Quantum Computing for Machine Learning and Physics Simulation

Citation

Zlokapa, Alexander (2021) Quantum Computing for Machine Learning and Physics Simulation. Senior thesis (Major), California Institute of Technology. doi:10.7907/q75q-zm20. https://resolver.caltech.edu/CaltechTHESIS:09272022-143825909

Abstract

Quantum computing is widely thought to provide exponential speedups over classical algorithms for a variety of computational tasks. In classical computing, methods in artificial intelligence such as neural networks and adversarial learning have enabled drastic improvements in state-of-the-art performance for a variety of tasks. We consider the intersection of quantum computing with machine learning, including the quantum algorithms for deep learning on classical datasets, quantum adversarial learning for quantum states, and variational quantum machine learning for improved physics simulation.

We consider a standard deep neural network architecture and show that conditions amenable to trainability by gradient descent coincide with those necessary for an efficient quantum algorithm. Considering the neural network in the infinite-width limit using the neural tangent kernel formalism, we propose a quantum algorithm to train the neural network with vanishing error as the training dataset size increases. Under a sparse approximation of the neural tangent kernel, the training time scales logarithmically with the number of training examples, providing the first known exponential quantum speedup for feedforward neural networks. Related approximations to the neural tangent kernel are discussed, with numerical studies showing successful convergence beyond the proven regime. Our work suggests the applicability of the quantum computing to additional neural network architectures and common datasets such as MNIST, as well as kernel methods beyond the neural tangent kernel.

Generative adversarial networks (GANs) are one of the most widely adopted machine learning methods for data generation. We propose an entangling quantum GAN (EQ-GAN) that overcomes some limitations of previously proposed quantum GANs. EQ-GAN guarantees the convergence to a Nash equilibrium under minimax optimization of the discriminator and generator circuits by performing entangling operations between both the generator output and true quantum data. We show that EQ-GAN has additional robustness against coherent errors and demonstrate the effectiveness of EQ-GAN experimentally in a Google Sycamore superconducting quantum processor. By adversarially learning efficient representations of quantum states, we prepare an approximate quantum random access memory and demonstrate its use in applications including the training of near-term quantum neural networks.

With quantum computers providing a natural platform for physics simulation, we investigate the use of variational quantum circuits to simulate many-body systems with high fidelity in the near future. In particular, recent work shows that teleportation caused by introducing a weak coupling between two entangled SYK models is dual to a particle traversing an AdS-Schwarzschild wormhole, providing a mechanism to probe quantum gravity theories in the lab. To simulate such a system, we propose the process of compressed Trotterization to improve the fidelity of time evolution on noisy devices. The task of learning approximate time evolution circuits is shown to have a favorable training landscape, and numerical experiments demonstrate its relevance to simulating other many-body systems such as a Fermi-Hubbard model. For the SYK model in particular, we demonstrate the construction of a low-rank approximation that favors a shallower Trotterization. Finally, classical simulations of finite-N SYK models suggest that teleportation via a traversable wormhole instead of random unitary scrambling is achievable with O(20) qubits, providing further indication that such quantum gravity experiments may realizable with near-term quantum hardware.

Thesis Files