A Benchmark for Local and Global Measurements on 2D Quantum States

We introduce a small benchmark for asking what measurement information is needed to classify 2D quantum states under a fixed number of copies. The states are graph states with local phase defects, and the label is whether the number of defects is even or odd. An exact stabilizer certificate shows that measuring one Pauli observable of weight at most three per copy reveals no label information on the tested $3\times3$ and $4\times4$ lattices. This is a restricted result: the same label is the parity of a known weight-$n$ Pauli, so a classical decoder told its product basis reads it perfectly from one copy. We add a data-driven search over basis-tagged product-measurement parity transcripts and report exact global-Pauli and Clifford-simulation controls. An 18-parameter QCNN trained with analytic expectations keeps its circuit weights fixed across sizes, but refits its labeled scalar readout per split and shot budget; a single finite-shot pilot is not evidence of noise-robust training. Together the results form a certificate-first measurement benchmark, not a quantum-advantage claim or a separation from unrestricted classical processing of global measurements.