Dynamic Quantum Optimal Communication Topology Design for Consensus Control in Linear Multi-Agent Systems

This paper proposes a quantum framework for the design of communication topologies in consensus-based multi-agent systems. The communication graph is selected online by solving a mixed-integer quadratic program (MIQP) that minimizes a cost combining communication and distance penalties with degree-regularization terms, while enforcing exact connectivity through a flow-based formulation. To cope with the combinatorial complexity of this NP-hard problem, we develop a three-block ADMM scheme that decomposes the MIQP into a convex quadratic program in relaxed edge and flow variables, a pure binary unconstrained subproblem, and a closed-form auxiliary update. The binary subproblem is mapped to a quadratic unconstrained binary optimization (QUBO) Hamiltonian and approximately solved via quantum imaginary time evolution (QITE). The resulting time-varying, optimizer-generated Laplacians are applied to linear first- and second-order consensus dynamics. Numerical simulations on networks demonstrate that the proposed method produces connected topologies that satisfy degree constraints, achieve consensus, and incur costs comparable to those of classical mixed-integer solvers, thereby illustrating how quantum algorithms can be embedded as topology optimizers within closed-loop distributed control architectures.

💡 Research Summary

This paper introduces a quantum‑enhanced framework for designing communication topologies in consensus‑based multi‑agent systems (MAS). Traditional approaches often fix the communication graph or optimize spectral metrics such as algebraic connectivity, which do not directly capture practical concerns like communication cost, link distance, or fairness in node degrees. To address this gap, the authors formulate the topology design as a mixed‑integer quadratic program (MIQP). The objective combines three physically meaningful terms: a cost proportional to the presence of each edge, a distance‑based penalty, and a degree‑regularization term that penalizes deviation from prescribed node degrees. Exact connectivity is enforced through a flow‑based formulation that guarantees the selected graph contains a spanning tree, thereby ensuring λ₂(L) > 0 at all times.

Because the MIQP is NP‑hard, the paper proposes a three‑block alternating direction method of multipliers (ADMM) decomposition. The first block solves a convex quadratic program over relaxed edge weights and flow variables, the second block isolates a pure binary subproblem, and the third block provides a closed‑form update for an auxiliary continuous variable. The binary subproblem is naturally expressed as a quadratic unconstrained binary optimization (QUBO) problem. Rather than using classical mixed‑integer solvers or hybrid variational quantum eigensolver (VQE)/QAOA loops, the authors map the QUBO to a Hamiltonian and solve it with the quantum imaginary‑time evolution (QITE) algorithm. QITE approximates imaginary‑time propagation directly on a quantum circuit, eliminating the outer classical optimization loop and reducing hyper‑parameter tuning.

The resulting time‑varying Laplacian matrices L(t) are fed into standard first‑order (ẋ = −Lx) and second‑order (ẍ + βLẋ + αLx = 0) consensus dynamics. Under the assumptions of lossless, delay‑free communication and piecewise‑constant topologies with a uniform dwell time, the authors prove that connectivity and non‑negative edge weights guarantee exponential convergence to consensus for both dynamics, independent of the optimality of the topology. Thus, the optimizer serves to improve performance (e.g., lower communication cost) while feasibility and stability are enforced by the flow constraints.

Numerical experiments on networks of 10–20 agents illustrate the approach. The quantum‑augmented ADMM produces connected, degree‑constrained graphs whose total cost is comparable to that obtained by commercial MIQP solvers (CPLEX, Gurobi). The consensus convergence rate remains essentially unchanged, and the average path length is reduced, indicating lower communication latency. The binary subproblem solved via QITE yields near‑optimal solutions despite the NISQ‑scale limitations of current quantum hardware; the authors rely on gate‑level simulators for validation.

The paper acknowledges current hardware constraints—limited qubit counts and gate errors—restricting direct deployment on large‑scale networks. Future work is suggested in three directions: (i) incorporating error‑mitigation and more efficient variational ansätze to improve QITE accuracy, (ii) developing hybrid quantum‑classical loops that delegate only the hardest combinatorial core to quantum processors while handling the convex part classically, and (iii) extending the methodology to nonlinear or higher‑order consensus models and to scenarios with communication delays or packet loss.

Overall, the work demonstrates a concrete integration of quantum optimization into a closed‑loop control architecture, showing that quantum algorithms can serve as real‑time topology optimizers for MAS, bridging the gap between theoretical quantum advantage and practical distributed control applications.