  We investigate the electronic and transport properties of topological and
trivial InAs$_{1-x}$Bi$_x$ quantum dots (QDs). By considering the rapid band
gap change within valence band anticrossing theory for InAs$_{1-x}$Bi$_x$, we
predicted that Bi-alloyed quantum wells become $\sim 30$meV gapped 2D
topological insulators for well widths $d>6.9$nm $(x = 0.15)$ and obtain the
$\boldsymbol{k.p}$ parameters of the corresponding Bernevig-Hughes-Zhang (BHZ)
model. We analytically solve this model for cylindrical confinement via
modified Bessel functions. For non-topological dots we find "geometrically
protected" discrete helical edge-like states, i.e., Kramers pairs with
spin-angular-momentum locking, in stark contrast with ordinary InAs QDs. For a
conduction window with four edge states, we find that the two-terminal
conductance ${\cal G}$ vs. the QD radius $R$ and the gate $V_g$ controlling its
levels shows a double peak at $2e^2/h$ for both topological and trivial QDs. In
contrast, when bulk and edge-state Kramers pairs coexist and are degenerate, a
single-peak resonance emerges. Our results blur the boundaries between
topological and non-topological phenomena for conductance measurements in small
systems such as QDs. Bi-based BHZ QDs should also prove important as hosts to
edge spin qubits.
