Károly F. Pál

THE GROUND STATE ENERGY OF THE EDWARDS-ANDERSON ISING SPIN GLASS WITH A HYBRID GENETIC ALGORITHM

Ground states of three-dimensional Edwards-Anderson ±J Ising spin glasses were calculated with a hybrid of genetic algorithm and local optimization. The algorithm was fast and reliable enough to allow extensive calculations for systems of linear size between 3 and 14 and determination of the average ground state energies with small errors. A linear dependence on 1/volume approximates the data very accurately in the whole range. The −1.7863 ± 0.0004 value for the ground state energy per spin of the infinite system was determined with extrapolation. The main source of uncertainty is that the functional form of the small but significant deviation from the linear 1/volume dependence is unknown.

Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems

The I3322 inequality is the simplest bipartite two-outcome Bell inequality beyond the ClauserHorne-Shimony-Holt (CHSH) inequality, consisting of three two-outcome measurements per party. In case of the CHSH inequality the maximal quantum violation can already be attained with local two-dimensional quantum systems, however, there is no such evidence for the I3322 inequality. In this paper a family of measurement operators and states is given which enables us to attain the largest possible quantum value in an infinite dimensional Hilbert space. Further, it is conjectured that our construction is optimal in the sense that measuring finite dimensional quantum systems is not enough to achieve the true quantum maximum. We also describe an efficient iterative algorithm for computing quantum maximum of an arbitrary two-outcome Bell inequality in any given Hilbert space dimension. This algorithm played a key role to obtain our results for the I3322 inequality, and we also applied it to improve on our previous results concerning the maximum quantum violation of several bipartite two-outcome Bell inequalities with up to five settings per party.

Genetic algorithm with local optimization

A hybrid of genetic algorithm and local optimization was tested on a massively multimodal spin-lattice problem involving a huge configuration space. The results are good, and global optima will probably be achieved in a sizeable proportion of cases, especially if a selection scheme is applied that maintains genetic diversity by introducing a spatial separation between the members of the population. If we use single-point cross-over, the performance of the algorithm depends strongly on the order of the units corresponding to individual spins in the bit strings that the genetic part of the algorithm processes. Due to some interplay between the genetic algorithm and local optimization, the best performance is achieved with a peculiar ordering, while the results with the most obvious ordering are much worse. I introduce an ordering-invariant crossover operation that gives excellent performance: it almost always yields states of the lowest energy. I expect this or some similar crossover operation to work well in the hybrid scheme for many other problems as well.

Quantum bounds on Bell inequalities

We have determined the maximum quantum violation of 241 tight bipartite Bell inequalities with up to five two-outcome measurement settings per party by constructing the appropriate measurement operators in up to six-dimensional complex and eight-dimensional real-component Hilbert spaces using numerical optimization. Out of these inequalities 129 have been introduced here. In 43 cases higher-dimensional component spaces gave larger violation than qubits, and in three occasions the maximum was achieved with six-dimensional spaces. We have also calculated upper bounds on these Bell inequalities using a method proposed recently. For all but 20 inequalities the best solution found matched the upper bound. Surprisingly, the simplest inequality of the set examined, with only three measurement settings per party, was not among them, despite the high dimensionality of the Hilbert space considered. We also computed detection threshold efficiencies for the maximally entangled qubit pair. These could be lowered in several instances if degenerate measurements were also allowed.

Selection Schemes with Spatial Isolation for Genetic Optimization

We tested genetic algorithms with several selection schemes on a massively multimodal spin-lattice problem. New schemes that introduce a spatial separation between the members of the population gave significantly better results than any other scheme considered. These schemes slow down considerably the flow of genetic information between different regions of the population, which makes possible for distant regions to evolve more or less independently. This way many distinct possibilities can be explored simultaneously and a high degree of diversity can be maintained, which is very important for most multimodal problems.

Hysteretic optimization for the Sherrington–Kirkpatrick spin glass

Hysteretic optimization is a heuristic optimization method based on the observation that magnetic samples are driven into a low-energy state when demagnetized by an oscillating magnetic field of decreasing amplitude. We show that hysteretic optimization is very good for finding ground states of Sherrington–Kirkpatrick spin glass systems. With this method it is possible to get good statistics for ground state energies for large samples of systems consisting of up to about 2000 spins. The way we estimate error rates may be useful for some other optimization methods as well. Our results show that both the average and the width of the ground state energy distribution converges faster with increasing size than expected from earlier studies.

Scaling laws of creep rupture of fiber bundles

We study the creep rupture of fiber composites in the framework of fiber bundle models. Two fiber bundle models are introduced based on different microscopic mechanisms responsible for the macroscopic creep behavior. Analytical and numerical calculations show that above a critical load the deformation of the creeping system monotonically increases in time resulting in global failure at a finite time t(f), while below the critical load the system suffers only partial failure and the deformation tends to a constant value giving rise to an infinite lifetime. It is found that approaching the critical load from below and above the creeping system is characterized by universal power laws when the fibers have long-range interaction. The lifetime of the composite above the critical point has a universal dependence on the system size.

Genetic algorithms for the traveling salesman problem based on a heuristic crossover operation

We apply strategies inspired by natural evolution to a classical example of discrete optimization problems, the traveling salesman problem. Our algorithms are based on a new knowledge-augmented crossover operation. Even if we use only this operation in the reproduction process, we get quite good results. The most obvious faults of the solutions can be eliminated and the results can further be improved by allowing for a simple form of mutation. If each crossover is followed by an affordable local optimization, we get the optimum solution for a 318-town problem, probably the optimum solutions for several different 100-town problems, and very nearly optimum solutions for 350-town and 1000-town problems. A new strategy for the choice of parents considerably speeds up the convergence of the latter method.

Bounding the dimension of bipartite quantum systems

Let us consider the set of joint quantum correlations arising from two-outcome local measurements on a bipartite quantum system. We prove that no finite dimension is sufficient to generate all these sets. We approach the problem in two different ways by constructing explicit examples for every dimension d, which demonstrates that there exist bipartite correlations that necessitate d-dimensional local quantum systems in order to generate them. We also show that at least 10 two-outcome measurements must be carried out by the two parties altogether so as to generate bipartite joint correlations not achievable by two-dimensional local systems. The smallest explicit example we found involves 11 settings.

Structure formation in binary colloids

A theoretical study of the structure formation observed very recently [W. D. Ristenpart, I. A. Aksay, and D. A. Saville, Phys. Rev. Lett. 90, 128303 (2003)] in binary colloids is presented. In our model solely the dipole-dipole interaction of the particles is considered, electrohydrodynamic effects are excluded. Based on molecular dynamics simulations and analytic calculations we show that the total concentration of the particles, the relative concentration, and the relative dipole moment of the components determine the structure of the colloid. At low concentrations the kinetic aggregation of particles results in fractal structures which show a crossover behavior when increasing the concentration. At high concentration various lattice structures are obtained in a good agreement with experiments.