Neural Quantum Graph Embedding

---
name: neural-quantum-graph-embedding
description: "Neural-enhanced optimization framework for quantum architecture embedding problems using Distance Encoder Networks. Solves constrained unit disk problems for neutral atom qubit positioning via modified autoencoder with custom Embedding Loss Function. Activation: quantum embedding, unit disk problem, neutral atom qubits, distance encoder network, qubit positioning, quantum architecture optimization."
---

# Neural Optimization for Quantum Graph Embedding

Research skill for solving quantum architecture embedding problems using neural-enhanced optimization, based on Vercellino et al. (arXiv: 2605.03565).

## Overview

This skill addresses the critical challenge of mapping real-world optimization problems onto quantum hardware through proper qubit positioning. Specifically, it solves the **constrained unit disk problem** that arises in neutral atom-based quantum computing architectures.

## Key Concepts

### 1. The Constrained Unit Disk Problem

- Quantum hardware (neutral atoms) requires qubits positioned within interaction range
- Not all problem graphs can be directly embedded
- Must find feasible qubit positions satisfying distance constraints
- Classical solvers struggle with this under fixed computation time

### 2. Distance Encoder Network (DEN)

- Modified autoencoder architecture
- Learns to compute Euclidean distances between points
- Custom embedding layer encodes spatial relationships
- Maps initial non-feasible solutions to feasible ones via non-linear transformation

### 3. Embedding Loss Function

- Custom loss function modeling the unit disk constraints
- Penalizes solutions that violate minimum/maximum distance requirements
- Guides network toward physically realizable qubit layouts
- Enables end-to-end gradient-based optimization

## Methodologies

### Distance Encoder Network Architecture

```python
import torch
import torch.nn as nn

class DistanceEncoderNetwork(nn.Module):
    """
    Modified autoencoder for learning spatial transformations
    that map non-feasible qubit positions to feasible ones.
    """
    def __init__(self, n_qubits, latent_dim):
        super().__init__()
        self.n_qubits = n_qubits
        
        # Encoder: maps positions to latent space
        self.encoder = nn.Sequential(
            nn.Linear(n_qubits * 2, 256),  # 2D coordinates per qubit
            nn.ReLU(),
            nn.Linear(256, 128),
            nn.ReLU(),
            nn.Linear(128, latent_dim)
        )
        
        # Decoder: reconstructs feasible positions
        self.decoder = nn.Sequential(
            nn.Linear(latent_dim, 128),
            nn.ReLU(),
            nn.Linear(128, 256),
            nn.ReLU(),
            nn.Linear(256, n_qubits * 2)  # Output 2D coordinates
        )
    
    def forward(self, positions):
        latent = self.encoder(positions)
        reconstructed = self.decoder(latent)
        return reconstructed

def embedding_loss(predicted_positions, min_dist, max_dist):
    """
    Custom loss enforcing unit disk constraints.
    
    Args:
        predicted_positions: (n_qubits, 2) tensor of positions
        min_dist: minimum separation between qubits
        max_dist: maximum interaction range
    
    Returns:
        Loss value penalizing constraint violations
    """
    # Compute pairwise distances
    diff = predicted_positions.unsqueeze(1) - predicted_positions.unsqueeze(0)
    distances = torch.norm(diff, dim=2)
    
    # Penalty for qubits too close
    too_close = torch.relu(min_dist - distances).sum()
    
    # Penalty for qubits that should interact but are too far
    # (based on problem graph adjacency)
    # too_far = ... depends on problem structure
    
    return too_close
```

### Training Pipeline

1. **Generate initial positions**: Random or heuristic-based starting configurations
2. **Encode through DEN**: Pass through the Distance Encoder Network
3. **Compute Embedding Loss**: Evaluate constraint satisfaction
4. **Backpropagate**: Update network weights
5. **Extract solution**: Use decoder output as feasible qubit layout

### Application Workflow

```python
def solve_quantum_embedding(problem_graph, n_qubits, hardware_constraints):
    """
    End-to-end pipeline for embedding optimization problems on quantum hardware.
    
    Args:
        problem_graph: Graph to embed (adjacency matrix or edge list)
        n_qubits: Number of available qubits
        hardware_constraints: Min/max distances, topology constraints
    
    Returns:
        Feasible qubit positions for the problem graph
    """
    # 1. Initialize with random/heuristic positions
    initial_positions = generate_initial_positions(n_qubits)
    
    # 2. Train Distance Encoder Network
    model = DistanceEncoderNetwork(n_qubits, latent_dim=64)
    optimizer = torch.optim.Adam(model.parameters(), lr=1e-3)
    
    for epoch in range(num_epochs):
        predicted = model(initial_positions)
        loss = embedding_loss(predicted, 
                            min_dist=hardware_constraints['min_dist'],
                            max_dist=hardware_constraints['max_dist'])
        optimizer.zero_grad()
        loss.backward()
        optimizer.step()
    
    # 3. Extract feasible solution
    final_positions = model(initial_positions)
    return final_positions
```

## Advantages Over Classical Solvers

- **Outperforms classical solvers** at fixed computation times
- **Learns spatial transformations** rather than solving from scratch
- **Generalizes** to similar embedding problems
- **Scalable** to larger qubit counts

## Use Cases

### Neutral Atom Quantum Computers
- Qubit positioning for Rydberg atom arrays
- Optimizing trap geometries
- Dynamic reconfiguration during computation

### Other Quantum Platforms
- Superconducting qubit layout optimization
- Ion trap positioning
- Photonic circuit routing

### General Graph Embedding
- Any problem requiring spatial constraint satisfaction
- Network design with geometric constraints
- Facility location problems

## Related Skills

- quantum-neural-barren-plateau: Mitigating barren plateaus in QNNs
- quantum-sparsity-edge-chaos: Quantum sparsity for robust VQA design