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Abstract
The Hopfield network is an Ising spin system with long-range binary interactions. The interaction coefficients are defined such that a particular set of possible states of the system are the stable states, and the system tends to evolve into one of these states, exhibiting associative memory. The model can be generalized to have multi-spin interactions, resulting in systems called modern Hopfield networks, which have much larger memory capacities scaling as higher powers of the system size. In this thesis, I have studied how the retrieval accuracy and the capacity of modern Hopfield networks are affected by noise in the interaction coefficients mediating the multi-spin interactions. I also study the effect of clipping the interaction coefficients to all have the magnitude and only differ in sign, which can be a more convenient or efficient way of designing or simulating Hopfield networks as the interactions can be stored with smaller precision. I find that in most of these cases the capacity is reduced but its scaling behavior with the system size remains unchanged. While most of my work focuses on one step of synchronous Hopfield dynamics at zero temperature, I also investigate the equilibrium behavior of the system with noise-free fourth-order interactions using replica theory and obtain the phases of this system. Finally, I consider a possible quantum generalization of the network and compare its behavior with the classical network.