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Evaluating a quantum-classical quantum Monte Carlo algorithm with Matchgate shadows
Solving the electronic structure problem of molecules and solids to high accuracy is a major challenge in quantum chemistry and condensed matter physics. The rapid emergence and development of quantum computers offer a promising route to systematically tackle this problem. Recent work by [Huggins et al ., Nature (London) 603 , 416 (2022)] proposed a hybrid quantum-classical quantum Monte Carlo (QC-QMC) algorithm using Clifford shadows to determine the ground state of a Fermionic Hamiltonian. This approach displayed inherent noise resilience and the potential for improved accuracy compared to its purely classical counterpart. Nevertheless, the use of Clifford shadows introduces an exponentially scaling postprocessing cost. In this work, we investigate an improved QC-QMC scheme utilizing the recently developed Matchgate shadows technique [Commun. Math. Phys. 404 , 629 (2023)], which removes the aforementioned exponential bottleneck. We observe from experiments on quantum hardware that the use of Matchgate shadows in QC-QMC is inherently noise robust. We show that this noise resilience has a more subtle origin than in the case of Clifford shadows. Nevertheless, we find that classical postprocessing, while asymptotically efficient, requires hours of runtime on thousands of classical CPUs for even the smallest chemical systems, presenting a major challenge to the scalability of the algorithm.
Turbulent Mixing Simulation via a Quantum Algorithm
Probability density function (PDF) methods have been very useful in describing many physical aspects of turbulent mixing. In applications of these methods, modeled PDF transport equations are commonly simulated via classical Monte Carlo techniques, which provide estimates of moments of the PDF at arbitrary accuracy. In this paper, recently developed techniques in quantum computing and quantum enhanced measurements (quantum metrology) are used to construct a quantum algorithm that accelerates the computation of such estimates. This quantum algorithm provides a quadratic speedup over classical Monte Carlo methods in terms of the number of repetitions needed to achieve the desired precision. This paper illustrates the power of this algorithm by considering a binary scalar mixing process modeled by means of the coalescence/dispersion (C/D) closure. The equation is first simulated using classical Monte Carlo methods, where error estimates for the computation of central moments are provided. Then the quantum algorithm for this problem is simulated by sampling from the same probability distribution as that of the output of a quantum computer, and it is shown that significantly fewer resources are required to achieve the same precision. Finally, the results demonstrate potential applications of future quantum computers for simulation of turbulent mixing, and large classes of related problems.
Classical trajectory Monte Carlo simulation of plasma fueling using magnetic plasma expulsion
The possibility of fueling a magnetically confined plasma using particle sources located inside of the plasma is studied by computer simulation. Magnetic plasma expulsion [R. E. Phillips and C. A. Ordonez, Phys. Plasmas 25, 012508 (2018)] would serve to keep the magnetically confined plasma away from the particle sources without adversely affecting plasma confinement. The simulations show how charged particles can be injected into a plasma by using particle sources located directly between two current-carrying wires that create a magnetic expulsion field. Plasma fueling with the average energy of injected particles greater than the average energy of plasma particles may serve for heating the plasma. Also, plasma fueling with positive and negative particles injected at different rates may serve for changing the neutrality of the plasma. Conditions for plasma fueling are investigated using a classical trajectory Monte Carlo simulation. Two types of particle sources are considered, and the fraction of emitted particles that reach (and fuel) the magnetically confined plasma is evaluated for each.
Classical trajectory Monte Carlo simulations of particle confinement using dual levitated coils
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Understanding Reentrance in Frustrated Magnets: The Case of the Er 2 Sn 2 O 7 Pyrochlore
Reentrance, the return of a system from an ordered phase to a previously encountered less-ordered one as a controlled parameter is continuously varied, is a recurring theme found in disparate physical systems, yet its microscopic cause is often not investigated thoroughly. Here, through detailed characterization and theoretical modeling, we uncover the microscopic mechanism behind reentrance in the strongly frustrated pyrochlore antiferromagnet Er 2 Sn 2 O 7 . We use single crystal heat capacity measurements to expose that Er 2 Sn 2 O 7 exhibits multiple instances of reentrance in its magnetic field B vs temperature T phase diagram for magnetic fields along three cubic high symmetry directions. Through classical Monte Carlo simulations, mean field theory, and classical linear spin-wave expansions, we argue that the origins of the multiple occurrences of reentrance observed in Er 2 Sn 2 O 7 are linked to soft modes. These soft modes arise from phase competition and enhance thermal fluctuations that entropically stabilize a specific ordered phase, resulting in an increased transition temperature for certain field values and thus the reentrant behavior. Our work represents a detailed examination into the mechanisms responsible for reentrance in a frustrated magnet and may serve as a template for the interpretation of reentrant phenomena in other physical systems.
