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arXiv:2312.10734

The degenerate vertices of the $2$-qubit $Λ$-polytope and their update rules

Selman Ipek, Cihan Okay

Published Dec 17, 2023• 0 citations
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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 $Λ_{2}$, the $2$-qubit $Λ$ polytope. Our approach exploits the fact that $Λ_{2}$ projects to a well-understood polytope $\text{MP}$ consisting of distributions on the Mermin square scenario. More precisely, we study the ``classical" polytope $\overline{\text{MP}}$, which is $\text{MP}$ 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 $\text{MP}$ we utilize a streamlined version of the double-description method for vertex enumeration to obtain certain vertices of $\overline{\text{MP}}$.

DOI
10.48550/arxiv.2312.10734

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The degenerate vertices of the 222-qubit ΛΛΛ-polytope and their update rules
Recently, a class of objects, known as ΛΛΛ-polytopes, were introduced for classically simulating universal quantum computation with magic states. In ΛΛΛ-simulation, the probabilistic update of ΛΛΛ vertices under Pauli measurement yields dynamics consistent with quantum mechanics. Thus, an important open problem in the study of ΛΛΛ-polytopes is characterizing its vertices and determining their update rules. In this paper, we obtain and describe the update of all degenerate vertices of Λ2Λ_{2}Λ2​, the 222-qubit ΛΛΛ polytope. Our approach exploits the fact that Λ2Λ_{2}Λ2​ projects to a well-understood polytope MP\text{MP}MP consisting of distributions on the Mermin square scenario. More precisely, we study the ``classical" polytope MP‾\overline{\text{MP}}MP, which is MP\text{MP}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\text{MP}MP we utilize a streamlined version of the double-description method for vertex enumeration to obtain certain vertices of MP‾\overline{\text{MP}}MP.
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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.
Bell inequalities characterize the boundary of the local-realist correlation polytope -- the set of joint probability distributions achievable by classical hidden-variable models. Quantum mechanics exceeds this boundary through non-commutativity, reaching the Tsirelson bound 222\sqrt{2}22​ for CHSH. We show that this polytope structure is not specific to quantum foundations: it appears identically in the causal inference literature, where the instrumental inequality, the Balke--Pearl linear programming bounds, and the Tian--Pearl probabilities of causation all arise as facets of the same marginal compatibility polytope. Fine's theorem -- that CHSH inequalities hold if and only if a joint distribution exists -- is precisely the pivot: the instrumental variable model in causal inference is structurally equivalent to the Bell local hidden-variable model, with the instrument playing the role of the measurement setting and the latent confounder playing the role of the hidden variable λλλ. We develop this correspondence in detail, extending it to algorithmic (Kolmogorov complexity) and entropic formulations of Bell inequalities, the NPA semidefinite programming hierarchy, and the MIP∗^*∗=RE undecidability result. We further show that the Born-rule / Bayes-rule duality underlying quantum Bayesian computation exploits the same non-commutativity that enables Bell violation, providing polynomial speedups for posterior inference. The framework yields a concrete dictionary between quantum information theory, causal econometrics, and Bayesian computation, and suggests new directions including NPA-based quantum causal inference algorithms and quantum architectures for function approximation.
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