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Quantum machine learning is an emerging field that combines techniques in the disciplines of machine learning (ML) and quantum physics. Research in this field takes three broad forms: applications of classical ML techniques to quantum physical systems, quantum computing and algorithms for classical ML problems, and new ideas inspired by the intersection of the two disciplines. We mainly focus on the power of artificial neural networks (NNs) in quantum-state representation and phase classification in this work.

In the first part of the dissertation, we study NN quantum states which are used as wave-function ans{" a}tze 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 efficiently representing particular states such as those with a stabilizer nature, more rigorous quantitative analysis about their expressibility and complexity is warranted. Here, our analysis of the restricted Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems provides new insight into their computational complexity. We define a class of long-range-fast-decay (LRFD) RBM states with quantifiable upper bounds on truncation errors and provide numerical evidence for a large class of 1D quantum systems that may be approximated by LRFD RBMs of at most polynomial complexities. These results lead us to conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them, even in cases where other state representations become less efficient. At last, we provide the relations between multiple typical state manifolds. Our work proposes a paradigm for doing complexity analysis for generic long-range RBMs which naturally yields a further classification of this manifold. This paradigm and our characterization of their nonlocal structures may pave the way for understanding the natural measure of complexity for quantum many-body states described by RBMs and are generalizable for higher-dimensional systems and deep neural-network quantum states.

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 enables us to identify the critical exponent where the maximal quasiparticle group velocity transits from finite to divergent in the thermodynamic limit numerically. In $D=1$, the results agree with an analysis using the field theory and semiclassical spin-wave theory. Furthermore, we generalize the RBM method for learning excited states in nonzero-momentum sectors from 1D to 2D systems. At last, we analyze and provide all possible values ($3/2$, $2$ and $3$) of the critical exponent for 1D generic quadratic bosonic and fermionic Hamiltonians with long-range hoppings and pairings which serves for understanding the speed of information propagation in quantum systems.

In the third part, we study deep NNs as phase classifiers. We analyze the phase diagram of a 2D topologically nontrivial fermionic model Hamiltonian with pairing terms at first and then demonstrate that deep NNs can learn the band-gap closing conditions only based on wave-function samples of several typical energy eigenstates, thus being able to identify the phase transition point without knowledge of Hamiltonians.