Puncturing Quantum Stabilizer Codes

2024-10-23

arXiv.org

AI Summary

Conventional puncturing techniques for quantum stabilizer codes suffer from limited degrees of freedom and lack systematic approaches for constructing codes with optimal parameters. Method: We propose a generalized puncturing method based on the stabilizer generator matrix, defining puncturing operations directly at the level of row and column manipulations on the matrix and explicitly deriving the resulting stabilizer generators—thereby overcoming the limitation of relying solely on code-space descriptions. This framework enables a natural generalization of the classical Griesmer bound to the quantum setting and facilitates a parameter-optimization search strategy. Results: Our approach constructs multiple new quantum stabilizer codes with parameters (n, k, d) surpassing the current best-known records. It establishes a novel paradigm and practical toolset for the systematic construction of high-performance quantum error-correcting codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Classical coding theory contains several techniques to obtain new codes from other codes, including puncturing and shortening. For quantum codes, a form of puncturing is known, but its description is based on the code space rather than its generators. In this work, we generalize the puncturing procedure to allow more freedom in the choice of which coded states are kept and which are removed. We describe this puncturing by focusing on the stabilizer matrix containing the generators of the code. In this way, we are able to explicitly describe the stabilizer matrix of the punctured code given the stabilizer matrix of the original stabilizer code. The additional freedom in the procedure also opens up new ways to construct new codes from old, and we present several ways to utilize this for the search of codes with good or even optimal parameters. In particular, we use the construction to obtain codes whose parameters exceed the best previously known. Lastly, we generalize the proof of the Griesmer bound from the classical setting to stabilizer codes since the proof relies heavily on the puncturing technique.

Problem

• Generalize puncturing procedure for quantum stabilizer codes
• Describe puncturing via stabilizer matrix of original code
• Extend Griesmer bound proof to quantum stabilizer codes

Innovation

• Generalizes puncturing for quantum stabilizer codes
• Describes puncturing via stabilizer matrix manipulation
• Extends Griesmer bound proof to stabilizer codes