quantum-probability-flow-hebbian-learning

name	quantum-probability-flow-hebbian-learning
category	ai_collection
description	Quantum probability-flow principle for deriving local Hebbian learning rules in associative memory networks using quantum annealer validation. arXiv:2606.02098
created	2026-06-19T00:00:00.000Z
version	1.0
tags	["quantum","hebbian-learning","associative-memory","quantum-annealing","probability-flow","attention","D-Wave"]
source	arXiv:2606.02098
trigger	quantum probability flow, Hebbian learning, associative memory, transverse field, survival loss, quantum annealer, D-Wave, softmax Hebbian, attention mechanism

Attention-Like Hebbian Learning from Quantum Probability Flow

Overview

A quantum probability-flow principle for deriving local learning rules in associative memory networks. Transverse field defines leakage channels from data states, and minimizing measured survival loss gives stability-driven updates. Validated on D-Wave quantum annealer.

Core Methodology

1. Quantum Probability Flow Principle

• Transverse field defines leakage channels from data states
• Survival loss: Measure probability leakage from target states
• Minimize survival loss: Derives local learning updates

2. Imaginary-Time Dephased Dynamics

For imaginary-time, dephased dynamics:

• Local leakage free energy = log-sum-exp of energy gaps
• Gradient = softmax-weighted Hebbian rule
• Attention-like weighting emerges naturally from quantum dynamics

3. Real-Time Dynamics

• Real-time stability yields power-law weighting
• Contrasts with softmax from imaginary-time dynamics

4. Experimental Validation

D-Wave standard- and fast-anneal tests of one-hot attention forward map:

• Better fitted by effective softmax than Lorentzian power law
• Confirms imaginary-time dynamics as better model

Implementation Pattern

1. Define transverse field Hamiltonian H = H_data + Γ·H_transverse
2. Measure survival probability of data states under evolution
3. Compute survival loss L = 1 - P_survival
4. Derive learning rule: Δw ∝ -∂L/∂w
5. For imaginary-time: softmax-weighted Hebbian update
6. For real-time: power-law weighted update
7. Validate on quantum annealer hardware

Applications

• Associative memory network design
• Quantum-inspired learning rules
• Attention mechanism derivation
• Quantum annealer validation
• Biologically plausible learning algorithms

Key Equations

• Leakage free energy: F_leak = log-sum-exp(ΔE_i)
• Imaginary-time gradient: ∂F_leak/∂w = softmax(ΔE) · Hebbian
• Real-time weighting: power-law(ΔE) · Hebbian

Pitfalls

• Imaginary-time vs real-time dynamics give qualitatively different learning rules
• D-Wave annealing approximates imaginary-time but not perfectly
• One-hot attention map is a simplified test case; general networks may differ