  The electronic structure of surfaces and interfaces plays a key role in the
properties of quantum devices. Here, we study the electronic structure of
realistic Al/InAs/Al heterojunctions using a combination of density functional
theory (DFT) with hybrid functionals and state-of-the-art quasi-particle $GW$
(QS$GW$) calculations. We find a good agreement between QS$GW$ calculations and
hybrid functional calculations which themselves compare favourably well with
ARPES experiments. Our study confirm the need of well controlled quality of the
interfaces to obtain the needed properties of InAs/Al heterojunctions. A
detailed analysis of the effects of spin-orbit coupling on the spin-splitting
of the electronic states show a linear scaling in $k$-space, related to the
two-dimensional nature of some interface states. The good agreement by QS$GW$
and hybrid functional calculations open the door towards trust-able use of an
effective approximation to QS$GW$ for studying very large heterojunctions.
