By name: quantum-probability-hebbian-learning description: "Quantum probability-flow methodology for deriving local Hebbian learning rules in associative memory networks. Use when: (1) quantum-inspired learning rules for neural networks, (2) attention mechanisms from quantum probability flow, (3) quantum annealer-based learning rule validation, (4) transverse-field leakage channels for stability-driven updates. Activation: quantum probability flow, Hebbian learning, quantum annealer, associative memory, transverse field, attention-like learning rule, softmax Hebbian, quantum annealing learning" license: Complete terms in LICENSE.txt metadata: arxiv_id: "2606.02098" published: "2026-06-01" authors: "Masayuki Ohzeki" tags: [quantum, hebbian-learning, associative-memory, attention, quantum-annealing]
Problem Statement
How to derive local learning rules for associative memory that exhibit attention-like behavior, using quantum mechanical principles.
Core Methodology
Quantum Probability-Flow Principle
- Transverse field defines leakage channels from data states
- Minimize measured survival loss → stability-driven weight updates
- Imaginary-time, dephased dynamics → local leakage free energy = log-sum-exp of energy gaps
- Gradient of leakage free energy → softmax-weighted Hebbian rule
Two Regimes
| Regime | Dynamics | Learning Rule |
|---|---|---|
| Imaginary-time | Dephased | Softmax-weighted Hebbian (log-sum-exp) |
| Real-time | Coherent | Power-law weighted Hebbian (Lorentzian) |
Empirical Validation
D-Wave standard-anneal and fast-anneal tests on one-hot attention forward map:
- Standard anneal: Better fitted by softmax than Lorentzian
- Fast anneal: Both regimes tested, softmax dominates
Reusable Patterns
Pattern 1: Stability-Driven Learning
Replace gradient descent with stability analysis:
- Define a transverse field (quantum fluctuation) on the energy landscape
- Measure state survival probability under perturbation
- Update weights to maximize survival (minimize leakage)
- Result: local learning rule emerges from global stability
Pattern 2: Softmax Hebbian from Quantum Flow
The leakage free energy gradient produces:
Δw_ij ∝ softmax(ΔE_k) · x_i · x_j
where ΔE_k are energy gaps between data and non-data states.
This recovers the softmax attention mechanism from first quantum principles.
Key Insights
- Attention-like weighting emerges naturally from quantum stability analysis
- Softmax is not an ad-hoc choice — it's the gradient of log-sum-exp leakage free energy
- Quantum annealer hardware provides physical validation of the learning rule
- Power-law alternative (from real-time dynamics) is less accurate empirically
Pitfalls
- 4-page paper: Core theory is concise; implementation details require extrapolation
- D-Ware specific: Validation on D-Wave hardware; results may vary on other quantum annealers
- One-hot mapping: Tested on one-hot attention; extension to distributed representations needed
References
- arXiv:2606.02098v1 — Attention-Like Hebbian Learning from Quantum Probability Flow and Quantum-Annealer Tests
- Related: cond-mat.dis-nn (disordered systems and neural networks)