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

• The paper introduces a novel construction of high-girth quantum LDPC codes using square-base hypergraph products combined with CPM lifts.
• It develops a methodology that enforces regular base matrix design and employs voltage-sum constraints to analyze forced 8-cycles under CSS orthogonality.
• Finite-length decoding experiments demonstrate extremely low frame error rates using degeneracy-aware BP decoding augmented by OSD post-processing.

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

Overview

The paper "High-Girth Regular Quantum LDPC Codes from Square-Base Hypergraph Products via CPM Lifts" (2604.27817) investigates the construction and properties of quantum LDPC codes derived from square-base hypergraph-products (HGP), focusing on finite-length instances with explicit control of degree-regularity and short-cycle structure. The approach leverages combinatorial design in the form of small-row/column-weight base matrices and analyzes the implications of circulant permutation matrix (CPM) lifts for girth and logical qubit count, culminating in rigorous finite-length decoding experiments. The study systematically develops explicit constructions for both $(3,6)$- and $(4,8)$-regular CSS (Calderbank–Shor–Steane) codes, pinpoints CSS-constrained forced 8-cycles, and demonstrates practical decoding performance for large lifted codes with high girth.

Theoretical Construction: Square-Base HGP and CPM Lifting

The foundation of the work is the specialization of the HGP construction—originally due to Tillich and Zémor—to square binary base matrices $B \in \mathbb{F}_2^{s\times s}$. The CSS check matrices are defined as:

• $$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$
• $$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$

By construction, these always satisfy the CSS orthogonality condition. The code's logical qubit yield is governed by the corank $c_B$ of $B$, with $k = 2c_B^2$. Critically, the base matrices are required to be regular (equal row and column weight $w$) and their overlap structure is strictly controlled to limit short cycles in the resulting Tanner graphs.

CPM lifts are then applied: each nonzero entry of the CSS matrices is replaced by a $P \times P$ circulant permutation block, potentially increasing blocklength by a factor $(4,8)$0. However, the voltage-sum constraints stemming from CSS orthogonality (i.e., on overlaps between $(4,8)$1 and $(4,8)$2 checks) induce forced short cycles, specifically unavoidable 8-cycles, that cannot be eliminated by increasing $(4,8)$3.

This is exemplified in the following formal result from the paper: for any $(4,8)$4-regular square base $(4,8)$5 of size $(4,8)$6, every CPM $(4,8)$7 lift contains at least $(4,8)$8 Tanner 8-cycles in each of $(4,8)$9 and $B \in \mathbb{F}_2^{s\times s}$0. This sets an inherent girth limitation—CPM lifts cannot raise the Tanner girth above 8 when these structures exist.

Base-Matrix Design and Explicit Examples

The authors detail criteria for base-matrix selection:

1. Square, binary matrices of fixed row/col weight $B \in \mathbb{F}_2^{s\times s}$1.
2. Positive corank to ensure nonzero logical qubits.
3. Exclusion of 4-cycles and simple 6-cycles in the base (except for explicit girth-6 examples).
4. Preference for regular, connected, high-distance, and low small-codeword support.

Principal classes of explicit constructions include:

• The Fano plane ($B \in \mathbb{F}_2^{s\times s}$2): a $B \in \mathbb{F}_2^{s\times s}$3 incidence structure yielding a $B \in \mathbb{F}_2^{s\times s}$4-regular base with girth 6 but containing triangles (hence 6-cycles).
• The generalized quadrangle $B \in \mathbb{F}_2^{s\times s}$5 ($B \in \mathbb{F}_2^{s\times s}$6): a $B \in \mathbb{F}_2^{s\times s}$7 girth-8 $B \in \mathbb{F}_2^{s\times s}$8-regular incidence matrix, yielding high corank with no simple 6-cycles.
• Enlarged and randomized matrices: e.g., $B \in \mathbb{F}_2^{s\times s}$9 (connected sum of two $$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$0 with edge swap for increased corank), $$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$1 and $$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$2 (low and moderate corank, regular, girth-8), and higher-degree cases like $$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$3 (projective plane, $$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$4-regular, girth-6) and $$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$5 ($$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$6, $$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$7-regular, girth-8).

Explicit construction properties can be visualized in the matrix plots below, displaying the support of various bases used.

