[Paper Review] Quantum Router with Network Coding

Michael Epping, Hermann Kampermann+1 more|arXiv (Cornell University)|May 26, 2016

Quantum Information and Cryptography38 references17 citations

TL;DR

This paper proposes a quantum router architecture that uses network coding via measurement-based quantum computation to distribute multipartite entangled qudit graph states across long-distance quantum networks. By generalizing classical linear network codes to quantum graph states, the scheme enables robust distribution of entanglement even under node outages, with error correction capabilities mapped to stabilizer codes, achieving fault-tolerant distribution of graph states like the Steane and five-qubit codes.

ABSTRACT

Many protocols of quantum information processing, like quantum key distribution or measurement-based quantum computation, "consume" entangled quantum states during their execution. When participants are located at distant sites, these resource states need to be distributed. Due to transmission losses quantum repeater become necessary for large distances (e.g. $\gtrsim$ 300 km). Here we generalize the concept of the graph state repeater to $D$-dimensional graph states and to repeaters that can perform basic measurement-based quantum computations, which we call quantum routers. This processing of data at intermediate network nodes is called quantum network coding. We describe how a scheme to distribute general two-colorable graph states via quantum routers with network coding can be constructed from classical linear network codes. The robustness of the distribution of graph states against outages of network nodes is analysed by establishing a link to stabilizer error correction codes. Furthermore we show, that for any stabilizer error correction code there exists a corresponding quantum network code with similar error correcting capabilities.

Motivation & Objective

• To extend the graph state repeater concept to D-dimensional qudits and multi-node quantum routers that perform local quantum computation.
• To enable efficient and robust distribution of two-colorable graph states across quantum networks using network coding.
• To establish a formal link between quantum network codes and stabilizer error correction codes for analyzing fault tolerance.
• To demonstrate that any stabilizer code can be realized as a quantum network code with equivalent error-correcting performance.

Proposed method

• Generalizes the graph state repeater scheme from qubits to D-dimensional qudits using the stabilizer formalism.
• Constructs quantum network codes (QNCs) from classical linear network codes, mapping them to quantum graph states via measurement-based quantum computation.
• Uses X-basis measurements on intermediate nodes to generate Bell pairs between source and destination, enabling entanglement distribution.
• Identifies main stabilizer operators after measurement to track post-measurement states and ensure correct entanglement distribution.
• Establishes a mathematical framework linking quantum network codes to stabilizer codes using adjacency matrices and error propagation rules.
• Demonstrates that error correction in the network corresponds to logical error correction in stabilizer codes, with error thresholds determined by code distance.

Experimental results

Research questions

• RQ1 How can network coding be applied to quantum networks to improve the distribution of multipartite entangled graph states?
• RQ2 What is the relationship between quantum network codes and stabilizer error correction codes in terms of error-correcting capability?
• RQ3 Can any stabilizer code be implemented as a quantum network code with equivalent fault tolerance in a networked setting?
• RQ4 How does the robustness of the network code against node outages relate to the code distance of the underlying stabilizer code?
• RQ5 Can the framework be generalized to non-two-colorable graph states or more complex network topologies?

Key findings

• A quantum network code (QNC) can be constructed from any classical linear network code to distribute two-colorable graph states across a network using measurement-based quantum computation.
• The network code's ability to correct up to t node failures corresponds directly to the distance d of the underlying stabilizer code, with the scheme tolerating up to ⌊(d−1)/2⌋ errors.
• For the 7-qubit Steane code and 5-qubit code, the network code successfully generates k Bell pairs between source and destination, with error correction verified via stabilizer tracking.
• The adjacency matrix Γ in Eq. (26) fully characterizes the network structure required to implement a given QNC, with entries defined by X- and Z-error propagation.
• The framework proves that every stabilizer code corresponds to a QNC with identical error-correcting capability, establishing a one-to-one correspondence.
• The scheme is robust against node outages: for example, the [[12,2,3]] QNC can tolerate two node failures while still preserving logical information.