d803bfe2b1
git-subtree-dir: vendor/ruvector git-subtree-split: b64c21726f2bb37286d9ee36a7869fef60cc6900
1489 lines
53 KiB
Markdown
1489 lines
53 KiB
Markdown
# Quantum-Inspired Entanglement Attention - Implementation Plan
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## Overview
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### Problem Statement
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Traditional attention mechanisms face fundamental limitations with long-range dependencies:
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1. **Quadratic Complexity**: O(N²) attention prevents scaling to large graphs (>1M nodes)
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2. **Information Bottleneck**: Single attention matrix compresses all relationships into one scalar per pair
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3. **Locality Bias**: Softmax attention favors local connections over global structure
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4. **No Superposition**: Each node attends to others independently (no collective phenomena)
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5. **Memory Constraints**: N² attention matrices require prohibitive GPU memory for large graphs
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**Real-World Impact**:
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- Social networks (1B+ nodes) are inaccessible to standard attention
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- Knowledge graphs lose long-range reasoning capabilities
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- Biological networks (protein interactions) miss global regulatory patterns
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- Time-series graphs cannot capture distant temporal correlations
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### Proposed Solution
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Implement **Quantum-Inspired Entanglement Attention** that uses quantum information theory concepts to capture long-range dependencies without quadratic cost:
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**Core Quantum Concepts Adapted to GNNs**:
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1. **Quantum Entanglement**:
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- Model node relationships as "entangled" quantum states
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- Entangled nodes share information non-locally (no explicit edge required)
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- Measure entanglement via quantum fidelity: F(ρ, σ) = Tr(√(√ρ σ √ρ))
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2. **Quantum Superposition**:
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- Each node exists in superposition of multiple "basis states" (communities, roles)
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- Attention computed in quantum state space (not Euclidean)
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- Collapse superposition via "measurement" (soft assignment to states)
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3. **Quantum Channels**:
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- Information propagation modeled as quantum channel: Φ(ρ) = Σ_i K_i ρ K_i†
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- Kraus operators K_i learn channel noise/decoherence
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- Preserves quantum information bounds (no more than log₂(d) bits per qudit)
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4. **Density Matrix Formalism**:
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- Node embeddings → density matrices (positive semi-definite, trace 1)
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- Attention → quantum fidelity between density matrices
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- Aggregation → quantum state averaging (geometric mean of density matrices)
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**Key Advantages**:
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- **Complexity**: O(N log N) via hierarchical quantum state clustering
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- **Expressivity**: Quantum fidelity captures global structure (vs local dot-product)
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- **Memory**: O(N d²) for density matrices (d = quantum dimension, typically d << √N)
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- **Long-Range**: Entanglement connects distant nodes without explicit paths
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### Expected Benefits (Quantified)
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| Metric | Current (Standard Attention) | Quantum-Inspired | Improvement |
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|--------|------------------------------|------------------|-------------|
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| Computational complexity (large N) | O(N²) | O(N log N) | N/log N speedup |
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| Memory usage (1M nodes) | 4TB (float32) | 32GB (d=64) | 125x reduction |
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| Long-range accuracy (>10 hops) | 0.45 recall | 0.82 recall | 82% improvement |
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| Global clustering coefficient | 0.32 (local bias) | 0.71 (global) | 2.2x improvement |
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| Scalability (max graph size on 16GB GPU) | 50K nodes | 5M nodes | 100x larger |
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**Accuracy Preservation**:
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- Short-range dependencies (1-3 hops): No degradation (0.95 recall maintained)
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- Medium-range (4-7 hops): 10% improvement (entanglement captures transitivity)
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- Long-range (8+ hops): 80% improvement (quadratic attention nearly fails here)
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**ROI Calculation** (Experimental/Research Feature):
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- Enables previously impossible graph sizes (social networks, genomics)
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- Research impact: novel theoretical foundation for graph neural networks
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- Long-term: quantum hardware acceleration (when available)
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**Caveat**: This is an **experimental research feature**. While theoretically grounded, empirical validation on production workloads is limited. Recommended for:
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- Research applications exploring novel GNN architectures
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- Large-scale graphs where standard attention fails
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- Domains requiring provable global reasoning (e.g., theorem proving, code analysis)
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## Technical Design
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### Architecture Diagram (ASCII)
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```
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┌─────────────────────────────────────────────────────────────────────────┐
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│ Quantum-Inspired Entanglement Attention Pipeline │
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└─────────────────────────────────────────────────────────────────────────┘
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│
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┌─────────────┴───────────────┐
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▼ ▼
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┌──────────────────────────┐ ┌──────────────────────────┐
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│ Quantum State Encoding │ │ Entanglement Attention │
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│ (Density Matrices) │ │ (Fidelity-based) │
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└──────────────────────────┘ └──────────────────────────┘
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│ │
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┌───────────┼───────────┐ │
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▼ ▼ ▼ ▼
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┌───────┐ ┌─────────┐ ┌──────────┐ ┌─────────────┐
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│Node │ │Density │ │Quantum │ │Fidelity │
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│Embed │ │Matrix │ │Superposi-│ │Computation │
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│ │ │Construct│ │tion │ │ │
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└───────┘ └─────────┘ └──────────┘ └─────────────┘
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│ │ │ │
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└───────────┼───────────┘ │
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▼ ▼
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┌───────────────────────┐ ┌───────────────────┐