Strategies for Automation of Model Tuning in Multifidelity Trajectory Uncertainty Propagation
Multi-model Monte Carlo methods are efficient strategies to perform forward uncertainty quantification studies in entry, descent, and landing (EDL) applications. These multi-model methods are based on the classical Monte Carlo estimator, but fuse predictions from several low-fidelity models to obtain estimators with greater precision given a prescribed computational budget. The effectiveness of these approaches relies on the magnitudes of correlations between the low-fidelity models and the high-fidelity model, as well as the relative computational costs of all models. Identifying and exploiting the best trade-off between correlation and cost, which ultimately depends on the selection of hyperparameters in the low-fidelity models, is a task often performed by hand or simply inspired by the deterministic understanding available for a specific application. This work extends a preliminary effort,
Multi Model Monte Carlo with Python (MXMCPy)
Multi Model Monte Carlo with Python (\mxmc {}) is a software package developed as a general capability for computing the statistics of outputs from an expensive, high-fidelity model by leveraging faster, low-fidelity models for speedup. Motivated by uncertainty propagation problems where classical Monte Carlo (MC) simulation is computationally intractable, various multi-model MC approaches have recently emerged that yield unbiased estimators with significantly reduced variance relative to MC for the same cost. These existing methods include multi-level Monte Carlo (MLMC), multi-fidelity Monte Carlo (MFMC), and approximate control variates (ACV). Given a fixed computational budget and a collection of models with varying cost/accuracy, each method seeks a sample allocation strategy across the models that results in an estimator with optimal variance reduction. \mxmc {} is a versatile tool that enables convenient access to many existing multi-model MC approaches within one modular and extensible package. With \mxmc {}, users can easily compare existing methods to determine the best choice for their particular problem, while developers have a basis for implementing and sharing new variance reduction approaches. This report introduces the \mxmc {} software, providing a summary of the problem-solving workflow for users as well as a brief overview of the code layout for developers.
Quantum advantage for differential equation analysis
Quantum algorithms for differential equation solving, data processing, and machine learning potentially offer an exponential speedup over all known classical algorithms. However, there also exist obstacles to obtaining this potential speedup in useful problem instances. The essential obstacle for quantum differential equation solving is that outputting useful information may require difficult postprocessing, and the essential obstacle for quantum data processing and machine learning is that inputting the data is a difficult task just by itself. In this study, we demonstrate that, when combined, these difficulties solve one another. We show how the output of quantum differential equation solving can serve as the input for quantum data processing and machine learning, allowing dynamical analysis in terms of principal components, power spectra, and wavelet decompositions. To illustrate this, we consider continuous-time Markov processes on epidemiological and social networks. These quantum algorithms provide an exponential advantage over existing classical Monte Carlo methods.
Parallelized traveling cluster approximation to study numerically spin-fermion models on large lattices
Lattice spin-fermion models are quite important to study correlated systems where quantum dynamics allows for a separation between slow and fast degrees of freedom. The fast degrees of freedom are treated quantum mechanically while the slow variables, generically referred to as the “spins,” are treated classically. At present, exact diagonalization coupled with classical Monte Carlo (ED + MC) is extensively used to solve numerically a general class of lattice spin-fermion problems. In this common setup, the classical variables (spins) are treated via the standard MC method while the fermion problem is solved by exact diagonalization. The “traveling cluster approximation” (TCA) is a real space variant of the ED + MC method that allows to solve spin-fermion problems on lattice sizes with up to 10 3 sites. In this paper, we present a novel reorganization of the TCA algorithm in a manner that can be efficiently parallelized. Finally, this allows us to solve generic spin-fermion models easily on 10 4 lattice sites and with some effort on 10 5 lattice sites, representing the record lattice sizes studied for this family of models.
