In this thesis, I will present mainly three directions of research: symmetric mass genera- tion (SMG), anomaly-constrained gapless quantum phases, and unconventional superconductivity. SMG is a new mass generation mechanism through strong interactions that preserve the symme- tries, different from the conventional Yukawa-Higgs mechanism. I will discuss its applications in chiral lattice gauge theory, symmetric gapping of Fermi liquids, Green’s function properties and the corresponding optical responses of SMG insulators. Then I will discuss about emergent anomalies in many-body systems, and its implications for low-energy phases. In particular, we will see how gapless symmetry protected topological phases and symmetry-enriched topological ordered phases can arise through through effective partial gauging. Last but not least, for the unconventional superconductivity part, I will focus on the exotic time-reversal symmetry breaking normal state of multi-gap superconductors, and the stabilization and topology of superconductors with pairings involving multiple orbitals.

Quantum mechanics is one of the most successful and striking theories in physics. It predicts atomic particles can have exotic properties, such as quantum entanglement, that any classical local theory cannot describe. This phenomenon dramatically increases the complexity of nature, and it indicates there is no classical algorithm that can universally simulate all quantum many-body states. On the other hand, as opposed to classical systems, we never observe quantum properties directly since the measurement for the quantum systems is destructive. People can only determine the quantum black box by the statistics of classical readouts. The complexity of quantum objects implies exponentially many measurements and classical data to figure out the quantum states fully. It is underlying those challenges to find an efficient classical representation of quantum many-body states. An efficient (classical) representation will require fewer classical data of quantum states and learn many of its properties. And an efficient representation can also be served as a classical simulation algorithm for the quantum states. The is no universal, efficient representation for all the quantum states, and it usually depends on the learning properties or underlying quantum states. This thesis will give two efficient representations: the classical shadow representation of quantum states and the hierarchical representation of quantum states. We will see that those efficient representations will help us learn and simulate quantum many-body states and lead to many critical applications in quantum information technology, condensed matter physics, and quantum field theory.

Symmetry, anomaly and duality put strong constraints on the low-energy physics of the quantum field theory. In this dissertation, we used generalized symmetry and symmetry anomaly to investigate the quantum phases of matter and their transitions. In particular, we studied the generalized exact duality in 1+1d systems and infrared duality for 2+1d gauge theories. We investigated the 't Hooft anomaly matching of non-linear sigma model with the Wess-Zumino-Witten term. We focus on the implications of these non-perturbative methods on 1+1d quantum phase transitions, 2+1d deconfined quantum criticality and Fermi surface symmetric mass generation.

The nonlocality of quantum information is an intrinsic consequence of the interactions in many-body systems. Quantifying it can reveal how information between parties propogates across space and time, but in general it is hard to keep track of the exponentially-large set of bipartite entanglement data. In this dissertation, we exploit statistic symmetries about local-basis invariant ensembles to efficiently characterize the bipartite entanglement data in the form of the average purity, or entanglement features. We are able to systematically study the entanglement features of a variety of locally-scrambled ensembles, and determine their entanglement velocities, volume-law coefficients, butterfly velocities, etc. Next, we propose a two-parameter, bond dimension two, matrix-product state ansatz for characterizing the full, multiregion entanglement data of typical locally-scrambled state ensembles. We are able to compare this two-parameter ansatz to a variety of different models and show that it performs well at capturing the full evolution from product to Page state. We also study the meaning of the parameters physically and how they determine different aspects of the multiregion entanglement. Next, we adapt these techniques to classical shadow tomography, where the goal is to construct efficient, classical approximations of quantum states. We show that the reconstruction map is solely a function of the entanglement features of the snapshot state ensemble and we utilize our efficient description of the entanglement data to perform shallow circuit tomography scalably and flexibly. We show that shallow circuit tomography can perform better than Pauli or Clifford measurements for predicting quasi-local operators. We also demonstrate how shallow circuit tomography can serve as a more feasible alternative to Clifford measurements for fidelity estimation, a desirable task on current quantum devices. Lastly, we study the interplay between measurements and entanglement generating unitaries by studying the entanglement features of classical snapshot states produced from hybrid circuits, and show that such schemes are optimal at the phase transition in the purity.

In many systems, both physical and mathematical, internal boundaries play an outsized role in dictating the large-scale behavior of the system. This theme is explored here in two parts: by identifying natural boundaries to diffusion on undirected graphs for the improvement of spectral graph methods for clustering and classification, and by imposing boundaries in two dimensional PEPS tensor networks to reduce their computational complexity for image generation. In the graph diffusion work, an algorithm is developed that identifies and removes vertices serving as bridges between well-connected clusters. With the use of this algorithm, the performance of unsupervised spectral clustering on graphs derived from synthetic point cloud data shows excellent cluster separation down to approximately two-thirds the point cloud gap size that standard spectral clustering alone tolerates. The same vertex-removal scheme applied to a diffusion-informed active learning classification algorithm shows an approximately order-of-magnitude best-case reduction in the classification error rate on benchmark hyperspectral imagery data in the low-label-query-limit versus the same active learning algorithm applied without vertex removal. The PEPS work proposes a scheme whereby such networks are cut into overlapping patch networks, whose individual contraction complexity is comparatively low. Feasibility testing is performed showing that in principle, this scheme could be used for image in-painting, and has intuitively appealing model features that are directly reflective of the data.

The holographic duality is a duality between boundary $d$-dimensional quantum field theories and bulk $(d+1)$-dimensional gravitational theories in asymptotically anti-de Sitter (AdS) space. It provides an appealing explanation for the emergence of spacetime geometry from quantum entanglement, in particular via the Ryu-Takayanagi (RT) formula which assumes the gravity theory is in the classical limit.Yet the assumption of classical geometry has lead to exponentially small mutual information between disjoint sub-regions, which is not true in many system such as free fermion. In this work, we study a generalized Random Tensor Network (RTN) model with fluctuating bond dimensions, which is mapped to a statistical gravity model consisting a massive scalar field on a fluctuating background geometry. A concrete algorithm is constructed to recover the underlying geometry fluctuation from multi-region entanglement entropy data by modelling its distribution as a generative neural network. To demonstrate its effectiveness, we train the model using entanglement entropy of a free fermion system and showed mutual information can be mediated effectively by geometric fluctuation. Remarkably, locality emerges from the learned geometric distribution.