Energy–Noise Trade-offs and Local Cycle Stability in a Quantum-State Distance Network Toy Model

We study a computational toy model in which a connected weighted graph is built fromrandom quantum states: each node carries a density matrix ρi ∈ Cd×d (here d = 4) drawnfrom an induced (Wishart) ensemble, and baseline edge lengths are the quantum Jensen–Shannon distance dJS(ρi, ρj ) [ 1– 3]. We then “engineer shortcuts” by adding non-edges witha discrete set of admissible shortcut lengths, under two global budgets: an energy budgetEmax and an noise (depolarization) budget Pmax. To prevent locally unstable configurations,each candidate shortcut is screened by a cycle stability score κL evaluated on the shortcycles (triangles and optionally 4-cycles) that the shortcut would create; only shortcuts withκmax ≤ κlim are allowed. Across 100 independent random instances (n = 16 nodes), wequantify: (i) how global navigability (average weighted shortest-path length, APL) scaleswith Emax under fixed Pmax; (ii) an “optimism bias” that appears when stability is enforcedon triangles only (ignoring 4-cycles); and (iii) a regime transition under Pmax-ablation whereincreasing Emax becomes effective only once additional noise can be tolerated. All figuresand aggregated CSVs required to reproduce the reported means and confidence intervals are included with the Overleaf project.