Dissertation Committee Chair: Mohammad Hafezi
Committee:
Alexey Gorshkov
Maissam Barkeshli
Victor Albert
Amin Gholampour
Abstract: First, we introduce the notion of "generalized bosons," whose exchange statistics resemble those of bosons, but the local bosonic commutator $[a_i,a_i^{\dagger}]=1$ is replaced by an arbitrary single-mode operator that is diagonal in the generalized Fock basis. Examples of generalized bosons include boson pairs and spins. We consider the analogue of the boson sampling task for these particles and observe that its output probabilities are still given by permanents, so the results regarding the difficulty of sampling carry over directly. Finally, we propose implementations of generalized boson sampling in circuit-QED and ion-trap platforms.
In the rest of the thesis, we move on to different topics. Firstly, we incorporate machine learning techniques in quantum information. We use machine learning to classify rational two-dimensional conformal field theories (CFTs). We first use the energy spectra of these minimal models to train a supervised learning algorithm. In contrast to conventional methods that are typically qualitative and involve system size scaling, our method quantifies the similarity of the spectrum of a system at a fixed size to candidate CFTs. Such an approach allows us to correctly predict the nature and value of critical points of several strongly correlated spin models using only their energy spectra. Our results are also relevant for the ground-state entanglement Hamiltonian of certain topological phases of matter described by CFTs. Remarkably, we achieve high prediction accuracy by only using the lowest few Rényi entropies as the input. Finally, using autoencoders, an unsupervised learning algorithm, we find a hidden variable that has a direct correlation with the central charge and discuss prospects for using machine learning to investigate other conformal field theories, including higher-dimensional ones.
Next, we demonstrate how machine learning techniques, especially unsupervised learning algorithms, can be used to study Symmetry-Protected Topological (SPT) phases of matter. SPT phases are short-range entangled phases of matter with a non-local order parameter that are preserved under a local symmetry group. Here, we use an unsupervised learning algorithm, namely diffusion maps, to differentiate between symmetry-broken phases and topologically ordered phases and between non-trivial topological phases in different classes. Specifically, we show that phase transitions associated with these phases can be detected in various bosonic and fermionic models in one dimension, including the interacting SSH model, the AKLT model and its variants, and weakly interacting fermionic models. Our approach provides a cost-effective computational method for detecting topological phase transitions associated with SPT systems, which can also be applied to experimental data obtained from quantum simulators.