Linear readout versus global decoding: the sample-complexity separation of critical-point learning

Machine-learning probes of phase transitions almost exclusively extract local, low-order statistics and read them out linearly. For the critical point of the one-dimensional Kitaev/Ising chain we show that it is the linear readout, not the local measurement, that fails. Distinguishing g=1 from g=1+δ in the joint critical-scaling limit requires Ω[1/(δ²LΛ²)] ground-state copies for any linear-readout protocol, while a global optimal protocol needs only ≍8/(δ²L²): an overhead Ω(L/polylog L) growing without bound. Yet the same information is present in purely local snapshots: the full-chain Fock-basis projection—a product of single-site measurements—is exactly QFI-optimal, provided the outcomes are decoded jointly. The mechanism is structural: 81% of the critical quantum Fisher information resides in a single lowest-momentum mode, invisible to local linear statistics but accessible to local sensors through global correlations among them.