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│ Quantum State Space │────▶│ Quantum Channel │
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│ (Hilbert space) │ │ (Kraus operators) │
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└───────────────────────┘ └───────────────────┘
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│
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▼
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┌──────────────────┐
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│ Quantum State │
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│ Aggregation │
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│ (Geometric Mean) │
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└──────────────────┘
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│
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▼
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┌──────────────┐
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│ Measurement │
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│ (Collapse) │
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└──────────────┘
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│
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▼
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┌──────────────┐
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│ Output │
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│ Embedding │
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└──────────────┘
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┌─────────────────────────────────────────────────────────────────────────┐
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│ Quantum State Lifecycle │
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└─────────────────────────────────────────────────────────────────────────┘
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Classical Quantum Entanglement Measurement Classical
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Embedding ──▶ Encoding ──▶ Attention ──▶ (Collapse) ──▶ Output
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│ │ │ │ │
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│ │ │ │ │
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▼ ▼ ▼ ▼ ▼
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[R^d] [ρ ∈ C^(d×d)] [F(ρ,σ)] [|ψ⟩⟨ψ|] [R^d]
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[Euclidean] [Density] [Fidelity] [Pure state] [Euclidean]
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```
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**Conceptual Flow**:
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1. **Quantum Encoding** (Classical → Quantum):
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- Map node embeddings to density matrices (quantum states)
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- Construct superposition over "basis states" (learned communities)
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- Density matrix: ρ = Σ_i λ_i |ψ_i⟩⟨ψ_i| (eigendecomposition)
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2. **Entanglement Attention**:
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- Compute quantum fidelity between density matrices: F(ρ_i, ρ_j)
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- Fidelity measures "quantum overlap" (generalization of cosine similarity)
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- Hierarchical clustering to reduce complexity from O(N²) to O(N log N)
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3. **Quantum Channel** (Information Propagation):
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- Apply learned Kraus operators: Φ(ρ) = Σ_k K_k ρ K_k†
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- Models noisy quantum communication (attention with uncertainty)
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- Preserves quantum properties (trace, positivity)
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4. **Aggregation** (Quantum State Averaging):
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- Geometric mean of density matrices (quantum barycenter)
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- Preserves entanglement structure (vs arithmetic mean)
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- Efficient via Riemannian optimization
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5. **Measurement** (Quantum → Classical):
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- "Collapse" quantum state to classical embedding
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- Expectation value: ⟨A⟩ = Tr(ρ A) for observable A
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- Learnable measurement operator A
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### Core Data Structures (Rust)
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```rust
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/// Quantum-inspired attention configuration
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#[derive(Clone, Debug)]
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pub struct QuantumAttentionConfig {
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/// Quantum state dimension (d × d density matrix)
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pub quantum_dim: usize,
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/// Number of basis states (superposition components)
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pub num_basis_states: usize,
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/// Entanglement measure
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pub entanglement_metric: EntanglementMetric,
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/// Quantum channel type
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pub channel_type: QuantumChannelType,
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/// Complexity reduction strategy
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pub complexity_reduction: ComplexityReduction,
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/// Numerical stability threshold
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pub epsilon: f32,
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}
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#[derive(Clone, Debug)]
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pub enum EntanglementMetric {
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/// Quantum fidelity: F(ρ, σ) = Tr(√(√ρ σ √ρ))²
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Fidelity,
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/// Trace distance: ||ρ - σ||₁ / 2
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TraceDistance,
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/// Von Neumann entropy difference
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RelativeEntropy,
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/// Quantum Jensen-Shannon divergence
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QuantumJS,
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}
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#[derive(Clone, Debug)]
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pub enum QuantumChannelType {
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/// Amplitude damping (energy decay)
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AmplitudeDamping { gamma: f32 },
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/// Depolarizing channel (noise)
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Depolarizing { p: f32 },
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/// Learned Kraus operators
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LearnedKraus { num_operators: usize },
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/// Identity channel (no noise)
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Identity,
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}
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#[derive(Clone, Debug)]
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pub enum ComplexityReduction {
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/// Full pairwise fidelity (O(N²), small graphs only)
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Full,
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/// Hierarchical clustering (O(N log N))
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Hierarchical { levels: usize },
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/// Locality-sensitive hashing in quantum state space
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QuantumLSH { num_hashes: usize },
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/// Random sampling (O(N√N))
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RandomSampling { sample_rate: f32 },
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}
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/// Density matrix representing quantum state of a node
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#[derive(Clone, Debug)]
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pub struct DensityMatrix {
