We introduce and describe a new heuristic method for finding an upper bound on the degree of contextuality and the corresponding unsatisfied part of a quantum contextual configuration with three-element contexts (i.e., lines) located in a multi-qubit symplectic polar space of order two. While the previously used method based on a SAT solver was limited to three qubits, this new method is much faster and more versatile, enabling us to also handle four- to six-qubit cases. The four-qubit unsatisfied configurations we found are quite remarkable. That of an elliptic quadric features 315 lines and has in its core three copies of the split Cayley hexagon of order two having a Heawood-graph-underpinned geometry in common. That of a hyperbolic quadric also has 315 lines but, as a point-line incidence structure, is isomorphic to the dual $\mathcal{DW}(5,2)$ of $\mathcal{W}(5,2)$. Finally, an unsatisfied configuration with 1575 lines associated with all the lines/contexts of the four-qubit space contains a distinguished $\mathcal{DW}(5,2)$ centered on a point-plane incidence graph of PG$(3,2)$. The corresponding configurations found in the five-qubit space exhibit a considerably higher degree of complexity, except for a hyperbolic quadric, whose 6975 unsatisfied contexts are compactified around the point-hyperplane incidence graph of PG$(4,2)$. The most remarkable unsatisfied patterns discovered in the six-qubit space are a couple of disjoint split Cayley hexagons (for the full space) and a subgeometry underpinned by the complete bipartite graph $K_{7,7}$ (for a hyperbolic quadric).

翻译：我们提出并描述了一种新的启发式方法，用于寻找位于二阶多量子比特辛极空间中的三元素上下文（即直线）所构成的量子上下文构型的上下文性度上界及其对应的不满足部分。以往基于SAT求解器的方法仅限于三量子比特情形，而新方法速度更快、适用性更广，使我们能够处理四至六量子比特的情况。我们发现的不满足四量子比特构型非常值得关注：椭圆二次曲面的构型包含315条直线，其核心由三个二阶分裂凯莱六边形的副本构成，这些副本共享一个基于Heawood图的几何结构；双曲二次曲面的构型同样包含315条直线，但作为点-线关联结构，它同构于$\mathcal{W}(5,2)$的对偶$\mathcal{DW}(5,2)$；最后，与四量子比特空间所有直线/上下文相关联的包含1575条直线的不满足构型中，存在一个以PG$(3,2)$的点-平面关联图为中心的显著$\mathcal{DW}(5,2)$结构。在五量子比特空间中发现的不满足构型展现出更高的复杂度，但双曲二次曲面除外——其6975个不满足上下文紧密围绕PG$(4,2)$的点-超平面关联图分布。在六量子比特空间中发现的最显著不满足模式包括：一对不相交的分裂凯莱六边形（针对完整空间），以及基于完全二分图$K_{7,7}$的子几何结构（针对双曲二次曲面）。