Advanced decoding techniques for quantum error correcting codes.

Quantum error-correcting codes offer a solution to mitigate quantum noise, which is inherent to quantum mechanics, and therefore they enable reliable quantum computation. Among these codes, quantum low-density parity-check codes are promising candidates for future quantum error correcting codes due to their ability to provide fault tolerance with constant overhead and recent advancements in designing asymptotically good quantum low-density parity-check codes. A notable challenge in implementing quantum low-density parity-check codes for practical quantum computers is the absence of a universal decoder that delivers good decoding performance across various quantum low-density parity check codes. Among the proposed solutions, the neural belief propagation decoder, leveraging machine and deep learning techniques, emerges as a promising approach. Neural belief propagation generalizes belief propagation decoding by incorporating trainable weights optimized through supervised learning, thereby accelerating the decoding process and enhancing accuracy in quantum error correction. In this thesis, we explore the application of machine learning and deep learning techniques in quantum error correction, focusing specifically on the toric code that is typically decoded using the minimum weight perfect matching algorithm. This algorithm has a worst-case complexity O(n^3) (where n is the number of qubits), and is considered to be too complex for a practical decoder. In particular, we investigate the quaternary neural belief propagation decoder proposed by Miao et al. which has linear complexity in n and we introduce techniques such as residual connections, weight-sharing, higher weight overcomplete check matrices, and weight reuse to potentially improve the decoder's performance. Our results demonstrate that some of these techniques (the use of weight-sharing and higher weight overcomplete check matrices) can lead to a significant performance improvement for small systems. In particular, for L=4 and L=6 we can largely outperform the minimum-weight perfect matching. However, these gains decrease as L increases, and they are negligible for L≥10. Furthermore, reusing the weights trained for a small toric code with larger toric codes significantly improves the performance with respect to standard belief propagation, although it fails to reach the performance of minimum-weight perfect matching.