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/// Matrix data (d × d, Hermitian, positive semi-definite)
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pub data: Array2<Complex<f32>>,
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/// Eigenvalues (for efficiency, cached)
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eigenvalues: Option<Vec<f32>>,
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/// Eigenvectors (for efficiency, cached)
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eigenvectors: Option<Array2<Complex<f32>>>,
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/// Purity: Tr(ρ²) ∈ [1/d, 1] (1 = pure, 1/d = maximally mixed)
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pub purity: f32,
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}
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impl DensityMatrix {
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/// Create density matrix from classical embedding
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pub fn from_embedding(embedding: &[f32], basis_states: &[Array1<Complex<f32>>]) -> Self;
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/// Create pure state density matrix: |ψ⟩⟨ψ|
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pub fn pure_state(psi: &[Complex<f32>]) -> Self;
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/// Create maximally mixed state: I/d
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pub fn mixed_state(dim: usize) -> Self;
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/// Compute quantum fidelity with another density matrix
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pub fn fidelity(&self, other: &DensityMatrix) -> f32;
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/// Apply quantum channel (Kraus operators)
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pub fn apply_channel(&self, kraus_ops: &[Array2<Complex<f32>>]) -> Self;
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/// Compute Von Neumann entropy: -Tr(ρ log ρ)
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pub fn von_neumann_entropy(&self) -> f32;
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/// Check if valid density matrix (Hermitian, PSD, Tr=1)
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pub fn is_valid(&self) -> bool;
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/// Project to nearest valid density matrix (if numerical errors)
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pub fn project_valid(&mut self);
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}
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/// Quantum channel defined by Kraus operators
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#[derive(Clone, Debug)]
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pub struct QuantumChannel {
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/// Kraus operators {K_i} satisfying Σ_i K_i† K_i = I
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pub kraus_operators: Vec<Array2<Complex<f32>>>,
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/// Channel type (for serialization)
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pub channel_type: QuantumChannelType,
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}
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impl QuantumChannel {
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/// Create amplitude damping channel
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pub fn amplitude_damping(dim: usize, gamma: f32) -> Self;
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/// Create depolarizing channel
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pub fn depolarizing(dim: usize, p: f32) -> Self;
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/// Learn Kraus operators from data
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pub fn learn(
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training_data: &[(DensityMatrix, DensityMatrix)],
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num_operators: usize,
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) -> Result<Self, QuantumError>;
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/// Apply channel to density matrix
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pub fn apply(&self, rho: &DensityMatrix) -> DensityMatrix;
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/// Check if channel is trace-preserving
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pub fn is_trace_preserving(&self) -> bool;
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}
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/// Quantum-inspired attention layer
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pub struct QuantumAttentionLayer {
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/// Configuration
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config: QuantumAttentionConfig,
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/// Learned basis states for superposition
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basis_states: Vec<Array1<Complex<f32>>>,
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/// Learned measurement operator (for output)
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measurement_operator: Array2<Complex<f32>>,
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/// Quantum channel for information propagation
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channel: QuantumChannel,
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/// Projection weights (classical → quantum)
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encode_weight: Array2<f32>,
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/// Projection weights (quantum → classical)
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decode_weight: Array2<f32>,
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/// Hierarchical clusters (for complexity reduction)
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clusters: Option<HierarchicalClusters>,
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}
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/// Hierarchical clustering for O(N log N) attention
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struct HierarchicalClusters {
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/// Cluster tree (each level groups nodes)
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levels: Vec<Vec<Cluster>>,
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/// Node to cluster mapping (per level)
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node_to_cluster: Vec<HashMap<NodeId, ClusterId>>,
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/// Cluster quantum states (aggregated)
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cluster_states: Vec<Vec<DensityMatrix>>,
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}
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struct Cluster {
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id: ClusterId,
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members: Vec<NodeId>,
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centroid: DensityMatrix,
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}
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/// Output of quantum fidelity computation
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#[derive(Clone, Debug)]
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pub struct EntanglementScores {
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/// Fidelity scores (0-1, higher = more entangled)
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pub scores: Vec<(NodeId, NodeId, f32)>,
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/// Total entanglement (summed fidelity)
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pub total_entanglement: f32,
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/// Entanglement entropy (measure of global correlation)
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pub entanglement_entropy: f32,
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}
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/// Quantum state aggregation (geometric mean of density matrices)
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pub struct QuantumAggregator {
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/// Convergence tolerance for Riemannian optimization
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pub tolerance: f32,
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/// Maximum iterations
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pub max_iterations: usize,
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}
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impl QuantumAggregator {
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/// Compute geometric mean of density matrices
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/// Solves: argmin_ρ Σ_i w_i D(ρ, ρ_i)² where D is Bures distance
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pub fn geometric_mean(
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&self,
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density_matrices: &[DensityMatrix],
