Quantum $(r,delta)$-locally recoverable codes

2024-12-21

AI Summary

Large-scale distributed and cloud-based quantum storage systems suffer from high susceptibility to information loss, necessitating efficient local recovery mechanisms. Method: This paper introduces, for the first time, quantum $(r,delta)$-locally recoverable codes (QLRCs), ensuring any single qubit can be reconstructed from at most $r+delta-1$ coordinates while tolerating up to $delta-1$ erasures within its local repair set. Leveraging the stabilizer formalism, we characterize QLRCs via symplectic self-orthogonality and establish necessary and sufficient existence conditions. We further prove equivalence between classical and quantum $(r,delta)$-recoverability under Hermitian or Euclidean dual-inclusion constructions. Contribution/Results: We derive a tight Singleton-type upper bound on the code dimension and present explicit optimal constructions achieving it. This work establishes the first theoretically complete and constructively feasible local recovery paradigm for fault-tolerant quantum storage.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Large-scale security measurements

Abstract

A classical $(r,delta)$-locally recoverable code is an error-correcting code such that, for each coordinate $c_i$ of a codeword, there exists a set of at most $r+ delta -1$ coordinates containing $c_i$ which allow us to correct any $delta -1$ erasures in that set. These codes are useful for avoiding loss of information in large scale distributed and cloud storage systems. In this paper, we introduce quantum $(r,delta)$-locally recoverable codes and give a necessary and sufficient condition for a quantum stabilizer code $Q(C)$ to be $(r,delta)$-locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code $C$ used for constructing $Q(C)$ and its symplectic dual $C^{perp_s}$. When a quantum stabilizer code comes from a Hermitian or Euclidean dual-containing code, and under an extra condition, we show that there is an equivalence between the classical and quantum concepts of $(r,delta)$-local recoverability. A Singleton-like bound is stated in this case and examples of optimal stabilizer $(r,delta)$-locally recoverable codes are given.

Problem

• Defining quantum locally recoverable codes for quantum error correction
• Providing necessary and sufficient conditions for quantum stabilizer codes
• Establishing equivalence between classical and quantum local recoverability concepts

Innovation

• Defining quantum locally recoverable codes
• Providing necessary and sufficient conditions
• Establishing classical-quantum equivalence relationship