Quantum metrology (QM) is expected to be a prominent use-case of quantum technologies. However, noise easily degrades these quantum probe states, and negates the quantum advantage they would have offered in a noiseless setting. Although quantum error correction (QEC) can help tackle noise, fault-tolerant methods are too resource intensive for near-term use. Hence, a strategy for (near-term) robust QM that is easily adaptable to future QEC-based QM is desirable. Here, we propose such an architecture by studying the performance of quantum probe states that are constructed from $[n,k,d]$ binary block codes of minimum distance $d \geq t+1$. Such states can be interpreted as a logical state of a CSS code whose logical $X$ group is defined by the aforesaid binary code. When a constant, $t$, number of qubits of the quantum probe state are erased, using the quantum Fisher information (QFI) we show that the resultant noisy probe can give an estimate of the magnetic field with a precision that scales inversely with the variances of the weight distributions of the corresponding $2^t$ shortened codes. If $C$ is any code concatenated with inner repetition codes of length linear in $n$, a quantum advantage in QM is possible. Hence, given any CSS code of constant length, concatenation with repetition codes of length linear in $n$ is asymptotically optimal for QM with a constant number of erasure errors. We also explicitly construct an observable that when measured on such noisy code-inspired probe states, yields a precision on the magnetic field strength that also exhibits a quantum advantage in the limit of vanishing magnetic field strength. We emphasize that, despite the use of coding-theoretic methods, our results do not involve syndrome measurements or error correction. We complement our results with examples of probe states constructed from Reed-Muller codes.

中文翻译：

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经典代码的权重分布影响稳健的量子计量

量子计量学 (QM) 有望成为量子技术的一个突出用例。然而，噪声很容易降低这些量子探测状态，并抵消它们在无噪声环境中提供的量子优势。尽管量子纠错 (QEC) 可以帮助解决噪声问题，但容错方法对于短期使用来说过于耗费资源。因此，需要一种易于适应未来基于 QEC 的 QM 的（近期）稳健 QM 策略。在这里，我们通过研究由最小距离 $d\geq t+1$ 的 $[n,k,d]$ 二进制块码构建的量子探测状态的性能，提出了这样一种架构。这种状态可以解释为CSS代码的逻辑状态，其逻辑$X$组由上述二进制代码定义。当常数 $t$ 量子探针状态的量子比特数被擦除时，使用量子费舍尔信息 (QFI)，我们表明所产生的噪声探针可以以与相应的 $2^t$ 缩短代码的权重分布的方差成反比的精度给出磁场的估计。如果 $C$ 是与 $n$ 中长度为线性的内部重复代码串联的任何代码，则 QM 中的量子优势是可能的。因此，给定任何恒定长度的 CSS 代码，与长度为 $n$ 的线性重复代码的串联对于具有恒定擦除错误数量的 QM 是渐近最优的。我们还明确地构建了一个可观察量，当在这种嘈杂的代码激发的探测状态上进行测量时，可以产生磁场强度的精度，这在磁场强度消失的极限中也表现出量子优势。我们强调，尽管使用了编码理论方法，但我们的结果不涉及综合症测量或纠错。我们用由 Reed-Muller 代码构建的探测状态示例来补充我们的结果。