Enabling quantum switches (QSs) to serve requests submitted by quantum end nodes in quantum communication networks (QCNs) is a challenging problem due to the heterogeneous fidelity requirements of the submitted requests and the limited resources of the QCN. Effectively determining which requests are served by a given QS is fundamental to foster developments in practical QCN applications, like quantum data centers. However, the state-of-the-art on QS operation has overlooked this association problem, and it mainly focused on QCNs with a single QS. In this paper, the request-QS association problem in QCNs is formulated as a matching game that captures the limited QCN resources, heterogeneous application-specific fidelity requirements, and scheduling of the different QS operations. To solve this game, a swap-stable request-QS association (RQSA) algorithm is proposed while considering partial QCN information availability. Extensive simulations are conducted to validate the effectiveness of the proposed RQSA algorithm. Simulation results show that the proposed RQSA algorithm achieves a near-optimal (within 5%) performance in terms of the percentage of served requests and overall achieved fidelity, while outperforming benchmark greedy solutions by over 13%. Moreover, the proposed RQSA algorithm is shown to be scalable and maintain its near-optimal performance even when the size of the QCN increases.

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore, we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representations variety.