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weights: &[f32],
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) -> Result<DensityMatrix, QuantumError>;
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/// Quantum barycenter via Riemannian gradient descent
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fn riemannian_optimize(
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&self,
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matrices: &[DensityMatrix],
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weights: &[f32],
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) -> DensityMatrix;
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}
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/// Error types for quantum operations
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#[derive(Debug, thiserror::Error)]
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pub enum QuantumError {
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#[error("Invalid density matrix: {0}")]
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InvalidDensityMatrix(String),
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#[error("Numerical instability in quantum operation")]
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NumericalInstability,
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#[error("Dimension mismatch: expected {expected}, got {actual}")]
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DimensionMismatch { expected: usize, actual: usize },
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#[error("Non-convergence in quantum aggregation")]
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NonConvergence,
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}
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```
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### Key Algorithms (Pseudocode)
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#### Algorithm 1: Quantum State Encoding (Classical → Quantum)
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```
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function encode_quantum_state(embedding, basis_states, encode_weight):
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"""
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Map classical embedding to quantum density matrix
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embedding: R^d (classical node embedding)
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basis_states: {|ψ_i⟩} (learned basis states)
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Returns: ρ ∈ C^(d_q × d_q) (density matrix)
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"""
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# Step 1: Project embedding to quantum dimension
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projected = encode_weight @ embedding # Shape: [d_q²]
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# Step 2: Construct superposition coefficients
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# Use softmax to ensure Σ_i |α_i|² = 1
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coefficients = softmax(projected[0:num_basis_states])
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# Step 3: Build superposition state
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psi = zeros(quantum_dim, dtype=complex)
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for i, alpha_i in enumerate(coefficients):
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psi += sqrt(alpha_i) * basis_states[i]
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# Step 4: Construct density matrix from pure state
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# ρ = |ψ⟩⟨ψ|
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rho = outer_product(psi, conjugate(psi))
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# Step 5: Add decoherence (mixed state)
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# ρ_mixed = (1 - ε) ρ + ε I/d (prevents pure states)
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epsilon = 0.01
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rho = (1 - epsilon) * rho + epsilon * eye(quantum_dim) / quantum_dim
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# Step 6: Ensure valid density matrix (numerical stability)
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rho = project_to_density_matrix(rho)
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return DensityMatrix(rho)
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function project_to_density_matrix(rho):
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"""
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Project matrix to nearest valid density matrix
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Properties: Hermitian, PSD, Tr(ρ) = 1
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"""
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# Make Hermitian
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rho = (rho + conjugate_transpose(rho)) / 2
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# Eigendecomposition
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eigenvalues, eigenvectors = eig_hermitian(rho)
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# Project eigenvalues to [0, ∞) (positive semi-definite)
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eigenvalues = maximum(eigenvalues, 0)
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# Normalize trace to 1
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eigenvalues = eigenvalues / sum(eigenvalues)
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# Reconstruct matrix
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rho = eigenvectors @ diag(eigenvalues) @ conjugate_transpose(eigenvectors)
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return rho
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```
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#### Algorithm 2: Quantum Fidelity Computation (Entanglement Attention)
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```
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function compute_quantum_fidelity(rho, sigma):
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"""
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Compute quantum fidelity between two density matrices
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F(ρ, σ) = [Tr(√(√ρ σ √ρ))]²
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Interpretation: probability that ρ and σ represent same quantum state
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Range: [0, 1] (1 = identical states)
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"""
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# Step 1: Compute √ρ via eigendecomposition
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eigenvalues_rho, eigenvectors_rho = eig_hermitian(rho)
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sqrt_eigenvalues_rho = sqrt(maximum(eigenvalues_rho, 0))
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sqrt_rho = eigenvectors_rho @ diag(sqrt_eigenvalues_rho) @ conjugate_transpose(eigenvectors_rho)
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# Step 2: Compute √ρ σ √ρ
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product = sqrt_rho @ sigma @ sqrt_rho
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# Step 3: Compute √(√ρ σ √ρ)
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eigenvalues_product, eigenvectors_product = eig_hermitian(product)
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sqrt_eigenvalues_product = sqrt(maximum(eigenvalues_product, 0))
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sqrt_product = eigenvectors_product @ diag(sqrt_eigenvalues_product) @ conjugate_transpose(eigenvectors_product)
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# Step 4: Compute fidelity
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fidelity = trace(sqrt_product) ** 2
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# Ensure fidelity ∈ [0, 1] (numerical errors)
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fidelity = clip(fidelity, 0, 1)
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return real(fidelity) # Fidelity is always real
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function hierarchical_entanglement_attention(nodes, quantum_states, config):
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"""
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Compute entanglement attention in O(N log N) via hierarchical clustering
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"""
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# Step 1: Build hierarchical clusters of quantum states
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clusters = hierarchical_cluster_quantum_states(
|
||
quantum_states,
|
||
num_levels = config.complexity_reduction.levels
|
||
)
|
||
|
||
# Step 2: Bottom-up attention computation
|
||
attention_scores = {}
|
||
|
||
for level in range(len(clusters.levels) - 1, -1, -1):
|
||
for cluster in clusters.levels[level]:
|
||
# Compute fidelity within cluster (fine-grained)
|
||
if level == 0:
|
||
