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

TL;DR

This paper introduces a new class of regular high-girth quantum LDPC codes based on square-base hypergraph products and analyzes their properties, including girth limitations and decoding performance.

Contribution

It provides explicit constructions and conditions for regularity and girth, and demonstrates the effectiveness of CPM lifts and decoding on a large finite-length code.

Findings

Explicit checkable conditions for regularity and girth in base matrices.

CPM lifts cannot increase Tanner girth beyond 8 when certain cycles are present.

A large finite-length code achieved zero decoding failures in extensive trials at a specific depolarizing probability.

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 $1.28\times 10^{-8}$.