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arXiv:2609.35601

Low-Weight Canonical Logical Bases from Pair-Partition Codes

Koki Okada, Kenta Kasai

quant-phcs.ITSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

We construct complete canonical logical bases for qubit CSS codes by assigning quaternary coefficients to binary circulant permutation matrix pair-partition (CPM-PP) checks, constructing and normalizing logical representatives, and expanding the result into binary matrices. When the two check systems have invertible submatrices on disjoint column sets, cofactor representatives are canonically paired by the inverse of a single pairing polynomial. The resulting pairs span the entire logical space. When the pairing polynomial is a cyclic-shift monomial with coefficient one, normalization preserves the binary weights of the representatives. We state the construction for general block dimensions and CPM size, work through a corresponding example, and report the parameters, check ranks, and basis weights of seven binary codes. Representative examples have parameters $[[320,80,14]]$, $[[448,112,18]]$, and $[[2048,512,24]]$. All three have maximum check weight 10 on both the X and Z sides. Their canonical logical representatives have binary weights 25, 27, and 27, respectively, on both the X and Z sides.

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13 pages, 2 tables

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We construct complete canonical logical bases for qubit CSS codes by assigning quaternary coefficients to binary circulant permutation matrix pair-partition (CPM-PP) checks, constructing and normalizing logical representatives, and expanding the result into binary matrices. When the two check systems have invertible submatrices on disjoint column sets, cofactor representatives are canonically paired by the inverse of a single pairing polynomial. The resulting pairs span the entire logical space. When the pairing polynomial is a cyclic-shift monomial with coefficient one, normalization preserves the binary weights of the representatives. We state the construction for general block dimensions and CPM size, work through a corresponding example, and report the parameters, check ranks, and basis weights of seven binary codes. Representative examples have parameters [[320,80,14]][[320,80,14]][[320,80,14]], [[448,112,18]][[448,112,18]][[448,112,18]], and [[2048,512,24]][[2048,512,24]][[2048,512,24]]. All three have maximum check weight 10 on both the X and Z sides. Their canonical logical representatives have binary weights 25, 27, and 27, respectively, on both the X and Z sides.
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.
Efficient fault-tolerant quantum computing architectures benefit from low-weight logical operators. However, reported canonical bases for high-rate, high-distance codes can be several times heavier than the code distance, and fundamental obstructions can prevent minimum-weight logical operators from forming a complete canonical basis. To address this issue, we develop design principles for group-valued quantum error-correcting codes with low-weight full regular canonical logical bases, generated by translating a set of seed operators. We introduce frame width fff, the smallest weight cutoff that permits a complete canonical logical basis, as an additional code parameter alongside distance ddd and check weight www. Cataloguing these parameters together with explicit logical bases can inform resource estimates and the compilation of fault-tolerant logical operations. First, we identify a structural obstruction: for fully populated binary monomial CSS checks over a group of odd order, full regularity forces distance two, while for groups of power-of-two order, full row rank of both check matrices after replacing every group element by 111 guarantees full regularity. Second, we demonstrate our principles through explicit high-rate lifted-product (LP) and pair-partition (PP) codes: an [[1088,128,22]][[1088,128,22]][[1088,128,22]] LP code with f=22f=22f=22 and an [[1024,256,24]][[1024,256,24]][[1024,256,24]] PP code with f≤27f \leq 27f≤27, both at w=10w=10w=10. At w=11w=11w=11, our PP constructions attain either d=25d=25d=25 with f≤27f \leq 27f≤27 or d=24d=24d=24 with the proved optimum f=25f=25f=25, demonstrating both a tradeoff among these parameters and a strict separation between distance and frame width. Finally, we propose a method combining symmetry reduction and stabilizer-based pruning for exhaustive distance certification, and use it to efficiently establish exact distances for LP examples with more than a thousand qubits.
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 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.
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 [[28800,62,18≤d≤192]][[28800,62,18\le d\le192]][[28800,62,18≤d≤192]] girth-8 (3,6)(3,6)(3,6)-regular CSS-LDPC code. Complete enumeration excludes nontrivial logical supports of weight at most 161616 on both sides, giving dX,dZ≥18d_X,d_Z\ge18dX​,dZ​≥18; explicit weight-192192192 representatives give dX,dZ≤192d_X,d_Z\le192dX​,dZ​≤192. Under degeneracy-aware belief-propagation decoding with optional ordered-statistics-decoding-lite post-processing, this code produced zero decoding failures in 2.993×1082.993\times 10^82.993×108 independent trials at depolarizing probability p=0.1402p=0.1402p=0.1402; the Wilson 95\% upper confidence bound is 1.28×10−81.28\times 10^{-8}1.28×10−8.
High-rate quantum low-density parity-check (qLDPC) codes encode many logical qubits with low physical-qubit overhead, but realizing efficient fault-tolerant computation on such dense encodings remains a major challenge. Generic, code-agnostic techniques such as code surgery and gate teleportation apply broadly, but are difficult to make modular, low-overhead, and fully certifiable on complex high-rate codes whose structure is left unexploited. Here we overcome these obstacles by co-designing the code together with its logical instruction set for a broad family of canonical lifted-product (LP) codes with cyclic symmetry. We show that these codes admit a canonical logical basis, in which conjugate logical operators are organized into rows and columns of cyclic orbits inherited directly from the underlying classical codes, analogous to the structure that makes hypergraph-product codes so tractable. This canonical basis enables universal logical processing using magic-state input and a constant number of reusable seed surgery gadgets---only two (resp. four) for a [[1122,148,≤20]][[1122,148,\leq 20]][[1122,148,≤20]] (resp. [[4350,1224,≤20]][[4350,1224,\leq 20]][[4350,1224,≤20]]) LP code. A fixed extractor smaller than half the data block also enables arbitrarily high-weight logical Pauli-product measurements (PPMs) while preserving the code's cyclic symmetry. These codes also enable rigorously fault-tolerant parallel logic, including PPMs across all logical qubits and magic-state injection from many surface codes into a full LP block. For example, a [[468,36,20]][[468,36,20]][[468,36,20]] (resp. [[952,112,17]][[952,112,17]][[952,112,17]]) code supports simultaneous measurements of arbitrary logically disjoint Pauli products with an ancilla overhead of roughly three (resp. four) times the data block size. These results advance the frontier of fault-tolerant quantum computation on ultra-high-rate quantum architectures.