Quantum Monte Carlo and Density Functional Theory Study of Strain and Magnetism in 2D 1T-VSe 2 with Charge Density Wave States
Two-dimensional (2D) 1T-VSe 2 has prompted significant interest due to the discrepancies regarding alleged ferromagnetism (FM) at room temperature, charge density wave (CDW) states, and the interplay between the two. We employed a combined Diffusion Monte Carlo (DMC) and density functional theory (DFT) approach to accurately investigate the magnetic properties, CDW states, and their responses to strain in monolayer 1T-VSe 2 . Our calculations show the delicate competition between various phases, revealing critical insights into the relationship between their energetic and structural properties. Here, we performed classical Monte Carlo simulations informed by our DMC and DFT results and found the magnetic transition temperature (T c ) of the undistorted (non-CDW) FM phase to be 228 K and the distorted (CDW) phase to be 68 K. Additionally, we studied the response of biaxial strain on the energetic stability and magnetic properties of various phases of 2D 1T-VSe 2 and found that small amounts of strain can increase the T c , suggesting a promising route for engineering and enhancing magnetic behavior. Finally, we synthesized 1T-VSe 2 and performed Raman spectroscopy measurements, which were in close agreement with our calculated results, validating our computational approach. Our work emphasizes the role of highly accurate DMC methods in advancing the understanding of monolayer 1T-VSe 2 and provides a robust framework for future studies of 2D magnetic materials.
Reexamination of the evidence for entanglement in a quantum annealer
A recent experiment claimed to provide evidence of up to eight-qubit entanglement in a D-Wave quantum annealing device. However, entanglement was measured using qubit tunneling spectroscopy, a technique that provides indirect access to the state of the system at intermediate times during the anneal by performing measurements at the end of the anneal with a probe qubit. In addition, an underlying assumption was that the quantum transverse-field Ising Hamiltonian, whose ground states are already highly entangled, is an appropriate model of the device and not some other (possibly classical) model. This begs the question of whether alternative classical or semiclassical models would be equally effective at predicting the observed spectrum and thermal state populations. To check this, we consider a recently proposed classical rotor model with classical Monte Carlo updates, which has been successfully employed in describing features of earlier experiments involving the device. Further, we consider simulated quantum annealing with quantum Monte Carlo updates, an algorithm that samples from the instantaneous Gibbs state of the device Hamiltonian. Finally, we use the quantum adiabatic master equation, which cannot be efficiently simulated classically and which has previously been used to successfully capture the open-system quantum dynamics of the device. We find that only the master equation is able to reproduce the features of the tunneling spectroscopy experiment, while both the classical rotor model and simulated quantum annealing fail to reproduce the experimental results. We argue that this bolsters the evidence for the reported entanglement.
MAGIC: M arching Cubes Isosurface Uncertainty Visualization for G auss i an Uncertain Data With Spatial C orrelation
Here, in this paper, we study the propagation of data uncertainty through the marching cubes algorithm for isosurface visualization for correlated uncertain data. Consideration of correlation has been shown paramount for avoiding errors in uncertainty quantification and visualization in multiple prior studies. Although the problem of isosurface uncertainty with spatial data correlation has been previously addressed, there are two major limitations to prior treatments. First, there are no analytical formulations for uncertainty quantification of isosurfaces when the data uncertainty is characterized by a Gaussian distribution with spatial correlation. Second, as a consequence of the lack of analytical formulations,existing techniques resort to a Monte Carlo sampling approach, which is expensive and difficult to integrate into visualization tools. To address these limitations, we present a closed-form framework to efficiently derive uncertainty in marching cubes level-sets for Gaussian uncertain data with spatial correlation (MAGIC). To derive closed-form solutions, we leverage the Hinkley's derivation on the ratio of Gaussian distributions. With our analytical framework, we achieve a significant speed-up and enhanced accuracy of uncertainty quantification over classical Monte Carlo methods. We further accelerate our analytical solutions using many-core processors to achieve speed-ups up to 585× and integrability with production visualization tools for broader impact. We demonstrate the effectiveness of our correlation-aware uncertainty framework through experiments on meteorology, urban flow, and astrophysics simulation datasets.
