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

TL;DR

This paper studies the structure of $\Lambda$-polytopes related to quantum computation with magic states, revealing how extremal points can generate others and improve classical simulation methods.

Contribution

It establishes how extremal points of $\Lambda$-polytopes can generate new vertices and reduce simulation complexity, extending classical simulation capabilities.

Findings

Extremal points can generate vertices in larger $\Lambda_n$ polytopes.

Simulation complexity can be reduced using preimages of vertices.

Discovered new vertices in $\Lambda_2$ outside known classifications.

Abstract

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.