Quantum error codes with strong local recovery and larger distances

Plotkin-optimal pure quantum LRCs with large minimum distances from NMDS codes

Information Theory

Summary

Quantum computers need special codes to fix errors so that information isn't lost or corrupted. This paper creates new types of quantum error-correcting codes that let you fix mistakes locally by looking at just a few parts, while still being very good at fixing big errors overall. The authors build these codes using advanced mathematical objects called NMDS and ESGRS codes and solve tricky problems to figure out how these codes behave. Their codes reach a theoretical limit called the Plotkin bound, which means they're as good as possible in certain ways. These new codes offer better options for protecting quantum information with both local and strong overall protections.

What this means in practice

  • For quantum hardware engineers: Protect quantum memory by using codes that quickly recover local errors while maintaining strong overall error correction.
  • For quantum communication teams: Implement error correction in quantum networks that can handle large-scale errors without sacrificing local repair efficiency.

Authors

Yang Li, Shitao Li, Gaojun Luo, San Ling

Abstract

Quantum locally recoverable codes (qLRCs) are a class of quantum error-correcting codes that can recover an erased qudit by accessing only a few other qudits. This paper develops a general framework for constructing optimal pure qLRCs that attain the Plotkin-like bound, referred to as Plotkin-optimal pure qLRCs, from Hermitian dual-containing near maximum-distance separable (NMDS) codes. Focusing on a special class of NMDS codes called extended subcodes of generalized Reed-Solomon (ESGRS) codes, we determine their localities by analyzing certain subset sum problems over finite fields. By further exploiting the algebraic structures of ESGRS codes and related codes, we construct four families of Plotkin-optimal pure qLRCs with large lengths and more flexible minimum distances than existing constructions. These constructions provide strong global error-correction capability while retaining local recovery. To the best of our knowledge, they give the first families of qLRCs attaining the pure Plotkin-like bound. In addition, by using the resulting locality property, we resolve in the negative a related problem on ESGRS codes supporting combinatorial designs.
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