On the Left-Right Lifted Product Quantum LDPC Code

Recent advances in both theory and experiment have identified quantum Low-Density Parity-Check (LDPC) codes as a prime candidate for fault-tolerant quantum computation in the near future. This thesis addresses various challenges related to the construction and decoding of quantum LDPC (qLDPC) codes. A major contribution of this research is the creation of a unified family of qLDPC codes called left-right lifted product codes. The proposed approach not only makes the construction process simpler but also brings different qLDPC constructions together into a single framework that greatly improves both understanding and practical use.

This thesis aims to highlight the practical and theoretical advantages of the left-right lifted product construction. As part of this, we present the first finite-length distance bounds for the general lifted product code, subject to certain mild technical conditions. Moreover, we also extend the finite-length result to asymptotic lengths and provide a conjecture for a new kind of asymptotically good quantum LDPC code that holds promise for more feasible and practical applications.

From a practical standpoint, this work includes the first detailed cycle analysis, specifically focusing on dominant 8-cycles, which significantly impact decoding performance. Our findings indicate that the lifted product construction substantially reduces the number of short 8-cycles compared to the hypergraph product code. Additionally, we establish an upper bound for the girth of the left-right lifted product code, demonstrating that, unlike classical QC-LDPC codes, the presence of 8-cycles is unavoidable, regardless of how the group algebra elements are selected in the base matrices.

Lastly, leveraging newly developed mathematical tools in spectral hypergraph theory, we offer a computable lower bound for the left-expansion property of the target factor graph. We believe that this contribution can be particularly valuable for code design, as the left-expansion property was previously difficult to quantify.