Quantum information science has made significant progress over the last several decades, but the eventual form a quantum computer will take has yet to be determined. Several physical systems have been shown to operate as quantum bits, or qubits, but each faces a central challenge: the qubit must be sufficiently isolated from its environment to maintain quantum coherence while simultaneously having sufficient coupling to the environment to allow quantum mechanical interactions for manipulation and measurement. An approach to achieve these conflicting requirements is to create qubits that are insensitive to small perturbing interactions within their environment by using topological properties of the physical system in which the qubits are formed. This dissertation presents studies on low-dimensional semiconductor heterostructures of InAs, GaSb and AlSb fabricated by molecular beam epitaxy with focus on relevant properties for their utilization in forming a topologically protected (TP) qubit.

The theoretical basis regarding the semiconductor characteristics suitable for realizing TP qubits stipulates the need for strong spin-orbit coupled semiconductors with high carrier mobility. A comparative study of InAs/AlSb heterostructures wherein structure parameters were systematically varied led to a greater understanding of the limits to mobility in InAs quantum wells. Magnetotransport measurements using a dual-gated device geometry and a comparison of experiment to models of carrier mobility as a function of carrier density were used to identify dominant scattering mechanisms in these heterostructures.

The development of dual-gated devices and high quality InAs channels with AlSb barriers led to a demonstration of the gate control of spin-orbit coupling in a high mobility InAs/AlSb quantum well in which the gate-tuned electron mobility exceeded 700,000 cm$^{2}$/V${\cdot}$s. Analysis of low temperature magnetoresistance oscillations indicated the zero field spin-splitting could be tuned via the Rashba effect while keeping the two-dimensional electron gas charge density constant.

Findings from the work on InAs quantum wells were applied to investigations on InAs/GaSb bilayers, a system predicted to be a two-dimensional topological insulator (TI). The temperature and magnetic field dependence of the resistance in dual-gated InAs/GaSb heterostructures gate-tuned to the predicted TI regime were found consistent with conduction through a disordered two-fluid system. The impact of disorder on the formation of topologically protected edge states and an insulating bulk was considered. Potential fluctuations in the band structure for realistic levels of disorder in state-of-the-art heterostructures were calculated using a gated heterostructure model. Potential fluctuations were estimated to be sufficiently large such that conduction in the predicted TI regime was likely dominated by tunneling between localized electron and hole charge fluctuations, corresponding to a symplectic metallic phase rather than a topological insulator. The implications are that future efforts must address defects and disorder in this system if the TI regime is to be achieved.