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: 2026-06-19
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
- Transverse field strength Γ must be carefully tuned
- Survival loss measurement requires multiple annealing runs