Learning a Local Hamiltonian from Local Measurements.

Analytical solution for one-dimensional advection–dispersion transport equation with distance-dependent coefficients

In the study of the predictability of great earthquakes in the perspective of seismicity analysis, two issues are presently controversial, and need more detailed studies based on real earthquake cases. The first issue is whether there exists pre-shock accelerating behavior of seismicity which is robust against the changing of spatio-temporal ranges for the sampling of seismic events, and the second is whether such an accelerating behavior is physically associated with an approach to the critical point. To answer these two questions, a retrospective case study was conducted on the 12 May 2008, Wenchuan earthquake, using the local earthquake catalogue in Sichuan and Yunnan Provinces, China, with cutoff magnitude M L3.0, from 1977 to 2008. The results show that the answer to the first question appears to be ‘yes’; that is, in a finite spatial domain within the last couple of years before the event, clear accelerating seismicity could be observed. The answer to the second question cannot be obtained merely by examining seismicity data. However, detailed analysis of the accelerating behavior reveals a potential spatial correlation between the accelerating region and a known asperity, which might be an evidence for that the observed acceleration may have a geometrical or mechanical rather than statistical origin.

Realizing ultra-stable SnO2 anodes via in-situ formed confined space for volume expansion

Working memory (WM) is limited in its temporal length and capacity. Classic conceptions of WM capacity assume the system possesses a finite number of slots, but recent evidence suggests WM may be a continuous resource. Resource models typically assume there is no hard upper bound on the number of items that can be stored, but WM fidelity decreases with the number of items. We analyze a neural field model of multi-item WM that associates each item with the location of a bump in a finite spatial domain, considering items that span a one-dimensional continuous feature space. Our analysis relates the neural architecture of the network to accumulated errors and capacity limitations arising during the delay period of a multi-item WM task. Networks with stronger synapses support wider bumps that interact more, whereas networks with weaker synapses support narrower bumps that are more susceptible to noise perturbations. There is an optimal synaptic strength that both limits bump interaction events and the effects of noise perturbations. This optimum shifts to weaker synapses as the number of items stored in the network is increased. Our model not only provides a circuit-based explanation for WM capacity, but also speaks to how capacity relates to the arrangement of stored items in a feature space.

This study formulates an open problem on spatial localization for quasilinear parabolic equations of pLaplacian type on the half-space R+N, with smooth, nonnegative initial data and boundary conditions that grow unbounded over time. While localization has been widely studied under homogeneous or decaying boundary conditions, its persistence under unbounded boundary input remains unresolved. We introduce two forms of spatial localization: effective localization, where the solution remains within a finite spatial domain, and strict localization, where it vanishes beyond a fixed boundary. The problem examines how the structure of the nonlinearities A(v) and B(v), and the growth rate of the boundary function ϕ influence solution confinement or spread. The study extends classical localization theory to degenerate and singular diffusion processes with diffusion exponent p > 1, offering new insight into spatial confinement under persistent external forcing.

The fully continuous and differentiable framework for performing molecular dynamics calculations introduced in parts I and II of this paper [1,2] requires the evaluation of rather complex force functions and their spatial partial derivatives. This paper presents an efficient interpolation scheme for the evaluation of these quantities over a finite spatial domain. The modified force function is approximated by a linear combination of Hermite cubic basis functions such that both the interpolant of the force and its spatial derivatives are continuous across the grid boundaries. In order to achieve better accuracy for a given grid size, a nonuniform rectilinear grid is constructed via iterative refinement procedure. The latter guarantees the accuracy of the force computed by interpolation within any specified tolerance > ε O. For many potential functions of practical interest, it is possible for polynomial interpolants to be constructed for parts of the force functions which are independent of the potential parameters and system density (the so-called “separable force functions”). In such cases, a single interpolation grid which is applicable for a wide range of potential parameters and system densities can be constructed a priori.

This paper considers the problem of multiobjective fuzzy control design for a class of nonlinear spatially distributed processes (SDPs) described by parabolic partial differential equations (PDEs), which arise naturally in the modeling of diffusion-convection-reaction processes in finite spatial domains. Initially, the modal decomposition technique is applied to the SDP to formulate it as an infinite-dimensional singular perturbation model of ordinary differential equations (ODEs). An approximate nonlinear ODE system that captures the slow dynamics of the SDP is thus derived by singular perturbations. Subsequently, the Takagi–Sugeno fuzzy model is employed to represent the finite-dimensional slow system, which is used as the basis for the control design. A linear matrix inequality (LMI) approach is then developed for the design of multiobjective fuzzy controllers such that the closed-loop SDP is exponentially stable, and an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm L}_{2}$</tex></formula> performance bound is provided under a prescribed <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm H}_{\infty}$</tex></formula> constraint of disturbance attenuation for the slow system. Furthermore, using the existing LMI optimization technique, a suboptimal fuzzy controller can be obtained in the sense of minimizing the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm L}_{2}$</tex></formula> performance bound. Finally, the proposed method is applied to the control of the temperature profile of a catalytic rod.

