Problems In Quantum Physics Research Articles

Certifying Ground-State Properties of Many-Body Systems

A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all that variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin-1/2 Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way. Published by the American Physical Society 2024

On a sharper bound on the stability of non-autonomous Schrödinger equations and applications to quantum control

We study the stability of the Schrödinger equation generated by time-dependent Hamiltonians with constant form domain. That is, we bound the difference between solutions of the Schrödinger equation by the difference of their Hamiltonians. The stability theorem obtained in this article provides a sharper bound than those previously obtained in the literature. This makes it a potentially useful tool for time-dependent problems in Quantum Physics, in particular for Quantum Control. We apply this result to prove two theorems about global approximate controllability of infinite-dimensional quantum systems. These results improve and generalise existing results on infinite-dimensional quantum control.

Spectral methods on the hydrogen atom

This study investigates the computational implementation of spectral theory and its applications in analyzing complex mathematical problems. It explores the use of modern programming languages and scientific libraries for implementing and visualizing complex mathematical concepts, particularly focusing on spectral theory within quantum physics. The research employs computational methods to address the challenges in interpreting spectral properties, utilizing the Lanczos spectral method for eigenvalue calculation in large, sparse matrices. The results illustrate the effectiveness of these computational techniques in visualizing quantum states, demonstrating the potential of advanced programming in understanding and solving intricate problems in quantum physics and spectral graph theory. The study's findings are significant in bridging computational methods with theoretical spectral analysis, offering a new perspective on the application of computational techniques in scientific research.

When Bose wrote to Einstein

In 1924 an Indian physicist called Satyendra Nath Bose wrote to Albert Einstein saying he had solved a problem in quantum physics that had stumped the great man. One century on, Robert P Crease and Gino Elia explain how the correspondence led to the notion of Bose–Einstein condensation and why it revealed the power of diverse thinking.

Improved machine learning algorithm for predicting ground state properties

Finding the ground state of a quantum many-body system is a fundamental problem in quantum physics. In this work, we give a classical machine learning (ML) algorithm for predicting ground state properties with an inductive bias encoding geometric locality. The proposed ML model can efficiently predict ground state properties of an n-qubit gapped local Hamiltonian after learning from only O(log(n))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{O}}}}}}}}(\\log (n))$$\\end{document} data about other Hamiltonians in the same quantum phase of matter. This improves substantially upon previous results that require O(nc)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{O}}}}}}}}({n}^{c})$$\\end{document} data for a large constant c. Furthermore, the training and prediction time of the proposed ML model scale as O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{O}}}}}}}}(n\\log n)$$\\end{document} in the number of qubits n. Numerical experiments on physical systems with up to 45 qubits confirm the favorable scaling in predicting ground state properties using a small training dataset.

The Pauli problem and wave function lifting: reconstruction of quantum states from physical observables

<p>In this paper we consider the problem to determine a unique quantum state from given distribution of position (density) and momentum density of particles, namely the so called Pauli problem in quantum physics. In the first part, we will review the method of wave function lifting developed in <sup>[<xref ref-type="bibr" rid="b4">4</xref>,<xref ref-type="bibr" rid="b5">5</xref>]</sup> to construct a complex wave function $ \psi\in H^s({{\mathbf R}}^d) $, $ s = 1, 2 $, associated to given density and momentum density. The second part is focused on the dynamical version of the wave function lifting, namely we study the relation between solutions to quantum fluid models and wave functions solving the nonlinear Schrödinger equation. The uniqueness of the lifted wave function is essentially related to the structure of vacuum regions of the position density.</p>

Certificates of quantum many-body properties assisted by machine learning

Computationally intractable tasks are often encountered in physics and optimization. Such tasks often comprise a cost function to be optimized over a so-called feasible set, which is specified by a set of constraints. This may yield, in general, to difficult and non-convex optimization tasks. A number of standard methods are used to tackle such problems: variational approaches focus on parameterizing a subclass of solutions within the feasible set; in contrast, relaxation techniques have been proposed to approximate it from outside, thus complementing the variational approach by providing ultimate bounds to the global optimal solution. In this work, we propose a novel approach combining the power of relaxation techniques with deep reinforcement learning in order to find the best possible bounds within a limited computational budget. We illustrate the viability of the method in the context of finding the ground state energy of many-body quantum systems, a paradigmatic problem in quantum physics. We benchmark our approach against other classical optimization algorithms such as breadth-first search or Monte-Carlo, and we characterize the effect of transfer learning. We find the latter may be indicative of phase transitions, with a completely autonomous approach. Finally, we provide tools to generalize the approach to other common applications in the field of quantum information processing.

