High-Girth Regular Quantum LDPC Codes from Square-Base Hypergraph Products via CPM Lifts

Abstract

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 $[[28800,62,\le 192]]$ girth-8 $(3,6)$-regular CSS-LDPC code. Explicit weight-$192$ non-stabilizer logical representatives give $d_X,d_Z\le 192$. 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 $1.28\times 10^{-8}$.

21 pages, 4 figures; ancillary TeX file included. v2: Added explicit weight-192 X- and Z-type non-stabilizer logical representatives for the randomized P=64 lift, giving d_X,d_Z,d<=192; updated the abstract, discussion, conclusion, and supplementary data