  We investigate the $\Lambda$-polytopes, a convex-linear structure recently
defined and applied to the classical simulation of quantum computation with
magic states by sampling. There is one such polytope, $\Lambda_n$, for every
number $n$ of qubits. We establish two properties of the family $\{\Lambda_n,
n\in \mathbb{N}\}$, namely (i) Any extremal point (vertex) $A_\alpha \in
\Lambda_m$ can be used to construct vertices in $\Lambda_n$, for all $n>m$.
(ii) For vertices obtained through this mapping, the classical simulation of
quantum computation with magic states can be efficiently reduced to the
classical simulation based on the preimage $A_\alpha$. In addition, we describe
a new class of vertices in $\Lambda_2$ which is outside the known
classification. While the hardness of classical simulation remains an open
problem for most extremal points of $\Lambda_n$, the above results extend
efficient classical simulation of quantum computations beyond the presently
known range.
