As measurements are costly and prone to errors on certain quantum computing devices, we should reduce the number of measurements and the number of measured qudits as small as possible in quantum erasure correction. It is intuitively obvious that a decoder can omit measurements of stabilizers that are irrelevant to erased qudits, but this intuition has not been rigorously formalized as far as the author is aware. In this paper, we formalize relevant stabilizers sufficient to correct erased qudits with a quantum stabilizer code, by using a recent idea from quantum local recovery. The minimum required number of measuring stabilizer observables is also clarified, which looks similar to the dimension length profile of classical linear codes. As an application, we also show that correction of $\delta$ erasures on a generalized surface code proposed by Delfosse, Iyer and Poulin requires at most $\delta$ measurements of vertexes and at most $\delta$ measurements of faces, independently of its code parameters.

翻译：由于在某些量子计算设备上测量成本高昂且易出错，在量子擦除纠错中，我们应尽可能减少测量次数及被测量量子比特的数量。直观上，解码器可以忽略与被擦除量子比特无关的稳定子测量，但据作者所知，这一直觉尚未被严格形式化。本文利用量子局域恢复的最新思想，形式化了量子稳定子码中纠正被擦除量子比特所需的充分相关稳定子集合。同时明确了所需测量的稳定子可观测量最小数量，其形式类似于经典线性码的维数长度分布。作为应用，我们还证明了在Delfosse、Iyer和Poulin提出的广义表面码上纠正δ个擦除错误时，最多需要δ次顶点测量和δ次面测量，且该结果与码参数无关。