We investigate the interplay between the longitudinal and lateral solitonic modes in perturbed ladder lattices in regard to the transmission of soliton wave packet. (1) In a longitudinal uniform field the lateral and longitudinal solitonic modes are shown to be independent. However, unlike in the unperturbed case the dynamics of the soliton center of mass becomes confined within a finite spatial domain via the Bloch-Zener mechanism in the longitudinal direction and due to the transverse finiteness of the ladder in the lateral one. (2) The segment of on-site impurities causes the soliton mode-mode mixing. As a result the soliton exhibits rather complex two- or three-dimensional dynamics accompanied by wave radiation which may give rise to soliton trapping. Nevertheless, under some specific conditions the soliton is able to bypass even the strong impurities slaloming between them. In particular, the slalom soliton dynamics is possible on a ladder lattice with a segment of zigzig-distributed on-site impurities. We formulate the conditions favorable to the case and show that their violation gives rise to either soliton trapping on or soliton reflection from the impure segment. (3) Finally, we study the effect of the modified transverse bond on the longitudinal soliton dynamics and reveal that it might act on the soliton as either an attractive or a repulsive potential, depending on the sign of the transverse energy of the ingoing soliton. The effect is essentially a solitonic one and becomes strictly pronounced for heavy solitons, when imperfection-induced radiation effects are exponentially suppressed. We expect that transverse-bond imperfection could serve as a filter selecting the solitons with prescribed properties. A similar function is feasible for zigzag-distributed on-site impurities too.

Many-body localized (MBL) systems are characterized by the absence of transport and thermalization, and therefore cannot be described by conventional statistical mechanics. In this paper, using analytic arguments and numerical simulations, we study the behaviour of local observables in an isolated MBL system following a quantum quench. For the case of a global quench, we find that the local observables reach stationary, highly non-thermal values at long times as a result of slow dephasing characteristic of the MBL phase. These stationary values retain the local memory of the initial state due to the existence of local integrals of motion in the MBL phase. The temporal fluctuations around stationary values exhibit universal power-law decay in time, with an exponent set by the localization length and the diagonal entropy of the initial state. Such a power-law decay holds for any local observable and is related to the logarithmic in time growth of entanglement in the MBL phase. This behaviour distinguishes the MBL phase from both the Anderson insulator (where no stationary state is reached), and from the ergodic phase (where relaxation is expected to be exponential). For the case of a local quench, we also find a power-law approach of local observables to their stationary values when the system is prepared in a mixed state. Quench protocols considered in this paper can be naturally implemented in systems of ultra cold atoms in disordered optical lattices, and the behaviour of local observables provides a direct experimental signature of many-body localization.

Local observables in generic periodically driven closed quantum systems are known to relax to values described by periodic infinite temperature ensembles. At the same time, ergodic static systems exhibit anomalous thermalization of local observables and satisfy a modified version of the eigenstate thermalization hypothesis (ETH), when disorder is present. This raises the question, how does the introduction of disorder affect relaxation in periodically driven systems? In this work, we analyze this problem by numerically studying transport and thermalization in an archetypal example. We find that thermalization is anomalous and is accompanied by subdiffusive transport with a disorder dependent dynamical exponent. Distributions of matrix elements of local operators in the eigenbases of a family of effective time-independent Hamiltonians, which describe the stroboscopic dynamics of such systems, show anomalous departures from predictions of ETH signaling that only a modified version of ETH is satisfied. The dynamical exponent is shown to be related to the scaling of the variance of these distributions with system size.

