Spontaneous Crystallisation of Q3 Octahedra from Unstructured Qubit Networks under Simulated Quantum Annealing

We demonstrate via simulated annealing that an unstructured network of N qubits,

subject only to a degree-3 regularity constraint and spectral energy minimisation, sponta-

neously partitions into ⌊N/8⌋copies of the Q3 hypercube graph — the unique 3-regular,

vertex-transitive graph on 8 vertices supporting a distance-4 error-correcting code realis-

able as the face-adjacency graph of a regular octahedron in three dimensions. Over 100

independent trials from random initial conditions with N = 24, perfect Q3 crystallisation

occurs in 94% of runs. We prove that Q3 is the unique optimal target by ruling out all

competing graphs on independent geometric and coding-theoretic grounds: the Petersen

graph fails both the convex polyhedral embedding test and the 4-cycle (distance-4 parity

check) requirement. Frustrated configurations (N ̸≡0 (mod 8)) produce high-energy partial

clusters that cannot close their parity-check circuits, providing a discrete model of quantum

vacuum fluctuations. When inter-cluster bonding is permitted, the isolated octahedra spon-

taneously form bridge connections, assembling into a connected lattice network. The code

and all simulation data are publicly available for independent reproduction.