The Quad- C 5 Graph: Maximum Contextuality Gap on Eight Vertices

This paper investigates graph-theoretic approaches to identifying strong contextuality witnesses, specifically focusing on maximizing the absolute contextuality gap, denoted as ∆(G) = ϑ(G) − α(G), for graphs with eight vertices. It addresses the challenge of finding compact and powerful noncontextual signatures in quantum systems by searching beyond analytically understood structures like cycles and benchmarks such as the Wagner graph. The discovery of a new structure, Quad-C5, which outperforms existing benchmarks in this metric, provides a compact bridge between minimal qutrit contextuality and stronger four-dimensional contextuality witnesses, offering a resource for understanding how contextuality can be organized and amplified.

Graph-Theoretic Framework for Contextuality

The study utilizes the graph-theoretic framework where each measurement event is represented as a vertex, and mutually exclusive events are connected by edges, forming an exclusivity graph G = (V, E). The core objective is to maximize the contextuality gap ∆(G) defined by Equation (2), which measures the separation between quantum and noncontextual descriptions within this graph model. This framework allows for a systematic search over all 11,117 connected non-isomorphic graphs with eight vertices. The analysis uses two primary quantities: the independence number α(G), representing the classical bound, and the Lovász theta number ϑ(G), representing the projective quantum bound derived from a semidefinite program (Eq. 3).

Identification of Quad-C5 as the Optimal Witness

An exhaustive search over all eight-vertex graphs identified a sparse ten-edge graph called Quad-C5 as the maximum-gap witness. This graph possesses a larger absolute contextuality gap than the Wagner graph, despite having fewer exclusivity relations. Its structure is described as being formed from four overlapping KCBS pentagons, with each edge shared by two pentagons. The numerical certification confirms that Quad-C5 achieves a gap of ∆ = 0.46784, surpassing the Wagner graph's gap of ∆ ≈ 0.41421.

Structural Properties and Spectral Fingerprint

The structure of Quad-C5 is defined by a vertex set V = [0, 7] and an edge set E with E = 10. It exhibits a degree sequence of (2, 2, 2, 2, 3, 3, 3, 3), featuring four hub nodes and four leaf nodes. A defining structural feature is that Quad-C5 contains exactly four induced five-cycles (KCBS pentagons), with the crucial property that every one of the 10 edges belongs to exactly two of the four C5 subgraphs—a perfect twofold edge coverage. Furthermore, its adjacency spectrum shows that six eigenvalues belong to the golden-ratio field Q(√5), which is a direct spectral fingerprint of the embedded C5 subgraphs.

Dimension Dependence and Physical Realization

The paper analyzes the Hilbert space dimension required to realize this advantage. It demonstrates that Quad-C5 is already contextual for a single qutrit by giving an explicit three-dimensional construction, where its contextuality margin equals that of KCBS. However, its full quantum advantage is reached numerically in four dimensions and exceeds that of the Wagner graph. The analysis shows that while the witness is qutrit-contextual at d=3, it requires d=4 to reach the full Lovász bound numerically.

Noise Robustness and Experimental Feasibility

The study evaluates noise robustness using a depolarizing channel, deriving a critical visibility threshold v∗ based on Equation (14). A key finding is that Quad-C5 at d = 3 shares the critical visibility v∗ = 1/(3√5 − 5) ≈ 0.585 of the KCBS witness, confirming an analytical equality due to a uniform shift in graph parameters. At d=4, Quad-C5 strictly outperforms the Wagner graph in noise robustness, requiring 2.6% less visibility. The paper concludes that this structure is experimentally more appealing than the Wagner benchmark because it requires a strictly lower critical visibility under depolarizing noise.

Algebraic Connection to KCBS

The paper establishes a deep algebraic link between Quad-C5 and the minimal qutrit witness, KCBS. Both structures share values in the field Q(√5): both ϑ(C5) = √5 and η3(Quad-C5) = 1 + √5 lie in Q(√5). This indicates that the advantage of Quad-C5 above the classical bound α=3 is exactly √5—one unit of KCBS advantage above the classical floor. The spectral analysis further reinforces this, showing that the contextuality advantage is inherited from the KCBS subgraphs through its eigenvalues.

Here are specific improvements to AI systems derived from the insights in this paper, categorized by application area:

)AI System Improvement: Quantum Contextuality Modeling & Simulation Engine (QCM-SE)

The QCM-SE can integrate the graph-theoretic framework of contextuality witnesses into its core architecture, moving beyond simple correlation checks to model complex resource constraints.

1. [] Perform exhaustive combinatorial optimization over non-isomorphic graphs for a fixed number of vertices (e.g., finding the optimal 8-vertex witness).

2. [] Implement SDP solvers (like CVXPY/SCS) to calculate Lovász Theta numbers, providing the upper quantum bound, and independence numbers for classical bounds across massive graph libraries in real-time.

