Explaining Robust Quantum Metrology by Counting Codewords

Quantum sensing holds great promise for high-precision magnetic field measurements. However, its performance is significantly limited by noise. The investigation of active quantum error correction to address this noise led to the Hamiltonian-Not-in-Lindblad-Span (HNLS) condition. This states that Heisenberg scaling is achievable if and only if the signal Hamiltonian is orthogonal to the span of the Lindblad operators describing the noise. In this work, we consider a robust quantum metrology setting where the probe state is inspired from CSS codes for noise resilience but there is no active error correction performed. After the state picks up the signal, we measure the code's $\hat{X}$ stabilizers to infer the magnetic field parameter, $θ$. Given $N$ copies of the probe state, we derive the probability that all stabilizer measurements return $+1$, which depends on $θ$. The uncertainty in $θ$ (estimated from these measurements) is bounded by a new quantity, the \textit{Robustness Bound}, which ties the Quantum Fisher Information of the measurement to the number of weight-$2$ codewords of the dual code. Through this novel lens of coding theory, we show that for nontrivial CSS code states the HNLS condition still governs the Heisenberg scaling in our robust metrology setting. Our finding suggests fundamental limitations in the use of linear quantum codes for dephased magnetic field sensing applications both in the near-term robust sensing regime and in the long-term fault tolerant era. We also extend our results to general scenarios beyond dephased magnetic field sensing.