Spin Glass Theory and its Applications to Learning in Classical and Quantum Neural Networks

Abstract

Spin glasses are, in essence, simply systems of magnetic moments with disordered interactions. Despite this basic premise, spin glasses have become relevant in a wide variety of fields, including condensed matter physics, complexity and optimization theory, error correction, quantum computing, machine learning, and even neuroscience. The pervasiveness of spin glasses stems from the fact that they are paradigmatic examples of how global complexity emerges from disordered interactions between a large number of basic constituents. The inclusion of quantum effects in spin glasses creates an even richer picture, due to the interplay between the energy barriers arising from disorder and quantum fluctuations which allow for tunneling.

This thesis initially focuses on quantum glasses and how their initial states explore their Hilbert spaces. Two complementary approaches are used. One is comparison of the spectral statistics, particularly the spectral form factor, with the spectral statistics of random matrix ensembles, a hallmark of quantum chaotic behavior. The other approach uses the Thouless-Anderson-Palmer mean-field equations to determine the number of local minima which appear in the free energy landscape.

This thesis then applies methods from spin glass theory to the field of machine learning. Neural networks, like spin glasses, are systems in which remarkable behavior emerges from interactions between basic constituents. By studying the phase diagrams of spin models analogous to snapshots of neural networks throughout their training process, insight is gained into the mechanisms of learning.

Finally, this thesis investigates how quantum effects can lead to advantages in machine learning. It is shown, through classical simulation and experiments on both trapped-ion and superconducting quantum computing hardware, that a natural quantization of a classical neural network can demonstrate superior generalization performance due to the injection of quantum uncertainty.