Metrologically advantageous states: long-range entanglement and asymmetric error correction

Quantum metrology aims to exploit many-body quantum states to achieve parameter-estimation precision beyond the standard quantum limit. For unitary parameter encoding generated by local Hamiltonians, such enhancement is characterized by superlinear scaling of the quantum Fisher information (QFI) with system size. Despite extensive progress, a systematic understanding of which many-body quantum states can exhibit this scaling has remained elusive. Here, we develop a general framework that connects metrological performance to long-range entanglement, state-preparation complexity, and quantum error-correction properties. We prove that super-linear QFI scaling necessarily requires long-range entanglement by deriving rigorous complexity-dependent upper bounds on the QFI. We further show that, for two broad classes of quantum error-correcting codes, nondegenerate codes and Calderbank--Shor--Steane quantum low-density parity-check codes, a nonconstant code distance precludes super-linear QFI scaling for a wide class of local Hamiltonians, revealing a fundamental incompatibility between metrological sensitivity and protection against local noise. Finally, we identify constructive routes that evade this obstruction by exploiting asymmetric code structures. In particular, we show that states associated with classical low-density parity-check codes, as well as asymmetric toric code states, both having asymmetric logical distances, can achieve Heisenberg-limited scaling. Together, our results establish long-range entanglement and asymmetric error correction as the essential resource underlying quantum metrology and clarify the interplay among state complexity, error correction, and metrological power.