  InAs-based two-dimensional electron systems grown on lattice mismatched InP
substrates offer a robust platform for the pursuit of topologically protected
quantum computing. We investigated strained composite quantum wells of
In$_{0.75}$Ga$_{0.25}$As/InAs/In$_{0.75}$Ga$_{0.25}$As with
In$_{0.75}$Al$_{0.25}$As barriers. By optimizing the widths of the
In$_{0.75}$Ga$_{0.25}$As layers, the In$_{0.75}$Al$_{0.25}$As barrier, and the
InAs quantum well we demonstrate mobility in excess of $1 \times
10^{6}\,$cm$^{2}/$Vs. Mobility vs. density data indicates that scattering is
dominated by a residual three dimensional distribution of charged impurities.
We extract the Rashba parameter and spin-orbit length as important material
parameters for investigations involving Majorana zero modes.
