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 measured stabilizer observables is also clarified. As an application, we also show that correction of $δ$ erasures on a generalized surface code proposed by Delfosse, Iyer and Poulin requires at most $δ$ measurements of vertexes and at most $δ$ measurements of faces, independently of its code parameters.

💡 Research Summary

The paper addresses a practical bottleneck in quantum error correction: the cost and error‑proneness of stabilizer measurements, especially on near‑term quantum devices where measurements are expensive or noisy. While it is well‑known that erasure errors (errors whose locations are known) can be corrected more efficiently than generic errors, existing decoding schemes for stabilizer codes still typically measure all n – k stabilizer generators regardless of which qudits have been erased. The authors formalize the intuition that only stabilizers “relevant” to the erased positions need to be measured, and they provide rigorous conditions and optimality proofs for this reduction.

The core technical contribution is the introduction of a subspace D ⊂ C, where C is the linear space of stabilizer exponent vectors (the self‑orthogonal subspace defining the code). For a given erasure set I ⊂ {1,…,n}, the code can correct the erasures if and only if the condition
C ∩ F_I = D^⊥ ∩ F_I
holds, where F_I denotes vectors that are zero outside I. This condition is equivalent to the standard erasure‑correctability criterion C ∩ F_I = C^⊥ ∩ F_I, but it isolates the part of C that actually interacts with the erased qudits. The authors show that measuring only the observables associated with a basis of D is sufficient for recovery, and that the minimal number of required measurements is

 min |measurements| = dim C – dim(C ∩ F_I).

Because dim C – dim(C ∩ F_I) ≤ 2|I|, the number of measurements never exceeds twice the number of erased qudits, and in many cases it is dramatically smaller than the full n – k measurements. The paper proves that this bound is tight: the minimum is attained when D ∩ F_I = {0}, i.e., when the chosen D contains no stabilizers that act trivially on the erased positions.

Two complementary optimality results are presented. Theorem 5 gives the exact minimal measurement count for a fixed erasure set I, while Theorem 8 characterizes the minimal measurement set that works uniformly for all erasure patterns of size δ. The latter is expressed in terms of the symplectic weight of the orthogonal complement D^⊥ \ C: a subspace D works for any δ erasures iff the smallest symplectic weight of vectors in D^⊥ \ C is at least δ + 1. This connects the measurement reduction problem to the code’s distance properties.

The authors then specialize the framework to CSS codes, separating X‑type and Z‑type stabilizers. For each type, the relevant subspace D_X or D_Z can be constructed from the corresponding classical parity‑check matrix, and the measurement count for each error type is bounded by the size of the erasure set. This leads directly to an application to generalized surface codes (the Delfosse‑Iyer‑Poulin construction). In these codes, vertex operators (X‑type) and face operators (Z‑type) correspond to the two CSS components. The paper proves that correcting δ vertex erasures requires at most δ vertex‑operator measurements, and similarly for δ face erasures, independent of the overall code distance or lattice size. This result dramatically reduces the measurement overhead for fault‑tolerant protocols based on surface codes, where measurement cycles dominate the runtime.

Algorithmically, the authors note that D can be obtained by Gaussian elimination on the generator matrix of C, with cubic complexity in n, which is acceptable for moderate‑size codes. They also discuss how to handle the situation where both erasures and unknown errors may occur on the same qudits: by additionally measuring the stabilizers in C ∩ F_I (one per qudit) one can detect whether an error is present, preserving the measurement savings while maintaining fault tolerance.

In summary, the paper provides a mathematically rigorous, measurement‑optimal decoding strategy for erasure errors in stabilizer codes, bridges the concept of quantum local recovery with erasure correction, and demonstrates concrete savings for generalized surface codes. The results have immediate relevance for near‑term quantum processors where measurement latency and error rates are limiting factors, and they open several avenues for future work, including extensions to mixed error‑and‑erasure scenarios, adaptive measurement schedules, and experimental validation on hardware platforms.