmNiO3 is in satisfactory agreement with the experimental value. Unlike empirical interpretations in literature, the present work indicates that the magnetic order-to-disorder transition in SmNiO3 is associated with the magnetic configurational entropy due to thermal mixture of multiple spin configurations at elevated temperatures, which can be quantitively captured by the first-principles based thermodynamic zentropy approach.

Towards the generation of mechanical Kerr-cats: awakening the perturbative quantum Moyal corrections to classical motion

The quantum to classical transition is determined by the interplay of a trio of parameters: dissipation, nonlinearity, and macroscopicity. Why is nonlinearity needed to see quantum effects? And, is not an ordinary pendulum quite nonlinear already? In this manuscript, we discuss the parameter regime where the dynamics of a massive oscillator should be quantum mechanical in the presence of dissipation. We review the outstanding challenge of the dynamical generation of highly quantum mechanical cat states of a massive ‘pendulum’, known as Kerr-cats. We argue that state-of-the-art cold atom experiments may be in a position to reach such a nonlinear regime, which today singles out superconducting quantum circuits. A way to stabilize Schrödinger cat superpositions of a mechanical atomic oscillator via parametric squeezing and further protected by an unusual form of quantum interference is discussed. The encoding of a neutral atom Kerr-cat qubit is proposed.

Generative model for learning quantum ensemble with optimal transport loss

Generative modeling is an unsupervised machine learning framework, that exhibits strong performance in various machine learning tasks. Recently, we find several quantum versions of generative model, some of which are even proven to have quantum advantage. However, those methods are not directly applicable to construct a generative model for learning a set of quantum states, i.e., ensemble. In this paper, we propose a quantum generative model that can learn quantum ensemble, in an unsupervised machine learning framework. The key idea is to introduce a new loss function calculated based on optimal transport loss, which have been widely used in classical machine learning due to its good properties; e.g., no need to ensure the common support of two ensembles. We then give in-depth analysis on this measure, such as the scaling property of the approximation error. We also demonstrate the generative modeling with the application to quantum anomaly detection problem, that cannot be handled via existing methods. The proposed model paves the way for a wide application such as the health check of quantum devices and efficient initialization of quantum computation.