On the Logical Error Rate of Sparse Quantum Codes

The quantum paradigm presents a phenomenon known as degeneracy that can potentially improve the performance of quantum error correcting codes. However, the effects of this mechanism are sometimes ignored when evaluating the performance of sparse quantum codes and the logical error rate is not always correctly reported. In this article, we discuss previously existing methods to compute the logical error rate and we present an efficient coset-based method inspired by classical coding strategies to estimate degenerate errors and distinguish them from logical errors. Additionally, we show that the proposed method presents a computational advantage for the family of Calderbank–Shor–Steane codes. We use this method to prove that degenerate errors are frequent in a specific family of sparse quantum codes, which stresses the importance of accurately reporting their performance. Our results also reveal that the modified decoding strategies proposed in the literature are an important tool to improve the performance of sparse quantum codes.