Shortcuts to quantum network routing

Communication is a key part of society that we make huge investments in. Each message has to both arrive quickly and use minimal network resources. Moreover, communications must sometimes also remain private, so encryption is used to prevent third parties from listening in. However, standard classical encryption protocols are proven not to be secure against quantum adversaries. With quantum communication we have a method that does provably attains perfect secrecy, even against adversaries with unlimited (quantum) computational power. It is, however, infeasible to have a direct quantum connection between all users of quantum communications. We thus look into the possibilities of a \emph{quantum network}, which is able to connect arbitrary users of the network as if they had direct quantum channel available. Once such a quantum network is built, we will need to route quantum messages effectively to their destination. In this work, we take the first step in studying network structures for quantum networks on a high level, and their accompanying routing algorithms. Current ongoing research into quantum communication with satellites and the possibility of a global network made the case of a quantum network of satellites relevant. We will consider a simplified model where each satellite can perform quantum communication with its immediate neighbours, and can create quantum virtual links that form shortcuts through the network. The first step towards a quantum network that we take is the structure of the network. Using virtual links as shortcuts we can reduce the maximum distance between nodes (the diameter) by an exponential factor. Let $|V|$ be the size of the set of all vertices in a graph, equal to the number of satellites. Then for the case of satellites we give a network structure with the following properties: \begin{itemize} \item The diameter of the graph is $4\log_2\left( \frac{|V|-2}{10} \right) +3 = O(\log |V|)$. This means that the distance between any two nodes in the graph will scale logarithmically in the number of nodes. \item The maximum degree of a vertex is $12\log_2\left( \frac{|V| -2}{10} \right) + 6 = O(\log |V|)$. The degree of a vertex is directly related to the size of its quantum memory. Since quantum memories are currently limited in size, it is important for feasibility reasons that the degree stays low. \end{itemize} With a logarithmic scaling, the diameter and degree scale well for large graphs. This is especially relevant for networks, which can become large. A lower distance between vertices also allows quantum communications to fail less often, in effect decreasing the communication time. We can thus achieve a much smaller diameter by using virtual links, while the degree of each vertex remains low. We furthermore give a routing algorithm for finding the shortest path on this network structure that only uses local information at each vertex to route, where local information is the nearby surroundings of a vertex. The routing algorithm has the following properties: \begin{itemize} \item The space complexity of the algorithm is $O(\log^5 |V|)$ per vertex. A small space complexity implies little memory usage of the algorithm. Since network routers are typically small computers, the small memory requirement allows them to run this algorithm. \item The time complexity of the algorithm is $O(\log^2 |V|)$ per vertex. A small time complexity allows network nodes to compute the next step in the path in minimal time. The time necessary to communicate decreases with decreasing time complexity. \end{itemize} For large enough $|V|$, our algorithm outperforms standard routing algorithms such as Dijkstra's algorithm by an exponential factor. Furthermore, we have proved that the local routing algorithm always gives a shortest path. Local routing transitions well into a real world implementation, so that every network router can make simple decisions for routing.