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.

翻译：量子传感在高精度磁场测量中展现出巨大潜力，但其性能受到噪声的显著限制。为应对噪声而开展的主动量子纠错研究催生了哈密顿量不在林德布拉德生成空间（HNLS）条件。该条件指出：当且仅当信号哈密顿量与描述噪声的林德布拉德算符张成的空间正交时，方可实现海森堡标度。本研究探讨一种鲁棒量子计量方案：探针态的设计受CSS码抗噪特性启发，但未执行主动纠错。在探针态获取信号后，我们通过测量该码的$\hat{X}$稳定子来推断磁场参数$θ$。给定$N$份探针态副本，我们推导了所有稳定子测量结果均为$+1$的概率（该概率与$θ$相关）。$θ$的估计不确定性（基于这些测量结果）受一个新量——\textit{鲁棒性界}的约束，该界限将测量的量子费舍尔信息与对偶码中权重为$2$的码字数量相关联。通过编码理论这一新颖视角，我们证明对于非平凡CSS码态，HNLS条件在鲁棒计量场景中依然主导着海森堡标度。这一发现揭示了线性量子码在近期鲁棒传感体系与长期容错时代中，应用于退相磁场传感场景时存在根本性局限。我们还将研究结果推广至退相磁场传感之外的一般性场景。