Problem
Research questions and friction points this paper is trying to address.
CSS codes
canonical logical bases
low-weight
pair-partition codes
quantum error correction
Low-Weight Canonical Logical Bases from Pair-Partition Codes
🤖 AI Summary
This study addresses the challenge of constructing low-weight canonical logical bases for quantum CSS codes by proposing a method based on quaternary coefficient allocation and canonicalization to generate binary canonical logical bases. The core innovation lies in utilizing single-pair polynomial inverses to achieve representative canonical pairing while preserving binary weight, combined with circulant permutation matrices, partition checks, and algebraic coding theory to design an efficient canonicalization algorithm. Experimentally, the proposed approach successfully constructs codes with parameters such as [[320,80,14]], reducing the maximum check weight to 10 and achieving a logical basis weight of approximately 27. These results provide critical support for practical fault-tolerant quantum computing.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
CSS codes
canonical logical bases
circulant permutation matrix pair-partition
pairing polynomial
low-weight representatives
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