When two layers of graphene are put on top of each another with a relative twist, their lattice mismatch gives rise to a moir {e} pattern. When the twist angle is near 1.1 degrees, a so-called magic angle, the band structure of twisted bilayer graphene becomes extremely flat around the Fermi energy, enhancing the effect of electron-electron interaction. Remarkably, magic-angle twisted bilayer graphene (MATBG) becomes superconducting at low temperatures. The origin of this superconducting behavior remains elusive.

Curiously, the superconducting behavior is usually accompanied by correlated insulating behavior in nearby parameter regions in the phase diagram. These correlated insulators cannot be described by non-interacting band theory, hinting at the importance of electron-electron interaction.

In this thesis, we attempt to understand these correlated insulators by a combination of numerical and analytical techniques. On the numerical side, we will utilize the density-matrix renormalization group (DMRG) extensively to obtain the ground states of MATBG. Owing to the complexity of the problem, this required us to develop a non-trivial routine for encoding the Hamiltonian. We find that DMRG often reproduces the findings from Hartree-Fock simulations.On the analytical side, we aim to understand different symmetry-breaking states of MATBG. Based on symmetry analysis, we propose using scanning tunneling microscopy (STM) to distinguish between different candidate ground states, and test our analytical prediction using numerical simulation.

Taken together, this thesis represents a significant step toward understanding the correlated insulating behavior of MATBG.

In recent years, the matrix product state (MPS) ansatz has become an indispensable tool for numerical studies of condensed matter systems in one spatial dimension (1D). The success of this ansatz can be attributed to the so-called area law of bipartite entanglement entropy which confines the possible ground states of 1D local, gapped Hamiltonians to the corner of Hilbert space which is described well by MPS. Further, the MPS ansatz can be expressed in a canonical form that allows for truncation and efficient variational optimization in an orthogonal basis. As a result, the application of MPS to obtain ground state wavefunctions in 1D is essentially solved. However, for two dimensional quantum states, the optimal representation remains unclear. Unlike MPS, the most direct generalization of MPS to 2D, the projected entangled pair state (PEPS) ansatz, lacks a canonical form. Further, there exist 2D quantum states, such as chiral topological states, which satisfy an area law but which do not admit PEPS descriptions. It therefore remains to be understood what is the optimal ansatz for numerically studying ground states of 2D Hamiltonians and whether representability as a tensor network is connected not only to bipartite entanglement but to tripartite entanglement.

This thesis primarily concentrates on results along these directions. We first show that a newly proposed restricted variant of the PEPS ansatz, the isometric tensor network state (isoTNS) ansatz, can represent string-net liquid states, which realize a wide variety of topological orders. We then develop a bulk measure of tripartite entanglement and show that it encodes universal data about the conformal field theory describing the gapless edge modes.

Finally, we pivot to experimental work in which we demonstrate in a system of superconducting qubits how continuous weak measurements can be applied to a common quantum information processing task: gate characterization. We experimentally demonstrate that an a priori unknown, time-dependent Hamiltonian can be reconstructed from continuous weak measurements concurrent with coherent time evolution in a system of two superconducting transmons coupled by a flux-tunable coupler. In contrast to previous work, our technique does not require interruptions, which would distort the recovered Hamiltonian. We introduce an algorithm which recovers the Hamiltonian and density matrix from an incomplete set of continuous measurements and demonstrate that it reliably extracts amplitudes of a variety of single qubit and entangling two qubit Hamiltonians. We further demonstrate how this technique reveals deviations from a theoretical control Hamiltonian which would otherwise be missed by conventional techniques.

It appears that the tensor network ansatz, while not quite complete, is an efficient coordinate system for the tiny subset of a many-body Hilbert space which can be realized as a low energy state of a local Hamiltonian.

However, we don't fully understand precisely which phases are captured by the tensor network ansatz, how to compute their physical observables (even numerically), or how to compute a tensor network representation for a ground state given a microscopic Hamiltonian. These questions are algorithmic in nature, but their resolution is intimately related to understanding the nature of quantum entanglement in many-body systems.

For this reason it is useful to compute the tensor network representation of various `model' wavefunctions representative of different phases of matter; this allows us to understand how the entanglement properties of each phase are expressed in the tensor network ansatz, and can serve as test cases for algorithm development.

Condensed matter physics has many illuminating model wavefunctions, such as Laughlin's celebrated wave function for the fractional quantum Hall effect, the Bardeen-Cooper-Schrieffer wave function for superconductivity, and Anderson's resonating valence bond ansatz for spin liquids.

This thesis presents some results on exact tensor network representations of these model wavefunctions.

In addition, a tensor network representation is given for the time evolution operator of a long-range one-dimensional Hamiltonian, which allows one to numerically simulate the time evolution of power-law interacting spin chains as well as two-dimensional strips and cylinders.