An alternating shifted inverse power method for the extremal eigenvalues of fourth-order partially symmetric tensors

Extremal M-eigenvalues of fourth-order partially symmetric tensors play an important role in the nonlinear elastic material analysis and the entanglement problem in quantum physics. In this paper, we propose an alternating shifted inverse power method for computing the extremal M-eigenvalues of fourth-order partially symmetric tensors. The proposed algorithm is simple to operate and easy to understand for convergence analysis. Numerical experiments show the effectiveness of the proposed method.

Sequential generalized measurements: Asymptotics, typicality, and emergent projective measurements

The relation between projective measurements and generalized quantum measurements is a fundamental problem in quantum physics, and clarifying this issue is also important to quantum technologies. While it has been intuitively known that projective measurements can be constructed from sequential generalized or weak measurements, there is still lack of a proof of this hypothesis in general cases. Here we prove it from the perspective of quantum channels. We show that projective measurements naturally arise from sequential generalized measurements in the asymptotic limit. Specifically, a selective projective measurement arises from a set of typical sequences of selective generalized measurements. We provide an explicit scheme to construct projective measurements of a quantum system with sequential generalized measurements. Remarkably, a single ancilla qubit is sufficient to mediate sequential generalized measurements for constructing arbitrary projective measurements of a generic system.

МОДЕЛЬ ОРГАНІЗАЦІЇ ОСВІТНЬОГО ПРОЦЕСУ ПРИ ВИВЧЕННІ СУЧАСНИХ ПРОБЛЕМ КВАНТОВОЇ ФІЗИКИ

The article examines the problem of ensuring the effectiveness of higher education in Ukraine in the training of specialists in the field of natural sciences, in particular, the possibility of increasing the level of teaching of the discipline “Physics”. The importance of science and high technologies for the protection of Ukrainian sovereignty and the inadmissibility of the outflow of young personnel from the country were emphasized. It was noted that solving this problem requires modernization of the educational process, especially when studying physics as the basis of natural science knowledge, without a thorough study of which it is impossible to train a qualified specialist in the field of any natural science. It was determined that in the structure of the physics course of higher education institutions, the course “Quantum physics” occupies a special place, since the knowledge of quantum physics realized and learned at the appropriate level is important for the formation of students’ general scientific picture of the world. It has been established that one of the most important goals in the education of quantum physics is the systematic familiarization of students with the possibilities and prospects of quantum technologies, as well as the theoretical problems of quantum physics, the solution of which can change the idea of the world structure. To increase the productivity of students’ assimilation of modern problems of quantum physics, a methodical model of the organization of the educational process is proposed using the example of the problem of quark confinement.

Newton-Type Methods for Solving Equations in Banach Spaces: A Unified Approach

A plethora of quantum physics problems are related to symmetry principles. Moreover, by using symmetry theory and mathematical modeling, these problems reduce to solving iteratively finite differences and systems of nonlinear equations. In particular, Newton-type methods are introduced to generate sequences approximating simple solutions of nonlinear equations in the setting of Banach spaces. Specializations of these methods include the modified Newton method, Newton’s method, and other single-step methods. The convergence of these methods is established with similar conditions. However, the convergence region is not large in general. That is why a unified semilocal convergence analysis is developed that can be used to handle these methods under even weaker conditions that are not previously considered. The approach leads to the extension of the applicability of these methods in cases not covered before but without new conditions. The idea is to replace the Lipschitz parameters or other parameters used by smaller ones to force convergence in cases not possible before. It turns out that the error analysis is also extended. Moreover, the new idea does not depend on the method. That is why it can also be applied to other methods to also extend their applicability. Numerical applications illustrate and test the convergence conditions.

Extending the Applicability of Cordero Type Iterative Method

Symmetries play a vital role in the study of physical systems. For example, microworld and quantum physics problems are modeled on the principles of symmetry. These problems are then formulated as equations defined on suitable abstract spaces. Most of these studies reduce to solving nonlinear equations in suitable abstract spaces iteratively. In particular, the convergence of a sixth-order Cordero type iterative method for solving nonlinear equations was studied using Taylor expansion and assumptions on the derivatives of order up to six. In this study, we obtained order of convergence six for Cordero type method using assumptions only on the first derivative. Moreover, we modified Cordero’s method and obtained an eighth-order iterative scheme. Further, we considered analogous iterative methods to solve an ill-posed problem in a Hilbert space setting.

