Canonical logical bases improve qubit error-correcting code efficiency
Low-Weight Canonical Logical Bases from Pair-Partition Codes
Information Theory
Summary
Building reliable quantum computers requires ways to protect qubits from errors. This paper shows how to create special sets of rules, called logical bases, that represent the information in certain quantum error-correcting codes. These bases are designed to be efficient by keeping the number of qubits involved in each rule low, making error correction simpler. The authors provide a general construction method, examples with real parameters, and demonstrate how their approach keeps the rules’ complexity manageable.
What this means in practice
- •For quantum hardware engineers: Design error-correcting schemes with lower qubit interaction complexity to improve physical qubit reliability and reduce error correction overhead.
- •For classical error-correcting code developers: Apply low-weight canonical logical bases design principles to improve decoding efficiency and logical operations in related classical codes.
Authors
Koki Okada, Kenta Kasai
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.