Neural Network Belief Propagation and Ordered Statistics Decoding for Quantum Error Correction Codes

Abstract

In the quest for fault-tolerant quantum computation, the delicate nature of qubits demands the need for robust quantum error correction. Quantum error correcting codes, the quantum counterpart to classical error correcting codes in information theory, have been developed to address the unique challenges quantum bits face against the noise of their environment in storing information. These codes include but are not limited to the Kitaev surface code, toric code, rotated surface code, generalized bicycle code, and bivariate bicycle code. Fast and reliable decoders for these quantum error correcting codes are needed to correct errors and achieve fault tolerance on these qubits. The belief propagation algorithm, which is a efficient and reliable heuristic for decoding classical low-density parity check codes, has been generalized to different types of quantum error correction codes. Paired with the ordered statistics decoding algorithm as a post-processing step from the soft outputs of the belief propagation decoder, belief propagation and ordered statistics decoding has been one of the most promising avenues of efficient quantum error correction. This thesis combines the state-of-the-art belief propagation and ordered statistics decoding algorithms with the exploration of the neural belief propagation algorithm to leverage classical deep learning machines to learn message passing patterns in decoding the newest quantum codes, the rotated surface code and bivariate bicycle code. The neural network exploits the local lattice structure of these codes to adjust the trainable weights and biases to more learn message update patterns more efficiently than the vanilla belief propagation algorithm. In this thesis, we prove a new threshold for the physical error rate on the rotated surface code using a neural belief propagation algorithm and ordered statistics decoding that consistently outperforms the threshold established by the vanilla belief propagation algorithm with ordered statistics decoding. We additionally show the results for similar efforts on the bivariate bicycle codes, and discuss the difficulties in achieving a threshold using the current neural belief propagation algorithms.