# Leaf level: compute pairwise fidelity
|
||
for i in cluster.members:
|
||
for j in cluster.members:
|
||
if i != j:
|
||
fidelity = compute_quantum_fidelity(
|
||
quantum_states[i],
|
||
quantum_states[j]
|
||
)
|
||
attention_scores[(i, j)] = fidelity
|
||
else:
|
||
# Higher level: approximate via cluster centroids
|
||
for child1 in cluster.children:
|
||
for child2 in cluster.children:
|
||
if child1.id != child2.id:
|
||
fidelity = compute_quantum_fidelity(
|
||
child1.centroid,
|
||
child2.centroid
|
||
)
|
||
# Distribute fidelity to members
|
||
for i in child1.members:
|
||
for j in child2.members:
|
||
attention_scores[(i, j)] = fidelity
|
||
|
||
return attention_scores
|
||
|
||
function hierarchical_cluster_quantum_states(quantum_states, num_levels):
|
||
"""
|
||
Build hierarchical clusters of quantum states
|
||
Uses fidelity as similarity metric
|
||
"""
|
||
clusters = HierarchicalClusters()
|
||
current_level_clusters = []
|
||
|
||
# Level 0: Each node is its own cluster
|
||
for i, state in enumerate(quantum_states):
|
||
cluster = Cluster {
|
||
id: i,
|
||
members: [i],
|
||
centroid: state
|
||
}
|
||
current_level_clusters.append(cluster)
|
||
|
||
clusters.levels.append(current_level_clusters)
|
||
|
||
# Build higher levels via agglomerative clustering
|
||
for level in 1..num_levels:
|
||
next_level_clusters = []
|
||
|
||
while len(current_level_clusters) > 0:
|
||
# Find most similar pair of clusters
|
||
max_fidelity = -inf
|
||
best_pair = (None, None)
|
||
|
||
for i, cluster1 in enumerate(current_level_clusters):
|
||
for j in range(i+1, len(current_level_clusters)):
|
||
cluster2 = current_level_clusters[j]
|
||
fidelity = compute_quantum_fidelity(
|
||
cluster1.centroid,
|
||
cluster2.centroid
|
||
)
|
||
if fidelity > max_fidelity:
|
||
max_fidelity = fidelity
|
||
best_pair = (i, j)
|
||
|
||
# Merge best pair
|
||
if best_pair[0] is not None:
|
||
cluster1 = current_level_clusters[best_pair[0]]
|
||
cluster2 = current_level_clusters[best_pair[1]]
|
||
|
||
# Compute new centroid (geometric mean)
|
||
merged_centroid = quantum_geometric_mean([
|
||
cluster1.centroid,
|
||
cluster2.centroid
|
||
])
|
||
|
||
merged_cluster = Cluster {
|
||
id: len(next_level_clusters),
|
||
members: cluster1.members + cluster2.members,
|
||
centroid: merged_centroid,
|
||
children: [cluster1, cluster2]
|
||
}
|
||
|
||
next_level_clusters.append(merged_cluster)
|
||
|
||
# Remove merged clusters
|
||
current_level_clusters.remove(best_pair[1])
|
||
current_level_clusters.remove(best_pair[0])
|
||
else:
|
||
# No more pairs to merge
|
||
break
|
||
|
||
clusters.levels.append(next_level_clusters)
|
||
current_level_clusters = next_level_clusters
|
||
|
||
return clusters
|
||
```
|
||
|
||
#### Algorithm 3: Quantum Channel Application (Information Propagation)
|
||
|
||
```
|
||
function apply_quantum_channel(rho, channel):
|
||
"""
|
||
Apply quantum channel to density matrix
|
||
|
||
Φ(ρ) = Σ_k K_k ρ K_k†
|
||
|
||
where {K_k} are Kraus operators satisfying Σ_k K_k† K_k = I
|
||
"""
|
||
output = zeros_like(rho, dtype=complex)
|
||
|
||
for kraus_op in channel.kraus_operators:
|
||
# Apply Kraus operator: K_k ρ K_k†
|
||
output += kraus_op @ rho @ conjugate_transpose(kraus_op)
|
||
|
||
# Ensure trace preservation (numerical stability)
|
||
trace_output = trace(output)
|
||
if abs(trace_output - 1.0) > 1e-6:
|
||
output = output / trace_output
|
||
|
||
return output
|
||
|
||
function learn_kraus_operators(training_data, num_operators, quantum_dim):
|
||
"""
|
||
Learn Kraus operators from training data
|
||
|
||
training_data: [(ρ_in, ρ_out)] pairs of density matrices
|
||
num_operators: number of Kraus operators to learn
|
||
|
||
Optimization problem:
|
||
minimize Σ_i ||Φ(ρ_in^i) - ρ_out^i||_F²
|
||
subject to: Σ_k K_k† K_k = I (trace preservation)
|
||
"""
|
||
# Initialize Kraus operators randomly
|
||
kraus_ops = [
|
||
random_unitary(quantum_dim) / sqrt(num_operators)
|
||
for _ in range(num_operators)
|
||
]
|
||
|
||
optimizer = Adam(parameters=kraus_ops, lr=0.001)
|
||
|
||
for epoch in range(num_epochs):
|
||
total_loss = 0
|
||
|
||
for (rho_in, rho_out) in training_data:
|
||
# Apply channel
|
||
rho_pred = zeros_like(rho_in, dtype=complex)
|
||
for K_k in kraus_ops:
|
||
rho_pred += K_k @ rho_in @ conjugate_transpose(K_k)
|
||
|
||
# Loss: Frobenius norm
|
||
loss = frobenius_norm(rho_pred - rho_out) ** 2
|
||
total_loss += loss
|
||
|
||
# Backward pass
|
||
loss.backward()
|
||
|
||
# Update Kraus operators
|
||
optimizer.step()
|
||
|
||
# Project to trace-preserving constraint
|
||
# Σ_k K_k† K_k = I
|
||
sum_ktk = sum([conjugate_transpose(K) @ K for K in kraus_ops])
|
||
eigenvalues, eigenvectors = eig_hermitian(sum_ktk)
|
||
sqrt_inv = eigenvectors @ diag(1.0 / sqrt(eigenvalues)) @ conjugate_transpose(eigenvectors)
|
||
|
||
# Correct Kraus operators
|
||
for i in range(num_operators):
|
||
kraus_ops[i] = kraus_ops[i] @ sqrt_inv
|
||
|
||
return QuantumChannel(kraus_ops, QuantumChannelType::LearnedKraus)
|
||
```
|
||
|
||
#### Algorithm 4: Quantum State Aggregation (Geometric Mean)
|
||
|
||
```
|
||
function quantum_geometric_mean(density_matrices, weights):
|
||
"""
|
||
Compute geometric mean of density matrices
|
||
|
||
Solves: argmin_ρ Σ_i w_i D_B(ρ, ρ_i)²
|
||
where D_B is Bures distance: D_B(ρ, σ) = √(2 - 2√F(ρ, σ))
|
||
|
||
Uses Riemannian gradient descent on manifold of density matrices
|
||
"""
|
||
# Initialize to arithmetic mean
|
||
rho = weighted_arithmetic_mean(density_matrices, weights)
|
||
rho = project_to_density_matrix(rho)
|
||
|
||
for iteration in range(max_iterations):
|
||
# Compute Riemannian gradient
|
||
gradient = zeros_like(rho, dtype=complex)
|
||
|
||
for i, rho_i in enumerate(density_matrices):
|
||
# Bures distance gradient
|
||
sqrt_rho = matrix_sqrt(rho)
|
||
sqrt_rho_inv = matrix_inverse(sqrt_rho)
|
||
|
||
inner = sqrt_rho @ rho_i @ sqrt_rho
|
||
sqrt_inner = matrix_sqrt(inner)
|
||
|
||
grad_i = sqrt_rho_inv @ sqrt_inner @ sqrt_rho_inv - eye(quantum_dim)
|
||
gradient += weights[i] * grad_i
|
||
|
||
# Riemannian gradient descent step
|
||
# Exponential map: ρ_new = √ρ exp(-α G) √ρ
|
||
step_size = 0.1
|
||
sqrt_rho = matrix_sqrt(rho)
|
||
update = sqrt_rho @ matrix_exp(-step_size * gradient) @ sqrt_rho
|
||
|
||
# Project to density matrix manifold
|
||
update = project_to_density_matrix(update)
|
||
|
||
# Check convergence
|
||
if frobenius_norm(update - rho) < tolerance:
|
||
break
|
||
|
||
rho = update
|
||
|
||
return rho
|
||
|
||
function matrix_sqrt(A):
|
||
"""
|
||
Compute matrix square root via eigendecomposition
|
||
"""
|
||
eigenvalues, eigenvectors = eig_hermitian(A)
|
||
sqrt_eigenvalues = sqrt(maximum(eigenvalues, 0))
|
||
return eigenvectors @ diag(sqrt_eigenvalues) @ conjugate_transpose(eigenvectors)
|
||
|
||
function matrix_exp(A):
|
||
"""
|
||
Compute matrix exponential via eigendecomposition
|
||
"""
|
||
eigenvalues, eigenvectors = eig_hermitian(A)
|
||
exp_eigenvalues = exp(eigenvalues)
|
||
return eigenvectors @ diag(exp_eigenvalues) @ conjugate_transpose(eigenvectors)
|
||
```
|
||
|
||
### API Design (Function Signatures)
|
||
|
||
```rust
|
||
// ============================================================
|
||
// Public API for Quantum-Inspired Attention
|
||
// ============================================================
|
||
|
||
pub trait QuantumAttention {
|
||
/// Create quantum attention layer
|
||
fn new(
|
||
config: QuantumAttentionConfig,
|
||
embedding_dim: usize,
|
||
) -> Result<Self, QuantumError> where Self: Sized;
|
||
|
||
/// Forward pass: compute quantum entanglement attention
|
||
fn forward(
|
||
&self,
|
||
node_embeddings: &[Vec<f32>],
|
||
graph: &Graph,
|
||
) -> Result<Vec<Vec<f32>>, QuantumError>;
|
||
|
||
/// Encode classical embedding to quantum state
|
||
fn encode(&self, embedding: &[f32]) -> Result<DensityMatrix, QuantumError>;
|
||
|
||
/// Decode quantum state to classical embedding
|
||
fn decode(&self, quantum_state: &DensityMatrix) -> Result<Vec<f32>, QuantumError>;
|
||
|
||
/// Compute entanglement scores between nodes
|
||
fn compute_entanglement(
|
||
&self,
|
||
quantum_states: &[DensityMatrix],
|
||
) -> EntanglementScores;
|
||
|
||
/// Get learned basis states
|
||
fn basis_states(&self) -> &[Array1<Complex<f32>>];
|
||
|
||
/// Save quantum model parameters
|
||
fn save(&self, path: &Path) -> Result<(), io::Error>;
|
||
|
||
/// Load quantum model parameters
|
||
fn load(path: &Path) -> Result<Self, io::Error> where Self: Sized;
|
||
}
|
||
|
||
// ============================================================
|
||
// Configuration Builders
|
||
// ============================================================
|
||
|
||
impl QuantumAttentionConfig {
|
||
/// Default configuration for research use
|
||
pub fn default_quantum() -> Self {
|
||
Self {
|
||
quantum_dim: 64,
|
||
num_basis_states: 16,
|
||
entanglement_metric: EntanglementMetric::Fidelity,
|
||
channel_type: QuantumChannelType::LearnedKraus { num_operators: 4 },
|
||
complexity_reduction: ComplexityReduction::Hierarchical { levels: 3 },
|
||
epsilon: 1e-6,
|
||
}
|
||
}
|
||
|
||
/// Large graph configuration (aggressive complexity reduction)
|
||
pub fn large_graph() -> Self {
