On the Generalization of Kitaev Codes as Generalized Bicycle Codes

Abstract

Surface codes have historically been the dominant choice for quantum error correction due to their superior error threshold performance. However, recently, a new class of Generalized Bicycle (GB) codes, constructed from binary circulant matrices with three non-zero elements per row, achieved comparable performance with fewer physical qubits and higher encoding efficiency. In this article, we focus on a subclass of GB codes, which are constructed from pairs of binary circulant matrices with two non-zero elements per row. We introduce a family of codes that generalizes both standard and optimized Kitaev codes for which we have a lower bound on their minimum distance, ensuring performance better than standard Kitaev codes. These codes exhibit parameters of the form $ [| 2n , 2, \geq \sqrt{n} |] $ where $ n$ is a factor of $ 1 + d^2 $. For code lengths below 200, our analysis yields $21$ codes, including $7$ codes from Pryadko and Wang's database, and unveils $14$ new codes with enhanced minimum distance compared to standard Kitaev codes. Among these, $3$ surpass all previously known weight-4 GB codes for distances $4$, $8$, and $12$.