3. [] Develop a Gap Maximization objective function tailored to maximize the absolute contextuality gap, bridging the gap between minimal qutrit tests (like KCBS) and higher-dimensional witnesses.

)AI System Improvement: Quantum Resource Allocation & State Preparation Optimizer (QR-ASPO)

The QR-ASPO will use the explicit orthogonal vector representations derived from the Quad-C5 construction to design optimal quantum states for specific measurement sequences.

1. [] Design and generate high-dimensional, rank-one projector sets (like those in Appendix A) that satisfy complex exclusivity constraints defined by arbitrary graphs (G = (V, E)).

2. [] Optimize the input state preparation protocol to yield a target maximum eigenvalue for the resulting measurement operator, ensuring the system realizes a specific dimension-constrained contextuality margin (e.g., targeting 1 + √5 for qutrit systems).

3. [] Implement non-injective representations (where projectors are repeated) to model practical experimental constraints in physical systems where vertex labels might be redundant or physically equivalent.

)AI System Improvement: Noise Robustness & Experimental Feasibility Predictor (NREFP)

The NREFP will use the derived analytical relationships between graph parameters and noise thresholds to predict the required hardware resources for a given measurement task.

1. [] Calculate the critical visibility threshold, using the analytically confirmed shift invariant relationship:

v∗(G, d) = α(G) - n/dηd(G) + (n/d - n/3).

2. [] Predict the minimum required Hilbert space dimension (d=3 vs. d=4) necessary to maintain contextuality certification under a specified depolarizing noise model, identifying when the system transitions from a qutrit-contextual regime to a stronger two-qubit regime.

3. [] Determine the optimal hardware platform (e.g., ion trap vs. photonic path) by comparing the required critical visibility against known platform capabilities, specifically noting where Quad-C5 outperforms Wagner in noise tolerance at d=4.

)AI System Improvement: Spectral Fingerprint Analysis Module (SFAM)

The SFAM will analyze the adjacency spectra of complex quantum correlation graphs to infer their underlying structural properties and contextuality heritage.

1. [] Compute the adjacency spectrum of a given exclusivity graph and perform spectral analysis to identify algebraic structures (e.g., membership in field extensions like Q(√5)).

2. [] Use spectral fingerprints (eigenvalues) as a diagnostic tool to determine if the contextuality advantage is inherited from known minimal constructions (like C5 pentagons) or arises from novel graph structures.

3. [] Automate the search for algebraic relations between system parameters and spectral coefficients, providing high-confidence numerical identification of closed-form solutions for Lovász Theta functions.

)AI System Improvement: Contextuality Benchmark & Hierarchy Navigator (CBHN)

The CBHN will serve as a knowledge base to guide experimentalists toward the most efficient contextuality tests based on resource constraints.

1. [] Maintain a structured hierarchy of contextuality witnesses, ranking them by absolute gap (∆(G)) and relative strength (ϑ(G)/α(G)), rather than just graph symmetry or edge count.

2. [] Provide actionable recommendations for experimental setups: If your hardware supports d=3 systems, Quad-C5 is the optimal 8-event test, offering a noise tolerance equal to KCBS.

3. [] Automatically compare candidate graphs against established benchmarks (Wagner graph) to quantify the efficiency gain (e.g., Quad-C5 achieves a 13% larger gap with two fewer edges).

Quantum measurements can exhibit contextuality: their outcomes cannot always be explained by assigning pre-existing values that are independent of which compatible measurements are performed together. The Klyachko-Can-Binicioğlu-Shumovsky (KCBS) inequality provides the canonical minimal test of this effect for a single three-level quantum system, or qutrit, using five measurement events arranged as a pentagon. Here we ask whether a larger but still compact set of measurement events can produce a stronger separation between quantum predictions and the corresponding noncontextual limit. We perform an exhaustive search over all 11,117 connected non-isomorphic graphs with eight vertices, where each vertex represents a measurement event and edges connect pairs of events that cannot occur together. We identify a sparse ten-edge graph, which we call Quad- C 5, as the unique maximizer of this separation at the reported numerical precision. The graph can be understood as four overlapping KCBS pentagons, with every edge shared by two pentagons. Quad- C 5 already demonstrates contextuality in a qutrit, for which we obtain an exact analytical result and find the same violation above the noncontextual bound as in the original KCBS test. The larger quantum-noncontextual separation allowed by the graph, however, becomes accessible in a four-level quantum system, where numerical optimization reaches the full graph-theoretic quantum bound and yields a larger contextuality gap than the standard eight-vertex Wagner-graph benchmark while requiring fewer pairwise constraints. Quad- C 5 therefore provides a compact connection between minimal qutrit contextuality and stronger contextuality tests available in higher-dimensional quantum systems.