Nearly-optimal CSS code subspace verification with local measurement

Abstract Quantum error correction (QEC) is the core mechanism enabling scalable and fault-tolerant quantum computation. Among various QEC schemes, Calderbank–Shor–Steane (CSS) codes play a central role. Efficiently verifying whether a quantum state remains within the target CSS code subspace is therefore essential both for understanding the theoretical performance of these codes and for assessing the error-correcting capability of practical quantum platforms. In this work, we develop an efficient framework for verifying whether a quantum system remains within the encoded subspace of a CSS code. We first show that the stabilizer structure of any CSS code can be mapped to a two-colorable graph, enabling the design of a hybrid verification strategy based solely on local Pauli measurements. We further analytically optimize the measurement probabilities and derive the resulting sample complexities for representative CSS codes. Specifically, for the Toric code, the efficiency of our hybrid strategy increases with lattice size and asymptotically approaches near-optimal performance. These results establish a unified and scalable approach to CSS code subspace verification, providing a foundation for certifiable and fault-tolerant quantum information processing.