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

Abstract

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, $Λ_n$, for every number $n$ of qubits. We establish two properties of the family $\{Λ_n, n\in \mathbb{N}\}$, namely (i) Any extremal point (vertex) $A_α\in Λ_m$ can be used to construct vertices in $Λ_n$, 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 $A_α$. In addition, we describe a new class of vertices in $Λ_2$ which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of $Λ_n$, the above results extend efficient classical simulation of quantum computations beyond the presently known range.