Decoding imperfect stabilizer codes with quantum neural networks
Error correction remains a key challenge for realizing fault-tolerant quantum computation in the noisy intermediate-scale quantum era. Stabilizer codes are among the leading candidates for the realization of fault-tolerant quantum computation. Stabilizer codes can be realized as ground states of physical systems. Although there has been much work on decoding ideal codes, the utility of imperfect codes is less understood. Motivated by practical realization of quantum codes on near-term devices, we develop a theoretical framework for decoding imperfect codes. Using Brillouin-Wigner perturbation theory, we prove that the scaling of the decoding error in standard procedure is independent of the underlying code distance. We corroborate the theory by numerical simulations of codes with various distances. We numerically find that the distant-independent scaling of the decoding error can be superseded by Quantum Neural Networks (QNNs). We provide theoretical guarantees by proving a QNN performance lower-bound on decoding imperfect codes from corrupted data, and show that QNNs are capable of achieving errors exponentially small in code distance, suggesting an scaling advantage over standard decoding procedures. We then numerically demonstrate that QNNs outperform standard procedures in decoding imperfect stabilizer codes. Our theoretical framework and numerical protocol provide a first step toward the understanding and practical decoding of imperfect stabilizer codes.