Geometry-Aware Quantum Error Mitigation via Quantum State Matrix Approximation

Learning-based quantum error mitigation (QEM) can operate directly on computational-basis output distributions, but such distributions do not uniquely identify the underlying quantum state. We establish a structural identifiability limit and quantify its cost. A computational-basis distribution uniquely determines only $2^b-1$ of the $4^b-1$ non-identity Pauli coordinates of a $b$-qubit block, a fraction $1/(2^b+1)$---20\% at $b=2$---while the remaining coordinates are not identifiable from that distribution alone without additional assumptions. In our probability-loss baseline, off-diagonal observable error remains unchanged in all twelve tested settings while diagonal observable error is reduced by up to 76\%. To access additional local state information, we introduce BlockGeo-QEM, which augments neural QEM with block-local Pauli observables and a Bures-geometric objective. The Bures term reduces local geometric error across all tested system-size/depth settings and both block layouts, and further improves off-diagonal observable estimation in the main overlapping layout. Locality nevertheless imposes its own limit: the strongest $b=2$ block-local lower bound reaches only 8--19\% of the global Bures distance. For the contiguous block families considered here, this local representation requires $3^b$ local Pauli-product measurement settings, compared with $3^n$ such settings for full-state tomography. We also identify a low-shot regime in which the geometric contribution becomes statistically indistinguishable from its ablation.