Constructing Quantum Network Coding Schemes from Classical Nonlinear Protocols

TL;DR

This paper develops a new quantum network coding protocol that can simulate nonlinear classical schemes over directed acyclic graphs, reducing classical communication needs and expanding applicability beyond linear solutions.

Contribution

It introduces a quantum network coding scheme that works with nonlinear classical protocols, overcoming previous linear-only limitations and reducing classical communication requirements.

Findings

Supports nonlinear classical network coding schemes in quantum settings

Reduces classical communication needed in quantum network protocols

Enables quantum solutions for networks with no linear classical solutions

Abstract

The k-pair problem in network coding theory asks to send k messages simultaneously between k source-target pairs over a directed acyclic graph. In a previous paper [ICALP 2009, Part I, pages 622--633] the present authors showed that if a classical k-pair problem is solvable by means of a linear coding scheme, then the quantum k-pair problem over the same graph is also solvable, provided that classical communication can be sent for free between any pair of nodes of the graph. Here we address the main case that remained open in our previous work, namely whether nonlinear classical network coding schemes can also give rise to quantum network coding schemes. This question is motivated by the fact that there are networks for which there are no linear solutions to the k-pair problem, whereas nonlinear solutions exist. In the present paper we overcome the limitation to linear protocols and describe a new communication protocol for perfect quantum network coding that improves over the previous one as follows: (i) the new protocol does not put any condition on the underlying classical coding scheme, that is, it can simulate nonlinear communication protocols as well, and (ii) the amount of classical communication sent in the protocol is significantly reduced.