Quantum Ballet of Classical Networks: Robust Quantum-Like States on Complex Synchronized Networks

This post reviews and translates Gregory D. Scholes' 2024 preprint "Quantum-like states on complex synchronized networks" (arXiv:2405.07950 [physics.soc-ph]), which argues that quantum-like (QL) behavior can emerge robustly from purely classical, complex networks—no cryogenic isolation required.

Key points

1. The core idea

• Quantum fragility (decoherence) makes conventional quantum states vulnerable to environmental noise. Scholes asks instead: can quantum-*like* probability laws, including interference effects, emerge in classical systems?
• The answer: yes, via k-regular random graphs whose synchronization dynamics produce emergent eigenstates with quantum-like properties.

2. Graph theory foundations

• Networks are modeled as graphs with adjacency matrices; for a k-regular graph the largest eigenvalue λ₀ = k.
• The isoperimetric constant h(G) measures expansion (communication efficiency), bounded by:
\[\frac{k - \lambda_1}{2} \leq h(G) \leq \sqrt{2k(k - \lambda_1)}\]
• Smaller second eigenvalue λ₁ ⟹ better expansion. Expander graphs (e.g., Ramanujan graphs) are ideal substrates.
• Robustness: numerically, a k=20, n=40 graph retains its isolated largest eigenvalue even after ~300 of 400 edges are randomly deleted.

3. Axioms for quantum-like states

QL states must satisfy five axioms: 1. Clearly distinguishable emergent states in the spectrum 2. Robustness to construction imprecision and node-frequency disorder 3. Two-state systems requiring network activation (no vacuum ground state) 4. Unitary operations on graphs/subgraphs 5. Coupling that yields tensor-product (Hilbert) state spaces

4. Building qubits and entanglement

• Partition a k-regular random graph into subgraphs G_a1, G_a2 with a tensor-sum structure A = A_a1 ⊕ A_a2; connecting edges (with tunable sign/probability) create superpositions like √(1/2)(a₁ ± a₂).
• Physically, vertices are oscillators (neurons, circuit pulses); positive edges = in-phase coupling, negative = anti-phase.
• Coupled qubit pairs with a Hamiltonian of the form H = ℏν_A|A⟩⟨A| + ℏν_B|B⟩⟨B| + σJ(|A⟩⟨B| + |B⟩⟨A|) can produce Bell states; N coupled qubits map onto the single-excitation subspace, homologous to H_A ⊗ H_B ⊗ ….

5. Contextual probability: the Växjö model

• Following Khrennikov, probabilities are context-dependent: P(b=β|C) = Σ_α p^a_C(α) p_{β|α} + δ(β|a, C), where δ is an interference term.
• This explains the question order effect in psychology surveys as context change via system–probe interaction under unitary evolution U_A on H ⊗ K.

6. Applications

• Superdense coding: two classical bits per qubit via four gate operations (identity, NOT, phase flip, both) implemented as network rewirings/phase flips of coupling edges.
• Quantum sensing & pattern recognition: one qubit senses, the other reads out via Bell-state measurement; networks can be trained to synchronize onto target states.

7. The brain hypothesis

• Neural oscillations (delta through gamma, 30–80 Hz) and long-range beta/gamma synchronization encode information coherently; phase synchronization solves the perceptual "binding" problem.
• Hypothesis: dense "hairball" brain networks, through feedback-driven synchronization, may host emergent states satisfying the QL axioms—robust precisely because they are classical.
• Testable predictions: non-classical correlations in synchronized neural networks; question order effects in neural decision-making; gate-like operations via connection-pattern control.

8. Open questions

1. Quantum advantage: can QL circuits outperform classical ones? Testable on electronic oscillator circuits or spiking/neuromorphic hardware. 2. Does the brain use QL states? Evidence would require measuring non-classical statistical signatures in vivo.

9. Why it matters

• The protective mechanism is an emergent energy gap—a spectrally isolated eigenvalue arising from collective coherence—not fragile microscopic quantum superposition.
• If realized, this could drastically simplify "quantum-like computation," offering a middle ground between classical and quantum hardware.

References

1. Gregory D. Scholes, *Quantum-like states on complex synchronized networks*, arXiv:2405.07950v1 [physics.soc-ph], 2024. 2. Andrei Khrennikov, *Contextual Probability and Quantum-like Modeling* (The Växjö Model), 2020. 3. Shlomo Hoory, Nathan Linial, Avi Wigderson, *Expander Graphs and Their Applications*, Bull. Amer. Math. Soc., 2006.