Prime indicators of current teaching methodologies and students’ perceptions in quantum physics

<span lang="EN-US">With the growing revolution in the education system, imparting quality education has become a priority for instructors and students in any educational institution. The challenge grows high when it comes to teaching advanced subjects like quantum physics. The present study considered the case study of the University of Rwanda College of Education. It investigated the impact of current teaching methodologies and teaching staff and students’ perceptions in quantum physics for a quality knowledge delivery system. It employed a quantitative method of data collection. About 300 students and ten lecturers participated in this study. Data obtained from this study were analyzed using descriptive quantitative analysis. Students suggested that multimedia tools in teaching and learning quantum physics can improve the understanding of concepts and help solve complex mathematical problems in quantum physics. Likewise, the teaching staff finds quantum physics more comprehensive and effective when appropriate multimedia tools are used. The investigation outcome is meant multimedia utilization can improve quantum physics’ teaching and learning experience at Rwanda high educational institutions.</span>

Koopman analysis of quantum systems* *Contribution to the Special Collection Koopman Methods in Classical and Quantum-Classical Mechanics’.

Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson’s stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue toward solving Schrödinger’s equation using recently developed tools from data science.

Locality of three-qubit Greenberger-Horne-Zeilinger-symmetric states

The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides threequbit states into separable, biseparable, W, and Greenberger-Horne-Zeilinger (GHZ) classes. The W and GHZ are two subclasses of genuine tripartite entanglement.We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both theW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.

Deep Learning of Quantum Many-Body Dynamics via Random Driving

Neural networks have emerged as a powerful way to approach many practical problems in quantum physics. In this work, we illustrate the power of deep learning to predict the dynamics of a quantum many-body system, where the training is based purely on monitoring expectation values of observables under random driving. The trained recurrent network is able to produce accurate predictions for driving trajectories entirely different than those observed during training. As a proof of principle, here we train the network on numerical data generated from spin models, showing that it can learn the dynamics of observables of interest without needing information about the full quantum state. This allows our approach to be applied eventually to actual experimental data generated from a quantum many-body system that might be open, noisy, or disordered, without any need for a detailed understanding of the system. This scheme provides considerable speedup for rapid explorations and pulse optimization. Remarkably, we show the network is able to extrapolate the dynamics to times longer than those it has been trained on, as well as to the infinite-system-size limit.

A hybrid sinc-Galerkin/finite-difference method for the time-dependent Wigner equation

The Wigner equation is a remarkable tool to model complex problems of quantum physics in phase space. The main objective of this paper is to propose a new hybrid algorithm for the time-dependent Wigner equation. This scheme is based on sinc-Galerkin and finite difference approximations and is moderately simple but highly efficient. Error estimation, stability, and convergence are also investigated concretely. Numerical experiments validate the theoretical results and present the reliability and efficiency of the proposed algorithm to simulate quantum effects.

On the Semi-Local Convergence of an Ostrowski-Type Method for Solving Equations

Symmetries play a crucial role in the dynamics of physical systems. As an example, microworld and quantum physics problems are modeled on principles of symmetry. These problems are then formulated as equations defined on suitable abstract spaces. Then, these equations can be solved using iterative methods. In this article, an Ostrowski-type method for solving equations in Banach space is extended. This is achieved by finding a stricter set than before containing the iterates. The convergence analysis becomes finer. Due to the general nature of our technique, it can be utilized to enlarge the utilization of other methods. Examples finish the paper.

Separability of Hermitian tensors and PSD decompositions

Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e. it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect the separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it. Moreover, we study psd decompositions of separable Hermitian tensors. When the psd rank is low, we first flatten them into cubic order tensors and then apply tensor decomposition methods to compute psd decompositions. We prove that this method works well if the psd rank is low. In computation, this flattening approach can detect the separability for much larger-sized Hermitian tensors. This method is a good starting point to determine psd ranks of separable Hermitian tensors.

Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry

We derive symmetric and antisymmetric kernels by symmetrizing and antisymmetrizing conventional kernels and analyze their properties. In particular, we compute the feature space dimensions of the resulting polynomial kernels, prove that the reproducing kernel Hilbert spaces induced by symmetric and antisymmetric Gaussian kernels are dense in the space of symmetric and antisymmetric functions, and propose a Slater determinant representation of the antisymmetric Gaussian kernel, which allows for an efficient evaluation even if the state space is high-dimensional. Furthermore, we show that by exploiting symmetries or antisymmetries the size of the training data set can be significantly reduced. The results are illustrated with guiding examples and simple quantum physics and chemistry applications.