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

2026-04-30

AI Summary

This work addresses the challenge of constructing practical-length quantum low-density parity-check (LDPC) codes with high girth and regular structure. Building upon the square-based Calderbank–Shor–Steane (CSS) hypergraph product construction and employing circulant permutation matrix (CPM) lifting, the authors derive verifiable conditions ensuring regularity, rank deficiency, and the absence of short cycles. They rigorously prove that the Tanner graph girth under CPM lifting is upper-bounded by 8, and thereby construct—for the first time—a girth-8 (3,6)-regular CSS-LDPC code [[28800,62]]. Under 299.3 million independent decoding trials at physical error rate p = 0.1402, this code exhibits zero decoding failures, yielding a 95% confidence upper bound on the logical error rate of 1.28 × 10⁻⁸, significantly advancing quantum error correction performance.

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 $[[28800,62]]$ 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 $1.28\times 10^{-8}$.

Problem

• quantum LDPC codes
• high-girth
• regular codes
• CSS codes
• Tanner graph

Innovation

• quantum LDPC codes
• high-girth
• hypergraph product
• CPM lifts
• CSS codes