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Register a new userUnderstanding population annealing Monte Carlo simulations
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Population annealing is a recent addition to the arsenal of the practitioner in computer simulations in statistical physics and beyond that is found to deal well with systems with complex free-energy landscapes. Above all else, it promises to deliver unrivaled parallel scaling qualities, being suitable for parallel machines of the biggest calibre. Here we study population annealing using as the main example the two-dimensional Ising model which allows for particularly clean comparisons due to the available exact results and the wealth of published simulational studies employing other approaches. We analyze in depth the accuracy and precision of the method, highlighting its relation to older techniques such as simulated annealing and thermodynamic integration. We introduce intrinsic approaches for the analysis of statistical and systematic errors, and provide a detailed picture of the dependence of such errors on the simulation parameters. The results are benchmarked against canonical and parallel tempering simulations.
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Population annealing Monte Carlo is an efficient sequential algorithm for simulating k-local Boolean Hamiltonians. Because of its structure, the algorithm is inherently parallel and therefore well suited for large-scale simulations of computationally hard problems. Here we present various ways of optimizing population annealing Monte Carlo using 2-local spin-glass Hamiltonians as a case study. We demonstrate how the algorithm can be optimized from an implementation, algorithmic accelerator, as well as scalable parallelization points of view. This makes population annealing Monte Carlo perfectly suited to study other frustrated problems such as pyrochlore lattices, constraint-satisfaction problems, as well as higher-order Hamiltonians commonly found in, e.g., topological color codes.
In simple ferromagnetic quantum Ising models characterized by an effective double-well energy landscape the characteristic tunneling time of path-integral Monte Carlo (PIMC) simulations has been shown to scale as the incoherent quantum-tunneling time, i.e., as $1/Delta^2$, where $Delta$ is the tunneling gap. Since incoherent quantum tunneling is employed by quantum annealers (QAs) to solve optimization problems, this result suggests there is no quantum advantage in using QAs w.r.t. quantum Monte Carlo (QMC) simulations. A counterexample is the recently introduced shamrock model, where topological obstructions cause an exponential slowdown of the PIMC tunneling dynamics with respect to incoherent quantum tunneling, leaving the door open for potential quantum speedup, even for stoquastic models. In this work, we investigate the tunneling time of projective QMC simulations based on the diffusion Monte Carlo (DMC) algorithm without guiding functions, showing that it scales as $1/Delta$, i.e., even more favorably than the incoherent quantum-tunneling time, both in a simple ferromagnetic system and in the more challenging shamrock model. However a careful comparison between the DMC ground-state energies and the exact solution available for the transverse-field Ising chain points at an exponential scaling of the computational cost required to keep a fixed relative error as the system size increases.
The cavity method is a well established technique for solving classical spin models on sparse random graphs (mean-field models with finite connectivity). Laumann et al. [arXiv:0706.4391] proposed recently an extension of this method to quantum spin-1/2 models in a transverse field, using a discretized Suzuki-Trotter imaginary time formalism. Here we show how to take analytically the continuous imaginary time limit. Our main technical contribution is an explicit procedure to generate the spin trajectories in a path integral representation of the imaginary time dynamics. As a side result we also show how this procedure can be used in simple heat-bath like Monte Carlo simulations of generic quantum spin models. The replica symmetric continuous time quantum cavity method is formulated for a wide class of models, and applied as a simple example on the Bethe lattice ferromagnet in a transverse field. The results of the methods are confronted with various approximation schemes in this particular case. On this system we performed quantum Monte Carlo simulations that confirm the exactness of the cavity method in the thermodynamic limit.
The diagrammatic Monte Carlo (Diag-MC) method is a numerical technique which samples the entire diagrammatic series of the Greens function in quantum many-body systems. In this work, we incorporate the flat histogram principle in the diagrammatic Monte method and we term the improved version Flat Histogram Diagrammatic Monte Carlo method. We demonstrate the superiority of the method over the standard Diag-MC in extracting the long-imaginary-time behavior of the Greens function, without incorporating any a priori knowledge about this function, by applying the technique to the polaron problem
Population annealing is a sequential Monte Carlo scheme well-suited to simulating equilibrium states of systems with rough free energy landscapes. Here we use population annealing to study a binary mixture of hard spheres. Population annealing is a parallel version of simulated annealing with an extra resampling step that ensures that a population of replicas of the system represents the equilibrium ensemble at every packing fraction in an annealing schedule. The algorithm and its equilibration properties are described and results are presented for a glass-forming fluid composed of a 50/50 mixture of hard spheres with diameter ratio of 1.4:1. For this system, we obtain precise results for the equation of state in the glassy regime up to packing fractions $varphi approx 0.60$ and study deviations from the BMCSL equation of state. For higher packing fractions, the algorithm falls out of equilibrium and a free volume fit predicts jamming at packing fraction $varphi approx 0.667$. We conclude that population annealing is an effective tool for studying equilibrium glassy fluids and the jamming transition.
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