Abstract: The high mobility two-dimensional electron gas (2DEG) confined in GaAs/AlGaAs heterostructures is a model system for the discovery of unique phases of electronic matter. For example the study of topological phases which now permeates condensed matter physics has its origin in attempts to understand behavior in the 2DEG at high magnetic fields in the quantum Hall regime. It is speculated that specific topological states in the quantum Hall regime may find utility in a special flavor of quantum computation protected against decoherence. The 2DEG in GaAs has also been a playground for mesoscopic physics in which individual electrons are isolated in quantum dots. These structures are now studied as spin-based quantum bits (qubits). In this talk, I will describe how we actually create such physical systems in the laboratory using molecular beam epitaxy and how each experiment places unique demands on the design and operation of the underlying heterostructure. In addition to our efforts to develop stable qubits, I will discuss how we attempt to push the limits of 2DEG quality to uncover new physics. Our recent efforts to understand the role of residual disorder, ever present in real samples, have led to some surprising results that we believe has important consequences for the study of topological phases and quantum computing.