SPECIAL-PURPOSE ISING MODEL RANDOM LATTICE PROCESSOR

1991 ◽  
Vol 02 (03) ◽  
pp. 805-816 ◽  
Author(s):  
V.B. ANDREICHENKO ◽  
VL.S. DOTSENKO ◽  
L.N. SHCHUR ◽  
A.L. TALAPOV
Keyword(s):  
Monte Carlo ◽  
Ising Model ◽  
Monte Carlo Step ◽  
Monte Carlo Algorithm ◽  
Price Ratio ◽  
Random Lattice ◽  
Spin Flip ◽  
Different Types ◽  
Memory Cycle Time ◽  
Memory Cycle

We have designed and built a special purpose processor with a very good performance to price ratio, which permits to propose a new way for parallel computing. A simple one spin flip Monte Carlo algorithm is realized in hardware, so the processor is suitable for studies of dynamic as well as thermodynamic properties of the two-dimensional Ising model with different types of inhomogeneities. The speed of the processor is defined completely by the speed of memories used in it: to perform an elementary Monte Carlo step the processor needs a time only several percent larger than one memory cycle time. So it realizes the fastest possible one spin flip Monte Carlo processor architecture.

scholarly journals Monte Carlo simulation of stoquastic Hamiltonians

2015 ◽  
Vol 15 (13&14) ◽  
pp. 1122-1140
Author(s):  
Sergey Bravyi
Keyword(s):  
Monte Carlo ◽  
Ising Model ◽  
Ground State ◽  
State Energy ◽  
Transverse Field ◽  
Variable Number ◽  
Monte Carlo Algorithm ◽  
Standard Product ◽  
Ferromagnetic Ising Model ◽  
Transverse Field Ising Model

Stoquastic Hamiltonians are characterized by the property that their off-diagonal matrix elements in the standard product basis are real and non-positive. Many interesting quantum models fall into this class including the Transverse field Ising Model (TIM), the Heisenberg model on bipartite graphs, and the bosonic Hubbard model. Here we consider the problem of estimating the ground state energy of a local stoquastic Hamiltonian $H$ with a promise that the ground state of $H$ has a non-negligible correlation with some `guiding' state that admits a concise classical description. A formalized version of this problem called Guided Stoquastic Hamiltonian is shown to be complete for the complexity class $\MA$ (a probabilistic analogue of $\NP$). To prove this result we employ the Projection Monte Carlo algorithm with a variable number of walkers. Secondly, we show that the ground state and thermal equilibrium properties of the ferromagnetic TIM can be simulated in polynomial time on a classical probabilistic computer. This result is based on the approximation algorithm for the classical ferromagnetic Ising model due to Jerrum and Sinclair (1993).

scholarly journals PROPERTIES OF ITERATIVE MONTE CARLO SINGLE HISTOGRAM REWEIGHTING

2005 ◽  
Vol 16 (12) ◽  
pp. 1943-1952 ◽  
Author(s):  
MARTIN GMITRA ◽  
DENIS HORVÁTH
Keyword(s):  
Monte Carlo ◽  
Distribution Function ◽  
Ising Model ◽  
Probability Distribution Function ◽  
Stationary Regime ◽  
Simple Relation ◽  
Monte Carlo Algorithm ◽  
Linear Filtering ◽  
Suggested Model ◽  
Temperature Variable

We present an iterative Monte Carlo algorithm for which the temperature variable is attracted by a critical point. The algorithm combines techniques of single histogram reweighting and linear filtering. The ferromagnetic 2D Ising model is studied numerically as an illustration. In that case, the iterations reach a stationary regime with an invariant probability distribution function of temperature which peaked near the pseudocritical temperature of the specific heat. The sequence of generated temperatures is analyzed in terms of stochastic autoregressive model. The error of histogram reweighting can be better understood within the suggested model. The presented model yields a simple relation, connecting the variance of pseudocritical temperature and the parameter of linear filtering.

scholarly journals FLAT HISTOGRAM METHOD OF WANG–LANDAU AND N-FOLD WAY

2002 ◽  
Vol 13 (04) ◽  
pp. 477-494 ◽  
Author(s):  
BEN JESKO SCHULZ ◽  
KURT BINDER ◽  
MARCUS MÜLLER
Keyword(s):  
Monte Carlo ◽  
Density Of States ◽  
Nearest Neighbor ◽  
Monte Carlo Algorithm ◽  
Tunneling Time ◽  
Histogram Method ◽  
Single Spin ◽  
Numerical Tests ◽  
Spin Flip ◽  
Surface Fields

We present a method for estimating the density of states of a classical statistical model. The algorithm successfully combines the Wang–Landau flat histogram method with the N-fold way in order to improve the efficiency of the original single-spin-flip version. We test our implementation of the Wang–Landau method with the two-dimensional nearest neighbor Ising model for which we determine the tunneling time and the density of states on lattices with sizes up to 50 × 50. Furthermore, we show that our new algorithm performs correctly at right edges of an energy interval over which the density of states is computed. This removes a disadvantage of the original single-spin-flip Wang–Landau method where results showed systematically higher errors in the density of states at right boundaries. In order to demonstrate the improvements made, we compare our data with the detailed numerical tests presented in a study by Wang and Swendsen where the original Wang–Landau method was tested against various other methods, especially the transition matrix Monte Carlo method (TMMC). Finally, we apply our method to a thin Ising film of size 32 × 32 × 6 with antiparallel surface fields. With the density of states obtained from the simulations we calculate canonical averages related to the energy such as internal energy, Gibbs free energy and entropy, but we also sample microcanonical averages during simulations in order to determine canonical averages of the susceptibility, the order parameter and its fourth order cumulant. We compare our results with simulational data obtained from a conventional Monte Carlo algorithm.

Markov Chain Analysis of Single Spin Flip Ising Simulations

1997 ◽  
Vol 08 (02) ◽  
pp. 207-227 ◽  
Author(s):  
Michael Hennecke
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Ising Model ◽  
Markov Chain Analysis ◽  
Transition Matrices ◽  
Single Spin ◽  
Spin Flip ◽  
Chain Analysis ◽  
Dynamical Properties ◽  
Versus Loop

The Markov processes defined by random and loop-based schemes for single spin flip attempts in Monte Carlo simulations of the 2D Ising model are investigated, by explicitly constructing their transition matrices. Their analysis reveals that loops over all lattice sites using a Metropolis-type single spin flip probability often do not define ergodic Markov chains, and have distorted dynamical properties even if they are ergodic. The transition matrices also enable a comparison of the dynamics of random versus loop spin selection and Glauber versus Metropolis probabilities.

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