The Four-Qubit Hyperbolic Quadric Has Contextuality Degree 315
Abstract
The contextuality degree of a set of Pauli contexts is the least number of contexts that every noncontextual ±1 assignment violates. For the four-qubit hyperbolic quadric 𝒬+(7,2), with $135$ points and $1575$ lines as contexts, Muller et al. conjectured from heuristic searches that the degree is $315$ and that the violated contexts of every optimal assignment form a dual polar space DW(5,2). We prove both conjectures. The lower bound has a short proof by hand: every doily (GQ(2,2)) of the quadric contains ten Peres-Mermin grids covering each of its lines four times, so every assignment violates at least three of its fifteen lines, and averaging over the doilies gives 1575/5=315. The equality case is local: an assignment is optimal exactly when it violates one ruling of one grid in every doily. Using this as a SAT encoding, we show that there are exactly $240$ optimal violated sets, one orbit of O+(8,2), each isomorphic to DW(5,2). The same argument gives the first lower bounds ⌈ L/5⌉ for all quadrics with L lines in N≥4 qubits. The degree theorem is kernel-checked in Lean 4, and the completeness of the enumeration is certified by a DRAT proof checked by the verified checker cake_lpr.
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BibTeX
@misc{li2026fourqubit,
title = {{The Four-Qubit Hyperbolic Quadric Has Contextuality Degree 315}},
author = {Zhengyu Li and Amandeep Gupta and Vijay Ganesh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26465},
eprint = {2610.26465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26465},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Four-Qubit Hyperbolic Quadric Has Contextuality Degree 315
AU - Zhengyu Li
AU - Amandeep Gupta
AU - Vijay Ganesh
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26465
UR - https://arxiv.org/abs/2610.26465
AB - The contextuality degree of a set of Pauli contexts is the least number of contexts that every noncontextual ±1 assignment violates. For the four-qubit hyperbolic quadric 𝒬^+(7,2), with $135$ points and $1575$ lines as contexts, Muller et al. conjectured from heuristic searches that the degree is $315$ and that the violated contexts of every optimal assignment form a dual polar space DW(5,2). We prove both conjectures. The lower bound has a short proof by hand: every doily (GQ(2,2)) of the quadric contains ten Peres-Mermin grids covering each of its lines four times, so every assignment violates at least three of its fifteen lines, and averaging over the doilies gives 1575/5=315. The equality case is local: an assignment is optimal exactly when it violates one ruling of one grid in every doily. Using this as a SAT encoding, we show that there are exactly $240$ optimal violated sets, one orbit of O^+(8,2), each isomorphic to DW(5,2). The same argument gives the first lower bounds ⌈ L/5⌉ for all quadrics with L lines in N≥4 qubits. The degree theorem is kernel-checked in Lean 4, and the completeness of the enumeration is certified by a DRAT proof checked by the verified checker cake_lpr.
ER -