Abstract
Recently, a class of objects, known as Λ-polytopes, were introduced for classically simulating universal quantum computation with magic states. In Λ-simulation, the probabilistic update of Λ vertices under Pauli measurement yields dynamics consistent with quantum mechanics. Thus, an important open problem in the study of Λ-polytopes is characterizing its vertices and determining their update rules. In this paper, we obtain and describe the update of all degenerate vertices of Λ₂, the 2-qubit Λ polytope. Our approach exploits the fact that Λ₂ projects to a well-understood polytope MP consisting of distributions on the Mermin square scenario. More precisely, we study the ``classical" polytope MP, which is MP intersected by the polytope defined by a set of Clauser-Horne-Shimony-Holt (CHSH) inequalities. Owing to a duality between CHSH inequalities and vertices of MP we utilize a streamlined version of the double-description method for vertex enumeration to obtain certain vertices of MP.