Quantum utility routing in continuous-variable QKD networks with trusted and untrusted relays

Abstract Continuous-variable quantum key distribution (CV-QKD) networks require routing strategies that account simultaneously for secret key generation, relay trust and finite network resources. We introduce Quantum Utility Routing (QUR), a weighted routing framework for networks combining trusted point-to-point links with untrusted CV Bell measurement relays. Transition level composable finite size secret key rates (SKRs) are incorporated directly into the route search, allowing QUR to balance bottleneck SKR against route geometry and resource use. We consider graph optimized QUR (QURgo) and a fixed weight implementation (QURfw) that uses machine learning to avoid repeated graph specific weight optimization, and benchmark both against A* and Max-Min routing. At the default network configuration, QURgo and QURfw improve the mean path SKR relative to A* while requiring substantially fewer network resources than Max-Min. The framework also maintains secure connectivity over a broad range of trusted node availability by exploiting untrusted infrastructure as measurement relays. These results demonstrate that routing in CV-QKD networks can be treated as a configurable trade off between cryptographic performance and network resource consumption.