Figure 1: Matrix plots of the smaller square base matrices: $$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$8, $$H_X(B) = [B \otimes I_s \mid I_s \otimes B^{\mathsf T}]$$9, $$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$0, $$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$1, $$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$2. Each visual displays the binary support pattern (entries equal to 1 in black).

Figure 2: Matrix plots of the larger square base matrices: $$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$3 and $$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$4. These larger, more complex structures yield higher corank and enable larger codes.

Figure 3: Matrix plots of the HGP check matrices ($$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$5, $$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$6) derived from the Fano-plane base $$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$7, demonstrating the expanded regular structure post-HGP.

CPM Lifts: Forced Cycles and Logical Qubits

A principal contribution is the detailed analysis of CPM $$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$8 lifts. The authors prove that certain patterns of overlap in the base matrix, enforced by the CSS orthogonality condition, result in forced 8-cycles in every possible CPM lift, regardless of $$H_Z(B) = [I_s \otimes B \mid B^{\mathsf T} \otimes I_s]$$9. As a consequence, it is impossible to achieve Tanner girth $c_B$0 for the lifted code graphs derived from these square-base designs, a phenomenon rigorously characterized via a voltage-sum criterion.

The logical qubit count in the lift, $c_B$1, satisfies $c_B$2 by construction, but further increases—beyond trivial disconnected lifts—require the emergence of new nonconstant left-kernel vectors, a nontrivial combinatorial condition not generically realized.

Decoding and Numerical Performance

The finite-length performance of the designed codes is empirically evaluated for a large lifted code derived from the generalized-quadrangle base $c_B$3 with a randomized CPM $c_B$4 lift, resulting in a $c_B$5 CSS-LDPC code with girth 8 and degrees $c_B$6. The decoding protocol uses a degeneracy-aware belief propagation (BP) decoder, augmented with an OSD-lite post-processing for additional syndrome-based repairs.

Figure 4: $c_B$7–FER plot for the randomized $c_B$8 lift derived from the generalized-quadrangle base $c_B$9. The plot shows the frame error rate (FER) vs. depolarizing probability $B$0, including confidence intervals. Vertical lines indicate the BP density-evolution threshold ($B$1) and the depolarizing-channel hashing limit ($B$2).

At $B$3, the code yielded zero observed decoding failures in $B$4 trials, corresponding to a Wilson 95% confidence upper bound on FER of $B$5. Notably, all recorded decoding failures were non-syndrome-matching, so the data do not address minimum-distance upper bounds. Degenerate success (correction up to stabilizers) dominated, as expected for CSS codes.

Implications and Future Directions

From a theoretical viewpoint, the paper clarifies inherent girth and degree limitations that arise in quantum LDPC codes constructed via square-base HGP with CPM lifts, notably highlighting indelible short cycles imposed by code commutation structure. This imposes a fundamental tradeoff in design: while degree-regularity and short-cycle suppression in the base can be strictly enforced, lift-induced short cycles limit potential for further improvements in girth or minimum pseudocodeword weight via simple permutation structure.

Practically, the explicit constructions and high-precision decoding experiments demonstrate that quantum LDPC codes of large blocklengths and moderate rate, with provably high girth and regular degree, can achieve extremely low finite-length FERs under efficient BP-type decoding regimes with modest OSD augmentation. However, the inherited short cycles represent a performance bottleneck, and additional logical qubit growth via lifting remains structurally hard to achieve in connected lifts.

Open challenges underscored by the work include:

• Certifying minimum distance for large constructed codes via failure-type analysis rather than exclusively decoding statistics;
• Systematic design of CPM or non-Cartesian lifts that generate nontrivial increases in $B$6 while retaining connectedness and orthogonality without impacting girth;
• Extending these construction paradigms to yield codes with positive design quantum rate (rather than exclusive reliance on corank);
• Comparative study across different base-matrix structure families (geometric, random, combinatorial designs) for tradeoffs in rate, distance, and decoding thresholds.

Conclusion

This paper provides a rigorous analysis and practical demonstration of high-girth, regular quantum LDPC codes from the square-base HGP construction with CPM lifts, establishing explicit and checkable design criteria for finite-length base matrices and characterizing forced short-cycle effects from code orthogonality. The decoding experiments for large lifted codes confirm the viability of the approach for high-performance quantum error correction under standard Pauli channels. Future advances depend on resolving minimum-distance certification and devising novel lifts or base constructions circumventing present girth and corank limitations.