quantum-probability-hebbian-learning

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

1. Transverse field defines leakage channels from data states
2. Minimize measured survival loss → stability-driven weight updates
3. Imaginary-time, dephased dynamics → local leakage free energy = log-sum-exp of energy gaps
4. 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:

1. Define a transverse field (quantum fluctuation) on the energy landscape
2. Measure state survival probability under perturbation
3. Update weights to maximize survival (minimize leakage)
4. 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)