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This research advances the design of finite-length quantum LDPC codes by establishing rigorous conditions to exclude short cycles in square-base CSS hypergraph-product constructions. The study demonstrates that while CPM lifting is a powerful technique, its ability to increase Tanner girth beyond 8 is fundamentally limited when specific orthogonality-forced cycles exist within the base matrices. By providing explicit examples and decoding performance data at high depolarizing probabilities, this work offers concrete benchmarks for building more robust quantum error correction schemes without relying on infinite-length limits.
High-Girth Regular Quantum LDPC Codes from Square-Base Hypergraph Products via CPM Lifts
arXiv:2604.27817v2 Announce Type: replace 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 $[[28800,62,\le192]]$ girth-8 $(3,6)$-regular CSS-LDPC code. Explicit weight-$192$ non-stabilizer logical representatives give $d_X,d_Z\le192$. 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 $1.28\times 10^{-8}$.
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 $[[28800,62,\le192]]$ girth-8 $(3,6)$-regular CSS-LDPC code. Explicit weight-$192$ non-stabilizer logical representatives give $d_X,d_Z\le192$. 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
Quantum Physics (quant-ph); Information Theory (cs.IT)
arXiv:2604.27817 [quant-ph]
(or arXiv:2604.27817v2 [quant-ph] for this version)
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