On the Generalization of Kitaev Codes as Generalized Bicycle Codes

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 $12$.