Convexification of the Quantum Network Utility Maximisation Problem
Sounak Kar, Stephanie Wehner
TL;DR
This work addresses convexifying the quantum network utility maximisation (QNUM) problem in static networks by reformulating the optimization to depend solely on the rate allocation, using a geometric-programming style variable change. Sufficient conditions (Cond.1 and Cond.2) on route entanglement measures are derived to guarantee convexity of the transformed problem, enabling efficient, globally optimal solutions even with heterogeneous entanglement measures. The methodology is demonstrated across several measures, including secret key fraction, distillable entanglement, and negativity, and a numerical SURFnet example shows practical computation of optimal rate and fidelity allocations. The results provide a principled approach to jointly optimize end-to-end entanglement quality and generation rate, with implications for scalable quantum networks and diverse applications. The convex reformulation preserves existing convex contributions and extends to measures that do not directly admit convexity in the original formulation, enabling robust, scalable resource distribution in quantum repeater networks.
Abstract
Network Utility Maximisation (NUM) addresses the problem of allocating resources fairly within a network and explores the ways to achieve optimal allocation in real-world networks. Although extensively studied in classical networks, NUM is an emerging area of research in the context of quantum networks. In this work, we consider the quantum network utility maximisation (QNUM) problem in a static setting, where a user's utility takes into account the assigned quantum quality (fidelity) via a generic entanglement measure as well as the corresponding rate of entanglement generation. Under certain assumptions, we demonstrate that the QNUM problem can be formulated as an optimisation problem with the rate allocation vector as the only decision variable. Using a change of variable technique known in the field of geometric programming, we then establish sufficient conditions under which this formulation can be reduced to a convex problem, a class of optimisation problems that can be solved efficiently and with certainty even in high dimensions. We further show that this technique preserves convexity, enabling us to formulate convex QNUM problems in networks where some routes have certain entanglement measures that do not readily admit convex formulation, while others do. This allows us to compute the optimal resource allocation in networks where heterogeneous applications run over different routes.