|
||
Self {
|
||
quantum_dim: 32,
|
||
num_basis_states: 8,
|
||
entanglement_metric: EntanglementMetric::Fidelity,
|
||
channel_type: QuantumChannelType::Depolarizing { p: 0.1 },
|
||
complexity_reduction: ComplexityReduction::QuantumLSH { num_hashes: 10 },
|
||
epsilon: 1e-5,
|
||
}
|
||
}
|
||
|
||
/// High accuracy configuration (minimal approximation)
|
||
pub fn high_fidelity() -> Self {
|
||
Self {
|
||
quantum_dim: 128,
|
||
num_basis_states: 32,
|
||
entanglement_metric: EntanglementMetric::Fidelity,
|
||
channel_type: QuantumChannelType::LearnedKraus { num_operators: 8 },
|
||
complexity_reduction: ComplexityReduction::Hierarchical { levels: 2 },
|
||
epsilon: 1e-7,
|
||
}
|
||
}
|
||
}
|
||
|
||
// ============================================================
|
||
// Density Matrix Operations
|
||
// ============================================================
|
||
|
||
impl DensityMatrix {
|
||
/// Validate quantum properties
|
||
pub fn validate(&self) -> Result<(), QuantumError>;
|
||
|
||
/// Compute quantum mutual information with another state
|
||
pub fn mutual_information(&self, other: &DensityMatrix) -> f32;
|
||
|
||
/// Partial trace over subsystem
|
||
pub fn partial_trace(&self, subsystem_dims: &[usize]) -> Self;
|
||
|
||
/// Quantum relative entropy (Kullback-Leibler divergence)
|
||
pub fn relative_entropy(&self, other: &DensityMatrix) -> f32;
|
||
|
||
/// Compute entanglement entropy (for bipartite systems)
|
||
pub fn entanglement_entropy(&self, partition: &[usize]) -> f32;
|
||
|
||
/// Visualize density matrix (export heatmap)
|
||
pub fn visualize(&self, path: &Path) -> Result<(), io::Error>;
|
||
}
|
||
|
||
// ============================================================
|
||
// Quantum Channel Operations
|
||
// ============================================================
|
||
|
||
impl QuantumChannel {
|
||
/// Compose two quantum channels
|
||
pub fn compose(&self, other: &QuantumChannel) -> Self;
|
||
|
||
/// Check complete positivity (required for valid channel)
|
||
pub fn is_completely_positive(&self) -> bool;
|
||
|
||
/// Compute channel capacity (information-theoretic bound)
|
||
pub fn capacity(&self) -> f32;
|
||
|
||
/// Choi matrix representation
|
||
pub fn choi_matrix(&self) -> Array2<Complex<f32>>;
|
||
}
|
||
|
||
// ============================================================
|
||
// Utilities
|
||
// ============================================================
|
||
|
||
/// Quantum state tomography (reconstruct density matrix from measurements)
|
||
pub fn quantum_state_tomography(
|
||
measurements: &[(Array2<Complex<f32>>, f32)], // (observable, expectation)
|
||
dim: usize,
|
||
) -> Result<DensityMatrix, QuantumError>;
|
||
|
||
/// Quantum process tomography (reconstruct quantum channel)
|
||
pub fn quantum_process_tomography(
|
||
input_states: &[DensityMatrix],
|
||
output_states: &[DensityMatrix],
|
||
) -> Result<QuantumChannel, QuantumError>;
|
||
|
||
/// Visualize quantum state on Bloch sphere (for qubit only)
|
||
pub fn visualize_bloch_sphere(
|
||
state: &DensityMatrix,
|
||
path: &Path,
|
||
) -> Result<(), io::Error>;
|
||
```
|
||
|
||
## Integration Points
|
||
|
||
### Affected Crates/Modules
|
||
|
||
1. **`ruvector-gnn` (Core GNN crate)**:
|
||
- Add `attention/quantum/` module
|
||
- Extend attention layer with quantum variant
|
||
- Add complex number support to tensor operations
|
||
|
||
2. **`ruvector-core`**:
|
||
- Add complex number tensor type (`ComplexTensor`)
|
||
- Extend linear algebra with Hermitian operations
|
||
- Add density matrix serialization
|
||
|
||
3. **`ruvector-math`** (New crate for numerical methods):
|
||
- Implement matrix square root, exponential, logarithm
|
||
- Riemannian optimization on quantum manifolds
|
||
- Eigendecomposition for complex Hermitian matrices
|
||
|
||
4. **`ruvector-gnn-node` (Node.js bindings)**:
|
||
- Expose quantum attention API
|
||
- Serialize complex numbers to JavaScript
|
||
- Provide visualization tools
|
||
|
||
5. **`ruvector-cli`**:
|
||
- Add `ruvector quantum-attention` command
|
||
- Visualize quantum states and entanglement
|
||
- Export Bloch sphere visualizations
|
||
|
||
### New Modules to Create
|
||
|
||
```
|
||
crates/ruvector-gnn/src/attention/quantum/
|
||
├── mod.rs # Public API
|
||
├── density_matrix.rs # DensityMatrix type
|
||
├── quantum_channel.rs # QuantumChannel + Kraus operators
|
||
├── fidelity.rs # Quantum fidelity computation
|
||
├── encoding.rs # Classical → quantum encoding
|
||
├── decoding.rs # Quantum → classical decoding
|
||
├── aggregation.rs # Quantum state geometric mean
|
||
├── hierarchical.rs # Hierarchical clustering
|
||
└── visualization.rs # Bloch sphere, density matrix plots
|
||
|
||
crates/ruvector-math/ (new)
|
||
├── mod.rs # Public API
|
||
├── complex.rs # Complex tensor type
|
||
├── linalg/
|
||
│ ├── eigen.rs # Eigendecomposition (Hermitian)
|
||
│ ├── matrix_functions.rs # sqrt, exp, log
|
||
│ └── svd.rs # Singular value decomposition
|
||
└── optimization/
|
||
├── riemannian.rs # Riemannian gradient descent
|
||
└── manifolds/
|
||
└── density_matrix.rs # Density matrix manifold
|
||
|
||
crates/ruvector-gnn-node/quantum/
|
||
├── bindings.rs # NAPI bindings
|
||
└── typescript/
|
||
└── quantum_attention.d.ts # TypeScript definitions
|
||
```
|
||
|
||
### Dependencies on Other Features
|
||
|
||
1. **Prerequisite: Attention Mechanisms (Tier 1, Feature #3)**:
|
||
- Quantum attention extends base attention framework
|
||
- **Action**: Refactor attention into trait for quantum variant
|
||
|
||
2. **Synergy: Sparse Attention (Tier 3, Feature #8)**:
|
||
- Quantum fidelity can guide sparse pattern learning
|
||
- **Integration**: Use fidelity scores as importance metric for pruning
|
||
|
||
3. **Synergy: Graph Condensation (Tier 3, Feature #7)**:
|
||
- Quantum states can be condensed (dimensional reduction)
|
||
- **Integration**: Learn low-dimensional quantum embeddings
|
||
|
||
4. **Complementary: Adaptive HNSW (Tier 2, Feature #5)**:
|
||
- Quantum entanglement defines natural graph structure
|
||
- **Integration**: Use entanglement scores to guide HNSW construction
|
||
|
||
## Regression Prevention
|
||
|
||
### Existing Functionality at Risk
|
||
|
||
1. **Numerical Stability**:
|
||
- **Risk**: Complex matrix operations (sqrt, exp) may diverge or produce NaN
|
||
- **Mitigation**:
|
||
- Use stabilized algorithms (Schur decomposition for matrix functions)
|
||
- Add epsilon to eigenvalues before sqrt
|
||
- Project to valid density matrix after each operation
|
||
- Extensive unit tests with edge cases (zero eigenvalues, near-singular matrices)
|
||
|
||
2. **Memory Consumption**:
|
||
- **Risk**: Density matrices (d × d complex) use 16d² bytes (vs 4d for embeddings)
|
||
- **Mitigation**:
|
||
- Default to small quantum dimensions (d=32-64)
|
||
- Use low-rank approximation for large graphs
|
||
- Lazy computation (don't materialize all density matrices)
|
||
|
||
3. **Interpretability**:
|
||
- **Risk**: Quantum concepts may confuse users (complex numbers, Hermitian matrices)
|
||
- **Mitigation**:
|
||
- Provide "classical mode" (real-valued fidelity approximation)
|
||
- Extensive documentation with intuitive analogies
|
||
- Visualization tools (Bloch sphere for qubits)
|
||
|
||
4. **Backward Compatibility**:
|
||
- **Risk**: Breaking existing attention API
|
||
- **Mitigation**:
|
||
- Keep standard attention as default
|
||
- Quantum attention is separate class (opt-in)
|
||
- Shared trait for unified interface
|
||
|
||
### Test Cases to Prevent Regressions
|
||
|
||
```rust
|
||
// Test 1: Density matrix validity
|
||
#[test]
|
||
fn test_density_matrix_properties() {
|
||
let psi = random_pure_state(dim=4);
|
||
let rho = DensityMatrix::pure_state(&psi);
|
||
|
||
// Test Hermiticity
|
||
assert!(is_hermitian(&rho.data, tol=1e-6));
|
||
|
||
// Test positive semi-definiteness
|
||
let eigenvalues = rho.eigenvalues.unwrap();
|
||
assert!(eigenvalues.iter().all(|&x| x >= -1e-6));
|
||
|
||
// Test trace = 1
|
||
let trace = rho.data.diag().sum();
|
||
assert!((trace.re - 1.0).abs() < 1e-6);
|
||
assert!(trace.im.abs() < 1e-6);
|
||
|
||
// Test purity for pure state
|
||
assert!((rho.purity - 1.0).abs() < 1e-5);
|
||
}
|
||
|
||
// Test 2: Fidelity bounds
|
||
#[test]
|
||
fn test_fidelity_properties() {
|
||
let rho1 = random_density_matrix(dim=4);
|
||
let rho2 = random_density_matrix(dim=4);
|
||
|
||
let f = rho1.fidelity(&rho2);
|
||
|
||
// Fidelity ∈ [0, 1]
|
||
assert!(f >= 0.0 && f <= 1.0);
|
||
|
||
// Fidelity symmetric
|
||
let f_reverse = rho2.fidelity(&rho1);
|
||
assert!((f - f_reverse).abs() < 1e-5);
|
||
|
||
// Fidelity = 1 iff identical
|
||
let f_self = rho1.fidelity(&rho1);
|
||
assert!((f_self - 1.0).abs() < 1e-5);
|
||
}
|
||
|
||
// Test 3: Quantum channel trace preservation
|
||
#[test]
|
||
fn test_channel_trace_preserving() {
|
||
let channel = QuantumChannel::amplitude_damping(dim=4, gamma=0.5);
|
||
|
||
assert!(channel.is_trace_preserving());
|
||
|
||
let rho_in = random_density_matrix(dim=4);
|
||
let rho_out = channel.apply(&rho_in);
|
||
|
||
let trace_out = rho_out.data.diag().sum();
|
||
assert!((trace_out.re - 1.0).abs() < 1e-6);
|
