Parity control of superconducting islands hosting Majorana zero modes (MZMs) is required to operate topological qubits made from proximitized semiconductor nanowires. We test this control by studying parity effects in hybrid InAs-Al single-Cooper-pair transistors (SCPTs) to evaluate the feasibility of this material system. In particular, we investigate the gate-charge modulation of the supercurrent and observe a consistent 2e-periodic pattern indicating a general lack of low-energy subgap states in these nanowires at zero magnetic field. In a parallel magnetic field, an even-odd pattern develops with a gate-charge spacing that oscillates as a function of field demonstrating that the modulation pattern is sensitive to the presence of a single bound state. In addition, we find that the parity lifetime of the SCPT decreases exponentially with magnetic field as the bound state approaches zero energy. Our work shows that aluminum is the preferred superconductor for future topological qubit experiments and highlights the important role that quasiparticle traps and superconducting gap engineering would play in these qubits. Moreover, we demonstrate a means by which bound states can be detected in devices with superconducting leads.

We show that edge-state transport in semiconductor-based quantum spin Hall systems is unexpectedly robust to magnetic fields. The origin for this robustness lies in an intrinsic suppression of the edge-state g-factor and the fact that the edge-state Dirac point is typically hidden in the valence band. A detailed k·p band-structure analysis reveals that both InAs/GaSb and HgTe/CdTe quantum wells exhibit such buried Dirac points for a wide range of well thicknesses. By simulating transport in a disordered system described within an effective model, we demonstrate that edge-state transport remains nearly quantized up to large magnetic fields, consistent with recent experiments.