Dissertation Committee Chair: Professor Mohammad Hafezi
Committee:
Professor Charles W. Clark, Co-Chair/Advisor
Professor Victor Yakovenko
Professor Alexey Gorshkov
Professor Christopher Jarzynski, Dean's representative
Abstract: Quantum machine learning is an emerging field that combines techniques in the disciplines of machine learning (ML) and quantum physics. Research in this subject takes three broad forms: applications of classical ML techniques to quantum physical systems; application of quantum computing to classical ML problems; new concepts inspired by the intersection of the two disciplines.
In the first part of the dissertation, we study neural network (NN) states which are used as wave-function ansatze in the context of quantum many-body physics. While these states have achieved success in simulating low-lying eigenstates and short-time unitary dynamics of quantum systems and representing particular states such as code words of a stabilizer code, more rigorous quantitative analysis about their expressibility and complexity is warranted. Here, we analyze the efficiency of the restricted Boltzmann machine (RBM) state representation and its use in approximating ground states of one-dimensional (1D) quantum spin systems. We define long-range-fast-decay RBM states and derive upper bounds on truncation errors with two measures of state differences. This enables us to identify distinct asymptotic scaling forms of spatial complexities for RBMs which depend on the quantified decaying rates and demonstrate their potentially high efficiency (at most polynomial) in representing ground states of a wide range of 1D systems, including critical systems. At last, we provide the relationships between multiple important state manifolds. The framework we build can benefit the understanding of the structure and natural complexities of RBMs and the relevant concepts can be generalized to other variants if it is necessary for future analysis of quantum deep NNs.
In the second part, we use RBMs to investigate, in dimensions D=1 and 2, the many-body excitations of long-range power-law interacting quantum spin models. We develop an energy-shift method to calculate the excited states of such spin models and obtain a high-precision momentum-resolved low-energy spectrum. This can help us to identify the critical exponent where the maximal quasiparticle group velocity transits from finite to divergent. In D=1, the results agree with an analysis using semiclassical spin-wave theory.
In the third part, we study deep NNs as phase classifiers. We analyze the phase diagram of a 2D topologically nontrivial fermionic model Hamiltonian and demonstrate that deep NNs can learn the band-gap closing conditions only based on wave-function samples of several typical bands, thus being able to identify the phase transition point without knowledge of Hamiltonians.
Location: John S. Toll Physics Building 2219