High-Girth Regular Quantum LDPC Codes from Square-Base Hypergraph Products via CPM Lifts
For quantum error correction, this work provides a finite-length code construction with record-low error rates under realistic noise, though the approach is incremental as it builds on known hypergraph-product and CPM-lift techniques.
The paper introduces high-girth regular quantum LDPC codes from square-base hypergraph products via CPM lifts, achieving a [[28800,62]] girth-8 (3,6)-regular CSS-LDPC code that produced zero decoding failures in 2.993×10^8 trials at depolarizing probability p=0.1402, with a Wilson 95% upper confidence bound of 1.28×10^{-8}.
We study square-base Calderbank--Shor--Steane (CSS) hypergraph-product codes as a finite-length class for regular high-girth quantum low-density parity-check (LDPC) design. For base matrices of small column weight, we give checkable conditions for regularity, rank deficiency, and short-cycle exclusion, and we present explicit column-weight-three and column-weight-four examples with Tanner girth 6 and 8. We also analyze circulant permutation matrix (CPM) lifts of this class. Using the standard voltage-sum criterion, we identify orthogonality-forced Tanner 8-cycles and show that CPM lifting cannot raise the Tanner girth beyond 8 when these cycles are present. As a representative finite-length instance, a randomized CPM lift of the girth-8 base construction gives a $[[28800,62]]$ girth-8 $(3,6)$-regular CSS-LDPC code. Under degeneracy-aware belief-propagation decoding with optional ordered-statistics-decoding-lite post-processing, this code produced zero decoding failures in $2.993\times 10^8$ independent trials at depolarizing probability $p=0.1402$; the Wilson 95% upper confidence bound is $1.28\times 10^{-8}$.