Reducing measurements in quantum erasure correction by quantum local recovery

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.