Noise-Symmetry Optimization of Quantum Error-Corrected Metrology

Quantum error correction (QEC) code has emerged as a powerful tool to protect quantum-enhanced metrology against noise. However, the ability to correct errors alone does not guarantee high metrological sensitivity, as the encoded states may become insensitive to the parameter of interest. Here, we show that this limitation can be overcome by exploiting an intrinsic freedom of QEC codes: for a fixed set of correctable errors, the Knill-Laflamme conditions admit an equivalence class of encodings. When the correctable noise possesses unitary symmetries, these symmetries generate continuous transformations within this class, allowing systematic optimization of the encoding to increase the quantum Fisher information while preserving the correctable set of noise. Based on this observation, we develop a symmetry-based optimization approach and derive criteria identifying when such optimization can enhance metrological sensitivity. In particular, for stabilizer-sum Hamiltonians, it shows that the symmetry optimization can convert a code with vanishing quantum Fisher information into one achieving the standard quantum limit in general or even the Heisenberg scaling in specific cases, illustrating the power of symmetry optimization for QEC-assisted quantum metrology.