Quantum Error-Correcting Codes Based on Orthogonal Arrays.

Area of Science:

• Quantum Information Science
• Coding Theory
• Discrete Mathematics

Background:

• Quantum error-correcting codes (QECCs) are crucial for protecting quantum information from noise.
• Existing methods for constructing QECCs often provide existence results rather than explicit constructions.
• The relationship between specific types of QECCs and orthogonal arrays is known but limited.

Purpose of the Study:

• To establish a generalized relationship between quantum error-correcting codes and orthogonal arrays with orthogonal partitions.
• To utilize this relationship for the constructive development of pure quantum error-correcting codes.
• To demonstrate the construction of infinite families of optimal quantum codes.

Main Methods:

• Employing the Hamming distance to connect quantum error-correcting codes ((N,K,d+1))s with orthogonal arrays featuring orthogonal partitions.
• Generalizing the known connection between ((N,1,d+1))s QECCs and irredundant orthogonal arrays.
• Applying the established relation for the explicit construction of pure quantum error-correcting codes.

Main Results:

• Numerous infinite families of optimal pure quantum error-correcting codes are explicitly constructed.
• Examples include codes like ((3,s,2))s, ((4,s2,2))s, and up to ((12,s2,6))s, where parameters involve prime powers.
• The method provides constructive results, not just existence proofs, yielding pure codes with simpler basis states.

Conclusions:

• The developed method offers a powerful and constructive approach to designing optimal pure quantum error-correcting codes.
• The findings extend the known connections between coding theory and quantum error correction.
• The methodology is adaptable for constructing quantum error-correcting codes over mixed alphabets.