  In recent years, experimental advances have made it possible to achieve an
unprecedented degree of control over the properties of subgap bound states in
hybrid nanoscale superconducting structures. This research has been driven by
the promise of engineering subgap states for quantum applications, which
includes Majorana zero modes predicted to appear at the interface of
superconductor and other materials, like topological insulators or
semiconductors. In this chapter, we revise the status of the field towards the
engineering of quantum devices in controllable semiconductor-superconductor
heterostructures. We begin the chapter with a brief introduction about subgap
states, focusing on their mathematical formulation. After introducing
topological superconductivity using the Kitaev model, we discuss the advances
in the search for Majorana states over the last few years, highlighting the
difficulties of unambiguously distinguish these states from nontopological
subgap states. In recent years, the precise engineering of bound states by a
bottom-up approach using quantum dots has led to unprecedented experimental
advances, including experimental demonstrations of an Andreev qubits based on a
quantum dot Josephson junction and a minimal Kitaev chain based on two quantum
dots coherently coupled by the bound states of an intermediate superconducting
segment. These experimental advances have revitalized the field and helped to
understand that, far from being a disadvantage, the presence of subgap bound
states can be exploited for new qubit designs and quantum coherence
experiments, including Majorana-based qubits.
