Scriva, Giuseppe (2024) Solving Optimization and Sampling Problems with Quantum Computers and Neural Networks. [Tesi di dottorato]

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In my doctoral research, I combine machine learning techniques and quantum computing to address optimization and sampling problems. My approach focuses on employing neural-enhanced algorithms to efficiently sample from low-temperature Boltzmann distributions, particularly for Ising spin glasses simulations. By augmenting spin-glass simulations, I also aim to solve intricate optimization problems, typically represented as spin models. Resources from CINECA granted me access to D-Wave quantum devices, facilitating the generation of datasets for sampling low-energy configurations. These turns out to be instrumental for training state-of-the-art autoregressive neural networks (NADE, MADE, PixelCNN), that generate new proposals for a neural Monte Carlo simulation. For boosting the sampling procedure, I introduce a hybrid scheme, combining both popular single spin-flip updates and neural proposals. This algorithm outperforms the single spin-flip algorithm, and it is competitive with parallel tempering, one of the most common techniques to study spin glasses. My research extends to optimization problems using gate-based quantum computers, exploiting the most used hybrid algorithms such as Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA). I systematically assess their performance in determining the ground state of basic Ising models. A precise computation of the number of iterations, both on the classical and the quantum component of the algorithm, shows challenges in realizing a quantum advantage over purely classical optimization algorithms. The primary bottleneck seems to be the shot noise, due to the limited number of function evaluations achievable with the quantum computers. I also shown that with a smart initialization of QAOA, namely an initialization inspired by quantum annealing, it is possible to achieve good performance and variational quantum algorithms becomes competitive with the classical ones. Furthermore, I investigate the properties of deep learning applied to density functional theory (DFT). In this particular field I show how to implement neural networks resilient to the noise of gradient descent optimization in DFT domain, and, alternatively, how to exploit variational autoencoders to constraint energy minimization. The doctoral thesis shows the possibilities and limits of quantum computing in the noisy intermediate-scale quantum (NISQ) era. On one hand, the proposed hybrid schemes such as the one with quantum annealers and neural networks seem to lead to advantages over the classical counterpart. On the other hand, the widely used variational quantum algorithms, which have been introduced to overtake the limit of the NISQ devices, cannot achieve a quantum advantage due to the difficulty in reconstructing the state with a finite number of measures.

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