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arXiv:2606.27130

Rate-2/3 Girth-8 (3,18)-Regular Quantum LDPC Codes from Two-Branch Finite-Field Bases and CPM Lifts

Koki Okada, Kenta Kasai

quant-phcs.ITSubmitted Jun 25, 2026• Announced Jun 26, 2026• 0 citations• Revised Sep 5, 2026 (v2)
v1 · Jun 25, 2026v2 · Sep 5, 2026
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Abstract

We construct a rate-$2/3$ quantum low-density parity-check (LDPC) code from a $(3,18)$-regular two-branch finite-field base and a circulant-permutation-matrix (CPM) lift of degree $P=101$. The resulting code is a Calderbank--Shor--Steane (CSS) code with parameters $[[34542,23032,18]]$. Its distance is established by a computer-assisted proof: an explicit nontrivial logical operator gives the upper bound, and a complete symmetry-reduced enumeration excludes every nonzero vector of weight below 18 in both parity-check kernels. We also prove that every affine-in-$t$ CPM lift of the same base satisfying CSS orthogonality has distance at most 18 for every prime lift degree $P>19$. The construction has row weight 18 and column weight 3, and the Tanner graphs of $H_X$ and $H_Z$ separately have girth 8. Decoder experiments with log-likelihood-ratio (LLR) joint belief propagation (BP) and deterministic post-processing show no failures in $10^8$ trials at $p=0.01$, and a finite-length frame error rate (FER) sweep estimates the transition near $p=0.029$.

Comments
12 pages, 2 figures

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We construct a quantum low-density parity-check code family from a length-512512512 Calderbank--Shor--Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are (3,8)(3,8)(3,8)-regular with girth 888. The base code has parameters [[512,174,8]][[512,174,8]][[512,174,8]]. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor P=32P=32P=32, which has parameters [[16384,4142,18≤d≤32]][[16384,4142,18\le d\le32]][[16384,4142,18≤d≤32]], under the code-capacity depolarizing model. Complete enumeration excludes nontrivial logical representatives of weight at most 161616 on both sides, giving dX,dZ≥18d_X,d_Z\ge18dX​,dZ​≥18. Explicit weight-323232 non-stabilizer logical representatives give the upper bounds dX,dZ≤32d_X,d_Z\le32dX​,dZ​≤32. A belief-propagation decoder with post-processing achieved frame error rate about 10−810^{-8}10−8 at p=0.085p=0.085p=0.085; an independently observed logical residual of weight 404040 is consistent with the sharper structural bound.
This paper develops a two-branch multiplicative-coset construction for regular Calderbank--Shor--Steane (CSS) quantum low-density parity-check base matrices. For a target column weight JJJ and an even row weight LLL, the method reduces regularity, CSS orthogonality, and same-type 4-cycle exclusion to explicit quotient-coset conditions over a finite field. A normalized exhaustive search for these conditions produces base matrices for several (J,L)(J,L)(J,L) pairs, so the construction is not tied to a single degree distribution. The construction separates the finite-length design into two stages: the base matrix fixes the degree distribution and the first girth constraints, and a cyclic lift randomizes edge connections subject to exact algebraic checks. As a detailed example, we carry one (3,10)(3,10)(3,10)-regular base through the lift and decoding stages. For this example, the selected 64-fold lift gives a code whose same-type Tanner graphs have girth at least eight, and it also excludes a specified weight-16 nondegenerate logical-support orbit. Complete support enumeration excludes all nontrivial logical representatives of weight at most 16 on both sides. The resulting instance is a [[10240,4108, 18≤d≤32]][[10240,4108,\,18\le d\le32]][[10240,4108,18≤d≤32]] CSS code. For decoding, we use joint log-domain belief propagation together with low-complexity deterministic post-processing rules for small residual syndromes, including repairs for residual patterns with two unsatisfied checks. The frame error rate (FER) measurements provide finite-length decoding data for this detailed example; at depolarizing probability p=0.058p=0.058p=0.058, the post-processing FER is 1.0×10−71.0\times10^{-7}1.0×10−7.
We introduce the pair-partition (PP) construction of binary Calderbank--Shor--Steane quantum low-density parity-check codes from circulant permutation matrices. A square array of pair partitions imposes linear paired-difference equations on the CPM exponents and thereby guarantees CSS orthogonality. Pairing graphs derived from this array allow the combinatorial design to be screened before exponent search. We further give and prove a complete algorithm for verifying lower bounds on the quantum minimum distance of a fixed CSS lift. The algorithm searches for zero-syndrome vectors outside the opposing stabilizer row space, uses only rigorously valid pruning rules, and finds no vector through a prescribed weight if and only if the corresponding distance exceeds that weight. For fixed column weight, row weight, and search limit, cyclic symmetry makes the combinatorial support-enumeration bound independent of the lift size, although matrix preprocessing and row-space tests can still depend on the lift size. Thus the construction stage and the distance-verification stage are both specified by directly checkable finite procedures.
Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. As a concrete demonstration, we construct a girth-8, (3,12)-regular [[9216,4612,≤48]][[9216,4612, \leq 48]][[9216,4612,≤48]] quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as 10−810^{-8}10−8 on the depolarizing channel with error probability 4%4 \%4%.
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