Quantum Lyapunov Functions for distributed quantum systems

Abstract. —
We study continuous, adversarially robust control of distributed quantum substrates that must operate without measurement-induced interruptions. Our first contribution is an ancilla-safe weak-monitoring layer: via a Naimark dilation, each agent couples a fresh monitor ancilla to its data qubit and measures only the ancilla, thereby reproducing a symmetric weak POVM on the data without collapsing the live register. These telemetry signals can be privately aggregated (e.g., via MPC) to trigger remediation. Our second contribution is a fully unitary, measurement-free greedy Lyapunov healing procedure that provably decreases a composite Lyapunov functional at every cycle. Working in the density-matrix model, we give closed-form angles for (i) local single-qubit alignment that minimizes each site’s excited-population term, and (ii) conditional pairwise controlled rotations that reduce joint |11〉 mass along an interaction graph. Each substep is an exact minimizer for its targeted term and leaves non-targeted terms invariant, implying monotone descent. We provide a two-agent worked example in density matrices and a general n-agent formulation.

Together, ancilla-safe monitoring and greedy Lyapunov healing yield a non-destructive, streaming-compatible framework for resilient quantum services—e.g., smart-vehicle substrates and censorship-resilient secure messaging—supporting continuous anomaly detection (telemetry) and autonomous recovery (unitary control)