Quantum Networks with Minimum Cost Aggregation

Submission for QIA 2025 Quantum Internet Application Challenge

Table of Contents

• Introduction
• Scientific Background
• Implementation Details
• Features
• Basic Usage
• Demo Output
• Key Advantages
• Current Limitations
• Future Directions
• References

Introduction

This project is a submission for the QIA Quantum Internet Application Challenge 2025. It demonstrates a working simulation of a Minimum Cost Flow routing protocol for quantum networks, built using SquidASM and NetSquid.

The application addresses a fundamental challenge in quantum networking: efficiently distributing entanglement across complex topologies while minimizing resource usage. By mapping the routing problem to a classical Min-Cost Flow optimization, we show how multi-path aggregation can be used to route entanglement optimally between a source and sink node.

Scientific Background

Background

Current quantum repeater protocols typically focus on optimizing linear chains [7]. However, as the field matures towards a Quantum Internet, the focus has shifted to network optimization [3, 4]. Azuma (2025) [1] and related works [2] argue that relying on single-path routing is insufficient for high-rate entanglement distribution in complex topologies.

Min-Cost Flow Formulation

The core innovation is minimizing network resource usage while meeting entanglement demand. This is achieved by modeling the quantum network as a flow graph and solving a classical Minimum Cost Flow (MCF) problem [1, 2, 5]. Multi-path routing naturally emerges from the MCF solution, allowing the optimizer to combine resources from multiple disjoint paths to overcome individual link bottlenecks [4, 11].

• Flow Conservation $\leftrightarrow$ Entanglement Swapping: The classical constraint $\sum f_{in} = \sum f_{out}$ maps to entanglement swapping at a repeater, which consumes two Bell pairs to fuse them [6, 7].
• Capacity $\leftrightarrow$ Generation Rate: Edge capacities correspond to the physical generation rate of Bell pairs [1, 12].
• Cost $\leftrightarrow$ Resource Usage: The cost function represents generic resource consumption per link. In this demonstration, costs are randomly assigned; in practice, they could be set as $\propto 1/F$ to prioritize high-fidelity links.

Implementation Details

The simulation maps the protocol to our python code as follows:

1. Network Generation & Flow Optimization (controller.py):

   • Models the quantum network as a directed graph where edge capacities represent generation rates.
   • Uses NetworkX Min-Cost Flow to route entanglement optimally.
   • Decomposes the abstract "flow" into specific lists of nodes (e.g., [0, 2, 1, 9]).

2. Quantum Simulation (app.py, run_simulation.py):

   • SquidASM/NetSquid simulates the physical layer (qubits, gates, noise).
   • Nodes executing FlowAggregationNode realize "Flow Conservation" by performing Bell State Measurements (BSM) on arriving qubits to forward them.
   • Verify results with Monte Carlo simulations to ensure robustness against depolarization noise.

Features

• Optimal Resource Allocation: Uses NetworkX native min-cost flow solvers to solve the Min-Cost Flow problem, ensuring optimal usage of network links with O(1) code complexity.
• Arbitrary Topology Support: Can process random graphs or specific network topologies, demonstrating the protocol's universality.
• Realistic Noise Simulation:
  • Models depolarization noise in both quantum links and local gate operations.

Basic Usage

Checkout the main simulation script to see the protocol in action:

python run_simulation.py

This will:

1. Generate a random quantum network, write it to network_config.yaml, and load it for the squidasm simulation.
2. Calculate the optimal Min-Cost Flow for a requested ebit demand.
3. Execute the quantum simulation (Source → Repeaters → Sink).
4. Output the Fidelity and Purity of the distributed keys.
5. Generate the visualizations shown above.

Demo Output

A sample output from running the simulation:

Fidelity Analysis:
  Scenario                  | End-to-End Fidelity 
  --------------------------|----------------------
  Ideal                     | 1.0000              
  Noisy                     | 0.8450              

The simulation compares an Ideal (noise-free) network against a Noisy network with 0.5% depolarization per gate, demonstrating fidelity degradation over multi-hop paths.

Key Advantages

• Optimality: Unlike greedy routing, Min-Cost Aggregation finds the globally optimal routing strategy.
• Robustness: By aggregating multiple paths, the protocol is resilient to individual link failures or bottlenecks.
• Scalability: The classical flow algorithm scales polynomially, making it suitable for large-scale quantum internetworks.

Current Limitations

• Static Routing: Flow is calculated once per session; dynamic re-routing for real-time fluctuations is not yet implemented.
• Single-Pair Protocol: This implementation focuses on optimizing a single Source-Sink pair, rather than multi-commodity flow (multiple simultaneous users).

Future Directions

• Dynamic Aggregation: Implementing real-time flow adjustment based on live link fidelity monitoring.
• Multiplexing: Explicitly modeling time/frequency multiplexing in the aggregations.
• Multi-User Support: Extending the flow problem to support multiple concurrent user pairs (Multi-commodity Flow).
• Finite-Size Statistics: Integrating results from Sharp finite statistics for QKD [5] to rigorously account for fluctuations in probabilistic ebit generation across links.

References

1. Azuma, K. (2025). Networking quantum networks with minimum cost aggregation. npj Quantum Information, s41534-025-01000-5.
2. Azuma, K. (2023). Minimum-cost aggregation of quantum repeaters. SPIE Security + Defence, 127380A.
3. Abane, A., et al. (2022). Entanglement Routing in Quantum Networks: A Comprehensive Survey. IEEE Communications Surveys & Tutorials, 24(4).
4. Pant, M., et al. (2019). Routing entanglement in the quantum internet. npj Quantum Information, 5, 25.
5. Li, T., et al. (2022). Quantum Network Utility Maximization. IEEE/ACM Transactions on Networking, 30(2).
6. Azuma, K., Tamaki, K., & Lo, H. K. (2015). All-photonic quantum repeaters. Nature Communications, 6, 6787.
7. Briegel, H.-J., Dür, W., Cirac, J. I., & Zoller, P. (1998). Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication. Physical Review Letters, 81, 5932.
8. Bennett, C. H., et al. (1996). Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels. Physical Review Letters, 76, 722.
9. Mannalath, V., Zapatero, V., & Curty, M. (2025). Sharp Finite Statistics for Quantum Key Distribution. Physical Review Letters, 135, 020801.
10. Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network Flows: Theory, Algorithms, and Applications. Prentice Hall.
11. Van Meter, R., et al. (2013). Path selection for quantum repeater networks. Networking Science, 3, 82–95.
12. Pirandola, S., et al. (2017). Fundamental limits of repeaterless quantum communications. Nature Communications, 8, 15043.