  Spin qubits based on Si and Si$_{1-x}$Ge$_{x}$ quantum dot architectures
exhibit among the best coherence times of competing quantum computing
technologies, yet they still suffer from charge noise that limit their qubit
gate fidelities. Identifying the origins of these charge fluctuations is
therefore a critical step toward improving Si quantum-dot-based qubits. Here we
use hybrid functional calculations to investigate possible atomistic sources of
charge noise, focusing on charge trapping at Si and Ge dangling bonds (DBs). We
evaluate the role of global and local environment in the defect levels
associated with DBs in Si, Ge, and \sige alloys, and consider their trapping
and excitation energies within the framework of configuration coordinate
diagrams. We additionally consider the influence of strain and oxidation in
charge-trapping energetics by analyzing Si and Ge$_{\rm Si}$ DBs in SiO$_2$ and
strained Si layers in typical \sige quantum dot heterostructures. Our results
identify that Ge dangling bonds are more problematic charge-trapping centers
both in typical \sige alloys and associated oxidation layers, and they may be
exacerbated by compositional inhomogeneities. These results suggest the
importance of alloy homogeneity and possible passivation schemes for DBs in
Si-based quantum dot qubits and are of general relevance to mitigating possible
trap levels in other Si, Ge, and Si$_{1-x}$Ge$_{x}$-based
metal-oxide-semiconductor stacks and related devices.
