Decoding and check-measurement costs of the sparse protected construction

Analyze the quantum decoding and check-measurement costs of the sparse protected construction.

Background

The sparse protected code has bounded check support and bounded qubit participation, but its checks are commuting projectors rather than ordinary Pauli checks. The paper establishes distance and transversal-gate properties but does not provide an operational complexity analysis for decoding or measuring these checks.

The authors explicitly identify both quantum decoding costs and check-measurement costs as unresolved aspects of the construction.