LDGM-Based Quantum Codes for Fault-Tolerant Quantum Computation

Quantum error correction requires codes that are both effective at correcting errors and practical for implementation. Traditional quantum LDPC codes often struggle with high encoding complexity or stabilizer weights that are too large for efficient fault-tolerant operations. This paper investigates whether LDGM-based codes can provide a flexible, high-performance alternative that simplifies the design of fault-tolerant quantum circuits.

The researchers utilize the generator and parity-check matrices of classical systematic LDGM codes to construct CSS quantum codes. By applying specific row operations to these sparse matrices, they satisfy the CSS commutativity constraints while maintaining a sparse structure. Decoding is performed using an iterative message-passing algorithm (belief propagation). To improve the convergence of the decoder, the authors introduce a 'doping' technique, which adds degree-1 syndrome nodes to the decoding graph to provide reliable initial information, and optimize the code structure using Discrete Density Evolution (DDE).

The proposed LDGM-based quantum codes offer significant design flexibility, allowing for the adjustment of the quantum rate while keeping stabilizer generator weights small. This property is critical for fault-tolerant computation, as lower-weight stabilizers reduce the number of physical gates required for logical operations, thereby limiting error propagation. Despite the theoretical limitation that LDGM codes have a small minimum distance, the authors demonstrate that these codes achieve excellent error-correction performance, effectively managing the trade-off between convergence thresholds and error floors.