Thermodynamic scaling under graph-dressed local control

Determine the optimal system-size dependence of the graph-dressed local feedback cost, including whether the factorized upper bound and marginal lower bound can be sharpened to a matching thermodynamic scaling law.

Background

The graph-dressed construction reduces the number of monitored observables to linear in the number of qubits and restricts each monitor and control term to bounded support. For fixed system size and positive normal-noise parameter, the paper proves inverse-accuracy scaling, but its all-controller lower bound is obtained only from single-site marginals, while the upper bound comes from a factorized policy. The authors show that marginal Wasserstein discrepancies do not universally tensorize, leaving the many-body scaling unresolved.