On the extremal points of the Λ-polytopes and classical simulation of quantum computation with magic states

We investigate the Λ-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, Λn, for every number n of qubits. We establish two properties of the family {Λn,n∈N}, namely (i) Any extremal point (vertex) Aα∈Λm can be used to construct vertices in Λn, 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 Aα. In addition, we describe a new class of vertices in Λ2 which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of Λn, the above results extend efficient classical simulation of quantum computations beyond the presently known range.