||
}
|
||
|
||
// Test 4: Quantum vs classical attention accuracy
|
||
#[test]
|
||
fn test_quantum_attention_accuracy() {
|
||
let classical_layer = StandardAttentionLayer::new(config);
|
||
let quantum_layer = QuantumAttentionLayer::new(quantum_config);
|
||
|
||
let node_embeddings = generate_test_embeddings(num_nodes=100, dim=64);
|
||
let graph = generate_test_graph(num_nodes=100, avg_degree=10);
|
||
|
||
let classical_output = classical_layer.forward(&node_embeddings, &graph).unwrap();
|
||
let quantum_output = quantum_layer.forward(&node_embeddings, &graph).unwrap();
|
||
|
||
// Quantum should preserve short-range accuracy
|
||
let short_range_accuracy = compute_short_range_accuracy(&classical_output, &quantum_output);
|
||
assert!(short_range_accuracy > 0.95);
|
||
|
||
// Quantum should improve long-range accuracy
|
||
let long_range_accuracy_classical = compute_long_range_accuracy(&classical_output, &graph);
|
||
let long_range_accuracy_quantum = compute_long_range_accuracy(&quantum_output, &graph);
|
||
assert!(long_range_accuracy_quantum > long_range_accuracy_classical * 1.2);
|
||
}
|
||
|
||
// Test 5: Complexity scaling
|
||
#[test]
|
||
fn test_hierarchical_complexity() {
|
||
let config = QuantumAttentionConfig {
|
||
complexity_reduction: ComplexityReduction::Hierarchical { levels: 3 },
|
||
..QuantumAttentionConfig::default_quantum()
|
||
};
|
||
|
||
let layer = QuantumAttentionLayer::new(config, 64).unwrap();
|
||
|
||
// Measure time for different graph sizes
|
||
let mut times = vec![];
|
||
for n in [100, 1000, 10000, 100000] {
|
||
let embeddings = generate_random_embeddings(n, 64);
|
||
let graph = generate_random_graph(n, 10);
|
||
|
||
let start = Instant::now();
|
||
layer.forward(&embeddings, &graph).unwrap();
|
||
let elapsed = start.elapsed();
|
||
|
||
times.push((n, elapsed.as_secs_f64()));
|
||
}
|
||
|
||
// Check O(N log N) scaling
|
||
for i in 1..times.len() {
|
||
let (n1, t1) = times[i-1];
|
||
let (n2, t2) = times[i];
|
||
|
||
let empirical_ratio = t2 / t1;
|
||
let theoretical_ratio = (n2 as f64 / n1 as f64) * ((n2 as f64).ln() / (n1 as f64).ln());
|
||
|
||
// Allow 2x margin for overhead
|
||
assert!(empirical_ratio < theoretical_ratio * 2.0);
|
||
}
|
||
}
|
||
|
||
// Test 6: Geometric mean convergence
|
||
#[test]
|
||
fn test_quantum_aggregation_convergence() {
|
||
let matrices = vec![
|
||
random_density_matrix(dim=4),
|
||
random_density_matrix(dim=4),
|
||
random_density_matrix(dim=4),
|
||
];
|
||
let weights = vec![0.5, 0.3, 0.2];
|
||
|
||
let aggregator = QuantumAggregator {
|
||
tolerance: 1e-6,
|
||
max_iterations: 100,
|
||
};
|
||
|
||
let result = aggregator.geometric_mean(&matrices, &weights).unwrap();
|
||
|
||
// Result should be valid density matrix
|
||
assert!(result.is_valid());
|
||
|
||
// Check optimality (first-order condition)
|
||
let gradient_norm = compute_riemannian_gradient_norm(&result, &matrices, &weights);
|
||
assert!(gradient_norm < 1e-4);
|
||
}
|
||
```
|
||
|
||
### Backward Compatibility Strategy
|
||
|
||
1. **API Level**:
|
||
- Keep `StandardAttentionLayer` as default
|
||
- Add `QuantumAttentionLayer` (opt-in experimental feature)
|
||
- Both implement common `AttentionLayer` trait
|
||
|
||
2. **Feature Flags**:
|
||
- Quantum attention behind `quantum` feature flag
|
||
- Requires `ndarray` with `blas` backend for performance
|
||
- Optional dependency on `lapack` for eigendecomposition
|
||
|
||
3. **Documentation**:
|
||
- Clearly mark as "Experimental Research Feature"
|
||
- Provide intuitive explanations (not just quantum mechanics)
|
||
- Examples comparing quantum vs classical attention
|
||
|
||
4. **Fallback**:
|
||
- If numerical issues occur, fall back to classical attention
|
||
- Emit warning to user with debugging info
|
||
|
||
## Implementation Phases
|
||
|
||
### Phase 1: Core Quantum Math (Weeks 1-3)
|
||
|
||
**Goals**:
|
||
- Implement complex number tensor type
|
||
- Build matrix functions (sqrt, exp, log)
|
||
- Density matrix operations
|
||
- Quantum fidelity computation
|
||
|
||
**Deliverables**:
|
||
```rust
|
||
// Week 1-2: Complex tensor + linear algebra
|
||
crates/ruvector-math/
|
||
✓ complex.rs (ComplexTensor type)
|
||
✓ linalg/eigen.rs (Hermitian eigendecomposition)
|
||
✓ linalg/matrix_functions.rs (sqrt, exp, log)
|
||
|
||
// Week 3: Density matrices
|
||
crates/ruvector-gnn/src/attention/quantum/
|
||
✓ density_matrix.rs (DensityMatrix type + validation)
|
||
✓ fidelity.rs (quantum fidelity computation)
|
||
```
|
||
|
||
**Success Criteria**:
|
||
- Complex tensor tests pass
|
||
- Matrix functions match NumPy/SciPy (< 1e-5 error)
|
||
- Fidelity computation validates on known cases
|
||
|
||
### Phase 2: Quantum Encoding & Channels (Weeks 4-6)
|
||
|
||
**Goals**:
|
||
- Classical → quantum encoding
|
||
- Quantum channel implementation
|
||
- Learn Kraus operators
|
||
- Channel application
|
||
|
||
**Deliverables**:
|
||
```rust
|
||
// Week 4: Encoding/decoding
|
||
crates/ruvector-gnn/src/attention/quantum/
|
||
✓ encoding.rs (classical → quantum)
|
||
✓ decoding.rs (quantum → classical)
|
||
|
||
// Week 5-6: Quantum channels
|
||
crates/ruvector-gnn/src/attention/quantum/
|
||
✓ quantum_channel.rs (QuantumChannel + Kraus ops)
|
||
✓ Amplitude damping, depolarizing channels
|
||
✓ Learned Kraus operator training
|
||
```
|
||
|
||
**Success Criteria**:
|
||
- Encoding produces valid density matrices
|
||
- Channels are trace-preserving
|
||
- Learned channels reconstruct test data
|
||
|
||
### Phase 3: Hierarchical Attention (Weeks 7-9)
|
||
|
||
**Goals**:
|
||
- Hierarchical clustering of quantum states
|
||
- O(N log N) fidelity computation
|
||
- Quantum state aggregation (geometric mean)
|
||
- Full attention layer integration
|
||
|
||
**Deliverables**:
|
||
```rust
|
||
// Week 7-8: Hierarchical attention
|
||
crates/ruvector-gnn/src/attention/quantum/
|
||
✓ hierarchical.rs (hierarchical clustering)
|
||
✓ Complexity reduction algorithms
|
||
|
||
// Week 9: Aggregation + integration
|
||
crates/ruvector-gnn/src/attention/quantum/
|
||
✓ aggregation.rs (geometric mean)
|
||
✓ mod.rs (full QuantumAttentionLayer)
|
||
```
|
||
|
||
**Success Criteria**:
|
||
- Hierarchical attention scales to 100K nodes
|
||
- Complexity is O(N log N) empirically
|
||
- Attention output is valid embeddings
|
||
|
||
### Phase 4: Evaluation & Hardening (Weeks 10-12)
|
||
|
||
**Goals**:
|
||
- Comprehensive testing (numerical stability, edge cases)
|
||
- Documentation + tutorials
|
||
- Visualization tools (Bloch sphere, density matrices)
|
||
- Benchmarks vs classical attention
|
||
|
||
**Deliverables**:
|
||
```rust
|
||
// Week 10: Testing
|
||
tests/quantum_attention/
|
||
✓ Numerical stability tests
|
||
✓ Edge case handling (zero eigenvalues, etc.)
|
||
✓ Property-based tests
|
||
|
||
// Week 11: Visualization + docs
|
||
crates/ruvector-gnn/src/attention/quantum/
|
||
✓ visualization.rs (Bloch sphere, heatmaps)
|
||
docs/
|
||
✓ Quantum Attention Guide (non-physicist friendly)
|
||
✓ Theoretical foundations
|
||
|
||
// Week 12: Benchmarks
|
||
benches/quantum_attention.rs
|
||
✓ Long-range dependency accuracy
|
||
✓ Complexity scaling
|
||
✓ Comparison vs classical attention
|
||
```
|
||
|
||
**Success Criteria**:
|
||
- 100% test coverage for core quantum math
|
||
- Documentation complete with 2+ examples
|
||
- Benchmarks show long-range improvement
|
||
|
||
## Success Metrics
|
||
|
||
### Performance Benchmarks
|
||
|
||
| Benchmark | Metric | Target | Measurement Method |
|
||
|-----------|--------|--------|-------------------|
|
||
| Complexity Scaling | Time vs N | O(N log N) | Fit log-log plot to runtime data |
|
||
| Memory Usage | Bytes per node | <1KB (d=64) | Track density matrix storage |
|
||
| Fidelity Computation | Time per pair | <0.1ms | `criterion` benchmark |
|
||
| Hierarchical Clustering | Time for 1M nodes | <10s | One-time offline cost |
|
||
| Encoding/Decoding | Throughput | >10K nodes/sec | Batch processing benchmark |
|
||
|
||
### Accuracy Metrics
|
||
|
||
| Metric | Target | Measurement Method |
|
||
|--------|--------|-------------------|
|
||
| Short-range accuracy (1-3 hops) | >=0.95 (vs classical) | Recall on link prediction |
|
||
| Medium-range accuracy (4-7 hops) | >=1.10 (vs classical) | Relative improvement |
|
||
| Long-range accuracy (8+ hops) | >=1.80 (vs classical) | 80% improvement target |
|
||
| Global clustering coefficient | >=0.70 | Compare to ground truth |
|
||
| Numerical stability (valid density matrices) | 100% | Validation checks |
|
||
|
||
### Research Impact Metrics
|
||
|
||
| Metric | Target | Notes |
|
||
|--------|--------|-------|
|
||
| Novel theoretical contributions | >=2 publications | Quantum GNN theory |
|
||
| Open-source citations | >=50 (2 years) | GitHub stars, papers |
|
||
| User adoption (experimental) | >=10 research groups | Academic/industrial |
|
||
| Quantum hardware readiness | Proof-of-concept | Future IBM/Rigetti integration |
|
||
|
||
## Risks and Mitigations
|
||
|
||
### Technical Risks
|
||
|
||
#### Risk 1: Numerical Instability in Matrix Square Root
|
||
|
||
**Description**:
|
||
Computing √ρ for ill-conditioned density matrices may produce NaN or Inf.
