Quantum Subspace Verification for Error Correction Codes

Quantum error correction is pivotal in advancing toward large-scale quantum computation, and efficient verification is crucial for ensuring the high fidelity of code states. Traditional methodologies, such as state tomography, direct fidelity estimation, and state verification, either fall short in measurement efficiency, especially for large-scale systems, or are restricted to some specific states. In this work, we introduce a general framework for quantum subspace verification, enabling efficient and measurement-noise-robust fidelity estimation between a given state and the target subspace with a specified confidence level. By integrating the proposed subspace verification with direct fidelity estimation, we develop a composite protocol that significantly improves the efficiency of verifying the fidelity of general magic logical states, as demonstrated by intuitive numerical results. This improvement stems from the use of subspace verification, which leverages the knowledge of code subspaces to significantly reduce measurement costs. Additionally, we detail the construction of verification operators for typical error correction codes, including general stabilizer codes and quantum low-density parity-check codes, enabling their efficient implementation using practical local measurements. Notably, for certain codes, such as the Calderbank-Shor-Steane codes and quantum low-density parity-check stabilizer codes, we reduce the number of required measurement settings and sample complexity to a constant level using graphical methods. Our approach facilitates efficient and feasible verification of error correction codes and generic magic logical states, advancing their practical implementations on quantum platforms.