A new heuristic approach for contextuality degree estimates and its four- to six-qubit portrayals

TL;DR

This work develops a fast, geometry-aware heuristic to bound the contextuality degree $d$ for quantum configurations with three-element contexts in multi-qubit symplectic polar spaces $\mathcal{W}(2N-1,2)$, enabling analysis beyond three qubits. By reframing the problem as a stochastic local search over sign assignments with a threshold $\theta$ and flip probability $\gamma$, the method substantially improves upper bounds on $d$ for $N=4$–$7$ and reveals rich finite-geometric structures underpinning the unsatisfied parts, such as $\mathrm{DW}(5,2)$ on hyperbolic quadrics and split Cayley hexagons on elliptic quadrics. The authors derive lower bounds on the full-space contextuality degree and relate them to quadrics and incidence graphs (e.g., PG$(d,2)$ and Heawood/Coxeter graphs), highlighting deep connections between contextuality and finite geometry. The results offer a scalable framework for exploring contextuality in larger qubit spaces and motivate future work on higher-arity contexts and cross-space intersections, with potential implications for contextuality-based quantum information processing.

Abstract

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).

Figures (14)

• Figure 1: Graphical illustration of the successive steps of the algorithm on a two-spread, showing the sign assigned to each observable and in the associated table the number of unsatisfied contexts (dashed lines) containing it. The negative contexts are represented by the doubled lines colored in red. The value $+1$ is first assigned to all 15 observables (a). The last step (c) reaches the minimal possible distance for this geometry. In this example, the threshold $\theta$ is $0.8$ and the sign flip probability $\gamma$ is $0.7$.
• Figure 2: Minimal Hamming distances (the $y$-axis) per total number of contexts, over the number of iterations (the $x$-axis), computed by the heuristic method running simultaneously on 200 threads shared between 20 cores of an Intel(R) Core(TM) i7-12700H processor, for the quantum configurations composed of all the lines of the three- to six-qubit symplectic polar spaces.
• Figure 3: Properties of the point-line geometry comprising 315 unsatisfied constraints of a particular elliptic quadric whose index is $IIIY$. For a point on a line, the number inside the circle corresponds to its restricted degree in the configuration consisting solely of lines of this particular type.
• Figure 4: An illustration of the Heawood graph (left) and the Coxeter one (right), both drawn with seven-fold rotational symmetry.
• Figure 5: A generic layering of each of the three split Cayley hexagons of order two encapsulating the core of ${E}^{{\tt uns}}_{4}$. The 14 solid points are colored blue, the 21 dotted points are gray and 28 out of 84 dashed points are represented by red color. The 21 blue lines are the Heawood lines, those colored red are the Coxeter ones. Removing from the hexagon the 21 gray points we indeed get a disjoint union of the Heawood graph and the Coxeter graph.

Theorems & Definitions (1)

• Definition 1