Measurement-Based Quantum Computation and Symmetry-Protected Topological Order

While quantum computers can achieve dramatic speedups over the classical computers familiar to us, identifying the origin of this quantum advantage in physical systems remains a major goal of quantum information science. A useful tool here is measurement-based quantum computation (MQC), a computational framework utilizing the quantum entanglement found in many-body resource states. Not all resource states are useful for quantum computation however, so an important question is what properties of many-body entanglement characterize universal resource states, which can implement any quantum computation. Many-body states are also studied in condensed matter physics, where the collective behavior of quantum many-body systems sometimes define topological phases of matter. These phases are defined by nonlocal many-body entanglement, making topologically-ordered states natural candidates for MQC. We might wonder if these topological phases could be organized as phases of quantum computation, so that every state within the phase is universal for MQC. While phases of symmetry-protected topological order (SPTO) have arisen as natural candidates, previous attempts to demonstrate an MQC-SPTO correspondence were mostly limited to nonuniversal 1D spin chains, leaving the important 2D setting wide open. In this dissertation, we explore the wide and varied connections between MQC and SPTO, and obtain new results for 1D and 2D systems. After identifying a new MQC-SPTO correspondence within 1D spin chains, we move up and explore the operational use of 2D states with two complementary forms of SPTO. We create a new Union Jack resource state, whose different form of SPTO than previous 2D resource states permits a hierarchical notion of MQC universality. This state leads us to consider an idealized model of 2D SPTO, where we show that an additional symmetry condition makes these model states form universal resources for MQC only when they have nontrivial SPTO. We finally study the intrinsic complexity of SPTO-inspired states for classically intractable sampling, and identify inherent advantages of MQC for this purpose. Our work highlights the rich complexity available in states of entangled quantum matter, providing new evidence which sharpens our understanding of the diverse connections between MQC and SPTO