On the Minimum Distances of Finite-Length Lifted Product Quantum LDPC Codes

Quantum error correction (QEC) is critical for the practical realization of fault-tolerant quantum computing and quantum communication systems. Recently proposed families of quantum low-density parity-check (QLDPC) codes are prime candidates for advanced QEC hardware architectures. This paper focuses on finite-length QLDPC code design criteria, specifically aimed at constructing degenerate quasi-cyclic symmetric lifted-product (LP-QLDPC) codes with high minimum distances. We describe necessary conditions to ensure that the designed LP-QLDPC codes have a minimum distance strictly greater than the minimum weight stabilizer generators, thus ensuring superior error correction performance. The focus of the paper is on LP-QLDPC codes built from quasi-cyclic base codes belonging to the class of type-1 protographs, and the necessary constraints are efficiently expressed in terms of row and column indices of the base parity check matrix. This gives us combinatorial constraints - termed as Row-Column Partition Constraints (RCPC) - on classical quasi-cyclic base matrices to construct degenerate LP-QLDPC codes. For type-1 protographs, RCPC constraints provide an efficient certification approach to designing good codes, and we also give exhaustive search evidence on small base matrices. Simulations with BP+OSD decoding demonstrate significant logical error-rate gains when RCPC is avoided. We discuss extensions to circuit-level noise and outline sufficient conditions and research directions for exact distance preservation.