Multi-fidelity Uncertainty Quantification for Homogenization Problems in Structure-Property Relationships from Crystal Plasticity Finite Elements
Crystal plasticity finite element method (CPFEM) has been an integrated computational materials engineering (ICME) workhorse to study materials behaviors and structure-property relationships for the last few decades. These relations are mappings from the microstructure space to the materials properties space. Due to the stochastic and random nature of microstructures, there is always some uncertainty associated with materials properties, for example, in homogenized stress-strain curves. For critical applications with strong reliability needs, it is often desirable to quantify the microstructure-induced uncertainty in the context of structure-property relationships. However, this uncertainty quantification (UQ) problem often incurs a large computational cost because many statistically equivalent representative volume elements (SERVEs) are needed. In this article, we apply a multi-level Monte Carlo (MLMC) method to CPFEM to study the uncertainty in stress-strain curves, given an ensemble of SERVEs at multiple mesh resolutions. By using the information at coarse meshes, we show that it is possible to approximate the response at fine meshes with a much reduced computational cost. We focus on problems where the model output is multi-dimensional, which requires us to track multiple quantities of interest (QoIs) at the same time. In conclusion, our numerical results show that MLMC can accelerate UQ tasks around 2.23x, compared to the classical Monte Carlo (MC) method, which is widely known as ensemble average in the CPFEM literature.
Slow spin dynamics and quantum tunneling of magnetization in the dipolar antiferromagnet DyScO 3
Here, we present a comprehensive study of static and dynamic magnetic properties in the Ising-like dipolar antiferromagnet (AFM) DyScO 3 by means of DC and AC magnetization measurements supported by classical Monte Carlo calculations. Our AC-susceptibility data show that the magnetic dynamics exhibit a clear crossover from an Arrhenius-like regime to quantum tunneling of magnetization (QTM) at T*=10K. Below T N =3.2K, DyScO 3 orders in an antiferromagnetic GxAy-type magnetic structure and the magnetization dynamics slow down to the minute timescale. The low-temperature magnetization curves exhibit complex hysteretic behavior, which depends strongly on the magnetic field sweep rate. We demonstrate that the low-field anomalies on the magnetization curve are related to the metamagnetic transition, while the hysteresis at higher fields is induced by a strong magnetocaloric effect. Our theoretical calculations, which take into account dipolar interaction between Dy 3+ moments, reproduce essential features of the magnetic behavior of DyScO 3 . We demonstrate that DyScO 3 represents a rare example of an inorganic compound, which exhibits QTM at a single-ion level and magnetic order due to classical dipolar interaction.
K 2 CoS 2 : A two-dimensional in-plane antiferromagnetic insulator
Recent discovery of two-dimensional (2D) magnetic materials has brought magnetism to the flatland and opened up exciting opportunities for the exploration of fundamental physics as well as novel device applications. Here, we predict a thermodynamically stable 2D magnetic material, ${\mathrm{K}}_{2}{\mathrm{CoS}}_{2}$, which retains its in-plane bulk antiferromagnetic (AFM) order down to the monolayer and bilayer limits. Magnetic moments $(2.5{{\mu}}_{B}/\mathrm{Co})$ are found to form a quasi-one-dimensional antiferromagnetically ordered chain of Co atoms. The nonmagnetic electronic spectrum of the monolayer film is found to host flatbands and van Hove singularities, which play a key role in stabilizing the magnetic ground state. Based on classical Monte Carlo simulations, we estimate the Neel temperature for the AFM monolayer to be ${\approx}15\mathrm{K}$. Our study demonstrates that ${\mathrm{K}}_{2}{\mathrm{CoS}}_{2}$ hosts a robust AFM state which persists from the monolayer limit to the bulk material.
Semiclassical dynamics on multiple electronic surfaces - Three-dimensional treatment of reactive F + H2
The role of electron transitions in collisions is studied for the F + H2 reaction by combining quasi-classical Monte Carlo trajectories with a semiclassical decoupling approximation for the electron transitions. Attention is directed at the reaction of excited state F atoms reacting to form ground state products; the reactants are initiated in either of two spin-orbit states of the atom with the diatom in the ground vibrational state and the lowest four rotational states, at relative translational energies of 0.1, 0.2 and 0.3 eV. Even if the reactants are initiated on the excited state surface, the reactive cross sections (which are classically forbidden) are significant. The major dynamical effects of the excited state reaction are the flow of reactant electronic energy into product internal energy.