Structural encoding with classical codes for computational-basis bit-flip correction in the early fault-tolerant regime

Achieving reliable performance on early fault-tolerant quantum hardware will depend on protocols that manage noise without incurring prohibitive overhead. We propose a novel framework that integrates quantum computation with the functionality of classical error correction. In this approach, quantum computation is performed within the codeword subspace defined by a classical error correction code. The correction of various types of errors that manifest as bit flips is carried out based on the final measurement outcomes. The approach leverages the asymmetric structure of many key algorithms, where problem-defining diagonal operators (e.g., oracles) are paired with fixed non-diagonal operators (e.g., diffusion operators). The proposed encoding maps computational basis states to classical codewords. This approach commutes with diagonal operators, obviating their overhead and confining the main computational cost to simpler non-diagonal components. Noisy simulations corroborate this analysis, demonstrating that the proposed scheme serves as a viable protocol-level layer for enhancing performance in the early fault-tolerant regime.