Local and global models of physics and computation

Classical computation is essentially local in time, yet some formulations of physics are global in time. Here I examine these differences, and suggest that certain forms of unconventional computation are needed to model physical processes and complex systems. These include certain forms of analogue computing, massively parallel field computing, and self-modifying computations

Graph Energies of Egocentric Networks and Their Correlation with Vertex Centrality Measures

Graph energy is the energy of the matrix representation of the graph, where the energy of a matrix is the sum of singular values of the matrix. Depending on the definition of a matrix, one can contemplate graph energy, Randi\'c energy, Laplacian energy, distance energy, and many others. Although theoretical properties of various graph energies have been investigated in the past in the areas of mathematics, chemistry, physics, or graph theory, these explorations have been limited to relatively small graphs representing chemical compounds or theoretical graph classes with strictly defined properties. In this paper we investigate the usefulness of the concept of graph energy in the context of large, complex networks. We show that when graph energies are applied to local egocentric networks, the values of these energies correlate strongly with vertex centrality measures. In particular, for some generative network models graph energies tend to correlate strongly with the betweenness and the eigencentrality of vertices. As the exact computation of these centrality measures is expensive and requires global processing of a network, our research opens the possibility of devising efficient algorithms for the estimation of these centrality measures based only on local information

Quantum Cellular Automata

Quantum cellular automata (QCA) are reviewed, including early and more recent proposals. QCA are a generalization of (classical) cellular automata (CA) and in particular of reversible CA. The latter are reviewed shortly. An overview is given over early attempts by various authors to define one-dimensional QCA. These turned out to have serious shortcomings which are discussed as well. Various proposals subsequently put forward by a number of authors for a general definition of one- and higher-dimensional QCA are reviewed and their properties such as universality and reversibility are discussed.Comment: 12 pages, 3 figures. To appear in the Springer Encyclopedia of Complexity and Systems Scienc

A simple analytical description of the non-stationary dynamics in Ising spin systems

The analytical description of the dynamics in models with discrete variables (e.g. Isingspins) is a notoriously difficult problem, that can be tackled only undersome approximation.Recently a novel variational approach to solve the stationary dynamical regime has beenintroduced by Pelizzola [Eur. Phys. J. B, 86 (2013) 120], where simpleclosed equations arederived under mean-field approximations based on the cluster variational method. Here wepropose to use the same approximation based on the cluster variational method also for thenon-stationary regime, which has not been considered up to now within this framework. Wecheck the validity of this approximation in describing the non-stationary dynamical regime ofseveral Ising models defined on Erdos-R ́enyi random graphs: westudy ferromagnetic modelswith symmetric and partially asymmetric couplings, models with randomfields and also spinglass models. A comparison with the actual Glauber dynamics, solvednumerically, showsthat one of the two studied approximations (the so-called ‘diamond’approximation) providesvery accurate results in all the systems studied. Only for the spin glass models we find somesmall discrepancies in the very low temperature phase, probably due to the existence of alarge number of metastable states. Given the simplicity of the equations to be solved, webelieve the diamond approximation should be considered as the ‘minimalstandard’ in thedescription of the non-stationary regime of Ising-like models: any new method pretending toprovide a better approximate description to the dynamics of Ising-like models should performat least as good as the diamond approximation

Parallel Tempering Simulation of the three-dimensional Edwards-Anderson Model with Compact Asynchronous Multispin Coding on GPU

Monte Carlo simulations of the Ising model play an important role in the field of computational statistical physics, and they have revealed many properties of the model over the past few decades. However, the effect of frustration due to random disorder, in particular the possible spin glass phase, remains a crucial but poorly understood problem. One of the obstacles in the Monte Carlo simulation of random frustrated systems is their long relaxation time making an efficient parallel implementation on state-of-the-art computation platforms highly desirable. The Graphics Processing Unit (GPU) is such a platform that provides an opportunity to significantly enhance the computational performance and thus gain new insight into this problem. In this paper, we present optimization and tuning approaches for the CUDA implementation of the spin glass simulation on GPUs. We discuss the integration of various design alternatives, such as GPU kernel construction with minimal communication, memory tiling, and look-up tables. We present a binary data format, Compact Asynchronous Multispin Coding (CAMSC), which provides an additional $28.4\%$ speedup compared with the traditionally used Asynchronous Multispin Coding (AMSC). Our overall design sustains a performance of 33.5 picoseconds per spin flip attempt for simulating the three-dimensional Edwards-Anderson model with parallel tempering, which significantly improves the performance over existing GPU implementations.Comment: 15 pages, 18 figure

Local Unitary Quantum Cellular Automata

In this paper we present a quantization of Cellular Automata. Our formalism is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. It is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: universality, strong connections to the circuit model, simple implementation on quantum hardware, and a wealth of applications.Comment: To appear in Physical Review