Vidros de spin e redes neurais: diagramas de fase das máquinas de Boltzmann clássicas e semiclássicas

Abstract

Classical and quantum Boltzmann machines have become useful models for complex systems in statistical physics, but also learning algorithms on neural networks and optimization problems outside of statistical physics. These stochastic neural networks embodying the Boltzmann distribution, are capable of learning and representing probability distributions over input data that enables them to capture the related correlations and dependencies inherent in the systems they represent. In this project, through a theoretical analysis based on the Replica and Suzuki Trotter methods, we study and compare the phase diagrams of the classical (in both architectures) and quantum Boltzmann machine, highlighting the fundamental similarities and differences between them. The two methods mentioned above allow us to make the semi classical approximation in which the correlations between the neurons are ignored, both classical and quantum. The phase diagrams of these machines are studied in order to provide deep insights into its equilibrium behaviour, showing how transitions between different network states depend on variations of some control parameters like temperature, transverse field (just for the quantum case), interaction energy (Synapses) or external field strength (bias), we focus in the effects of the bias which is important in the context of neural networks. In the case of a classical Boltzmann machine, these transitions are described by changes in terms of the temperature that is directly related with the transition probabilities between microscopic states; while for a quantum Boltzmann Machine, it is necessary consider additionally effects from the transverse field that allows quantum tunneling, coherence and superposition.