We study the task of localizing multipartite entanglement in pure quantum states onto a subsystem by measuring the remaining systems. To this end, we fix a multipartite entanglement measure and consider two quantities: the multipartite entanglement of assistance (MEA), defined as the entanglement measure averaged over the post-measurement states and maximized over arbitrary measurements; and the localizable multipartite entanglement (LME), defined in the same way but restricted to only local single-system measurements. Both quantities generalize previously considered bipartite entanglement localization measures. In our work we choose the n-tangle, the genuine multipartite entanglement concurrence and the concentratable entanglement (CE) as the underlying seed measure, and discuss the resulting MEA and LME quantities. First, we prove easily computable upper and lower bounds on MEA and LME and establish Lipschitz-continuity for the n-tangle and CE-based LME and MEA. Using these bounds we investigate the typical behavior of entanglement localization by deriving concentration inequalities for the MEA evaluated on Haar-random states and performing numerical studies for small tractable system sizes. We then turn our attention to protocols that transform graph states. We give a simple criterion based on a matrix equation to decided whether states with a specified n-tangle value can be obtained from a given graph state, providing no-go theorems for a broad class of such graph state transformations beyond the usual “local Clifford plus local Pauli measurement” framework. This analysis is generalized to weighted graph states, which provide a realistic error model in current experiments preparing graph state. Our entanglement localization framework certifies the near-optimality of recently discussed local-measurement protocols to transform uniformly weighted line graph states into GHZ states, even when considering arbitrary entangled measurements. Finally, we demonstrate how our MEA and LME quantities can be used to detect critical phenomena such as phase transitions in transversal field Ising models. Since entanglement localization is operationally relevant throughout quantum networking and measurement-based quantum computation, our framework of results based on the MEA and LME has the potential for broad applications in these fields.

Simulating quantum systems using classical computers encounters inherent challenges due to the exponential scaling with system size. To overcome this challenge, quantum simulation uses a well-controlled quantum system to simulate another less controllable system. Over the last 20 years, many physical platforms have emerged as quantum simulators, such as ultracold atoms, Rydberg atom arrays, trapped ions, nuclear spin, superconducting circuits, and integrated photonics. Trapped ions, with induced spin interactions and universal quantum gates, have demonstrated remarkable versatility, capable of both analog and digital quantum simulation. Recent experimental results, covering a range of research areas including condensed matter physics, quantum thermodynamics, high-energy physics, and quantum chemistry, guide this introductory review to the growing field of quantum simulation.

Programmable quantum devices are now able to probe wave functions at unprecedented levels. This is based on the ability to project the many-body state of atom and qubit arrays onto a measurement basis which produces snapshots of the system wave function. Extracting and processing information from such observations remains, however, an open quest. One often resorts to analyzing low-order correlation functions—that is, discarding most of the available information content. Here, we introduce wave-function networks—a mathematical framework to describe wave-function snapshots based on network theory. For many-body systems, these networks can become scale-free—a mathematical structure that has found tremendous success and applications in a broad set of fields, ranging from biology to epidemics to Internet science. We demonstrate the potential of applying these techniques to quantum science by introducing protocols to extract the Kolmogorov complexity corresponding to the output of a quantum simulator and implementing tools for fully scalable cross-platform certification based on similarity tests between networks. We demonstrate the emergence of scale-free networks analyzing experimental data obtained with a Rydberg quantum simulator manipulating up to 100 atoms. Our approach illustrates how, upon crossing a phase transition, the simulator complexity decreases while correlation length increases—a direct signature of buildup of universal behavior in data space. Comparing experiments with numerical simulations, we achieve cross-certification at the wave-function level up to timescales of 4 μs with a confidence level of 90% and determine experimental calibration intervals with unprecedented accuracy. Our framework is generically applicable to the output of quantum computers and simulators with access to the system wave function and requires probing accuracy and repetition rates accessible to most currently available platforms. Published by the American Physical Society 2024

Opportunity for regulating the collective effect of random expansion with manifestations of finite size effects in a moderate number of finite systems

Quantum simulators have made a remarkable progress towards exploring the dynamics of many-body systems, many of which offer a formidable challenge to both theoretical and numerical methods. While state-of-the-art quantum simulators are in principle able to simulate quantum dynamics well outside the domain of classical computers, they are noisy and limited in the variability of the initial state of the dynamics and the observables that can be measured. Despite these limitations, here we show that such a quantum simulator can be used to in-effect solve for the dynamics of a many-body system. We develop an efficient numerical technique that facilitates classical simulations in regimes not accessible to exact calculations or other established numerical techniques. The method is based on approximations that are well suited to describe localized one-dimensional Fermi-Hubbard systems. Since this new method does not have an error estimate and the approximations do not hold in general, we use a neutral-atom Fermi-Hubbard quantum simulator with $L_{\text{exp}}\simeq290$ lattice sites to benchmark its performance in terms of accuracy and convergence for evolution times up to $700$ tunnelling times. We then use these approximations in order to derive a simple prediction of the behaviour of interacting Bloch oscillations for spin-imbalanced Fermi-Hubbard systems, which we show to be in quantitative agreement with experimental results. Finally, we demonstrate that the convergence of our method is the slowest when the entanglement depth developed in the many-body system we consider is neither too small nor too large. This represents a promising regime for near-term applications of quantum simulators.