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.