|
||
|
||
**Probability**: High (70%)
|
||
|
||
**Impact**: Critical (blocks fidelity computation)
|
||
|
||
**Mitigation**:
|
||
1. **Regularization**: Add epsilon to eigenvalues before sqrt (λ_i → λ_i + ε)
|
||
2. **Condition Number Check**: Warn if cond(ρ) > 10⁶
|
||
3. **Schur Decomposition**: Use more stable algorithm than eigendecomposition
|
||
4. **Fallback**: Use approximate fidelity (Hellinger distance) if exact fails
|
||
|
||
**Contingency Plan**:
|
||
If numerical issues persist, switch to "pseudo-quantum" mode using real-valued approximations (no complex numbers).
|
||
|
||
#### Risk 2: Geometric Mean Non-Convergence
|
||
|
||
**Description**:
|
||
Riemannian optimization for quantum barycenter may not converge.
|
||
|
||
**Probability**: Medium (40%)
|
||
|
||
**Impact**: High (blocks aggregation)
|
||
|
||
**Mitigation**:
|
||
1. **Adaptive Step Size**: Use Armijo line search for step size
|
||
2. **Multiple Initializations**: Try 3 random starts, pick best
|
||
3. **Convergence Monitoring**: Detect divergence and restart
|
||
4. **Fallback**: Use arithmetic mean (not geometrically optimal, but valid)
|
||
|
||
**Contingency Plan**:
|
||
If convergence rate <50%, replace geometric mean with weighted arithmetic mean (sacrifice optimality for reliability).
|
||
|
||
#### Risk 3: Interpretability Gap
|
||
|
||
**Description**:
|
||
Users may not understand quantum concepts (density matrices, fidelity, etc.).
|
||
|
||
**Probability**: Very High (90%)
|
||
|
||
**Impact**: Medium (adoption barrier)
|
||
|
||
**Mitigation**:
|
||
1. **Intuitive Documentation**: Use classical analogies (fidelity ≈ cosine similarity in Hilbert space)
|
||
2. **Visualization**: Provide Bloch sphere visualizations (for qubits)
|
||
3. **Classical Mode**: Offer real-valued approximation (no complex numbers)
|
||
4. **Educational Content**: Tutorials, blog posts, videos
|
||
|
||
**Contingency Plan**:
|
||
If user confusion is high, rename to "Entanglement Attention" and hide quantum terminology from API (internal implementation detail).
|
||
|
||
#### Risk 4: Limited Empirical Validation
|
||
|
||
**Description**:
|
||
Quantum attention is theoretically sound but lacks extensive empirical validation.
|
||
|
||
**Probability**: High (80%)
|
||
|
||
**Impact**: High (may not work in practice)
|
||
|
||
**Mitigation**:
|
||
1. **Benchmark Suite**: Test on 10+ diverse datasets (social, bio, citation networks)
|
||
2. **Ablation Studies**: Isolate contribution of each component
|
||
3. **Comparison**: Compare vs classical attention, sparse attention, graph transformers
|
||
4. **User Studies**: Collaborate with research groups for validation
|
||
|
||
**Contingency Plan**:
|
||
If accuracy is not competitive, pivot to "hybrid mode" (quantum for long-range, classical for short-range).
|
||
|
||
#### Risk 5: Scalability Bottleneck
|
||
|
||
**Description**:
|
||
Even O(N log N) may be too slow for billion-node graphs.
|
||
|
||
**Probability**: Medium (50%)
|
||
|
||
**Impact**: High (limits applicability)
|
||
|
||
**Mitigation**:
|
||
1. **Approximations**: Use random sampling instead of hierarchical clustering
|
||
2. **Distributed**: Parallelize across multiple GPUs/nodes
|
||
3. **Caching**: Cache quantum states between epochs
|
||
4. **Quantization**: Use low-precision (FP16) for density matrices
|
||
|
||
**Contingency Plan**:
|
||
If scalability is insufficient, limit to graphs <10M nodes and recommend sparse attention for larger graphs.
|
||
|
||
### Operational Risks
|
||
|
||
#### Risk 6: Dependency on Advanced Linear Algebra
|
||
|
||
**Description**:
|
||
Requires BLAS/LAPACK for efficient eigendecomposition. May not be available on all systems.
|
||
|
||
**Probability**: Medium (30%)
|
||
|
||
**Impact**: Medium (performance degradation)
|
||
|
||
**Mitigation**:
|
||
1. **Optional Dependency**: Make `lapack` optional (fall back to pure-Rust eigen solver)
|
||
2. **Clear Documentation**: List BLAS/LAPACK requirements prominently
|
||
3. **Pre-built Binaries**: Provide binaries with static BLAS linking
|
||
|
||
#### Risk 7: Patent/Legal Issues
|
||
|
||
**Description**:
|
||
Quantum computing is heavily patented. Risk of IP infringement.
|
||
|
||
**Probability**: Low (10%)
|
||
|
||
**Impact**: Critical (legal liability)
|
||
|
||
**Mitigation**:
|
||
1. **Prior Art Search**: Ensure algorithms are published in academic literature
|
||
2. **Legal Review**: Consult IP lawyer before release
|
||
3. **Open License**: Use permissive license (MIT/Apache 2.0) to clarify terms
|
||
|
||
---
|
||
|
||
## Appendix: Quantum Information Theory Primer
|
||
|
||
**For Non-Physicists**:
|
||
|
||
1. **Density Matrix**: Generalization of probability distribution to quantum mechanics. Represents uncertainty about quantum state.
|
||
|
||
2. **Quantum Fidelity**: Measures "closeness" of two quantum states. Analogous to cosine similarity, but in Hilbert space.
|
||
|
||
3. **Quantum Channel**: Noisy communication channel for quantum information. Models decoherence and information loss.
|
||
|
||
4. **Entanglement**: Non-local correlation between quantum systems. Two entangled nodes "share information" without direct connection.
|
||
|
||
5. **Geometric Mean**: Optimal averaging in quantum state space. Preserves quantum structure better than arithmetic mean.
|
||
|
||
**Key Intuition**:
|
||
Quantum-inspired attention treats nodes as quantum systems that can be "entangled" (correlated at a distance). This enables capturing long-range dependencies without explicit paths in the graph.
|
||
|
||
## Appendix: Related Research
|
||
|
||
This design is based on:
|
||
|
||
1. **Quantum Machine Learning** (Biamonte et al., 2017): Quantum algorithms for ML
|
||
2. **Quantum Graph Neural Networks** (Verdon et al., 2019): Quantum circuits for GNNs
|
||
3. **Quantum Attention** (Li et al., 2021): Quantum-inspired transformers
|
||
4. **Density Matrix Formalism** (Nielsen & Chuang, 2010): Standard QM textbook
|
||
5. **Riemannian Optimization** (Absil et al., 2008): Optimization on manifolds
|
||
|
||
Key differences from prior work:
|
||
- **Novel**: Hierarchical quantum state clustering for O(N log N) complexity
|
||
- **Novel**: Learned Kraus operators for quantum channels
|
||
- **Engineering**: Production-ready Rust implementation (no quantum hardware required)
|
||
- **Integration**: Seamless integration with classical GNN layers
|