On the Generalization of Kitaev Codes as Generalized Bicycle Codes
🤖 AI Summary
This work addresses the high physical resource overhead and low encoding efficiency in quantum error correction by proposing a novel class of generalized bicycle (GB) codes. Methodologically, it introduces a unified framework based on pairs of binary circulant matrices with exactly two non-zero entries per row/column, systematically generalizing both standard and optimized Kitaev toric codes; this enables the first rigorous construction and analysis of such codes, establishing a strict lower bound of minimum distance ≥ √n. The approach yields 21 GB codes of length < 200, including 14 previously unknown constructions; among these, three new codes achieve distances 4, 8, and 12—surpassing all known weight-4 GB codes and setting new records. Results demonstrate that the proposed GB codes simultaneously attain higher code rates, stronger fault-tolerance capabilities, and reduced physical qubit requirements, thereby significantly improving quantum hardware resource efficiency.
Application Category
📝 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 $12$.
Problem
Research questions and friction points this paper is trying to address.
Generalizing Kitaev codes as efficient Generalized Bicycle codes
Improving minimum distance bounds for better quantum error correction
Discovering new high-performance codes surpassing existing weight-4 GB codes
Innovation
Methods, ideas, or system contributions that make the work stand out.
Generalized Bicycle codes from binary circulant matrices
Lower bound on minimum distance ensures better performance
New codes with enhanced minimum distance unveiled
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