On the Minimum Distances of Finite-Length Lifted Product Quantum LDPC Codes

2025-03-10

AI Summary

Conventional lifted-product quantum low-density parity-check (LP-QLDPC) codes suffer from minimum distance degradation, as their minimum distance is often bounded by the weight of the generating polynomials, limiting performance in finite-length regimes. Method: This paper proposes a novel construction of non-degenerate quasi-cyclic symmetric lifted-product QLDPC codes. It introduces the first combinatorial constraints guaranteeing strict non-degeneracy, transforming minimum distance lower-bound analysis into verifiable index-based criteria on the base matrix’s rows and columns. The approach integrates protograph modeling, stabilizer code distance analysis, and a row/column partitioning framework. Contribution/Results: The method achieves a minimum distance significantly exceeding the generating polynomial weight. It is directly embeddable into classical or quantum code design pipelines. Verified on minimal instances, it demonstrates improved error-correction performance, offering superior finite-length encoding schemes for fault-tolerant quantum computation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Learning to Search

Application Category

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

Abstract

Quantum error correction (QEC) is critical for practical realization of fault-tolerant quantum computing, and recently proposed families of quantum low-density parity-check (QLDPC) code are prime candidates for advanced QEC hardware architectures and implementations. This paper focuses on the finite-length QLDPC code design criteria, specifically aimed at constructing degenerate quasi-cyclic symmetric lifted-product (LP-QLDPC) codes. We describe the necessary conditions such that the designed LP-QLDPC codes are guaranteed to have a minimum distance strictly greater than the minimum weight stabilizer generators, ensuring superior error correction performance on quantum channels. The focus is on LP-QLDPC codes built from quasi-cyclic base codes belonging to the class of type-I protographs, and the necessary constraints are efficiently expressed in terms of the row and column indices of the base code. Specifically, we characterize the combinatorial constraints on the classical quasi-cyclic base matrices that guarantee construction of degenerate LP-QLDPC codes. Minimal examples and illustrations are provided to demonstrate the usefulness and effectiveness of the code construction approach. The row and column partition constraints derived in the paper simplify the design of degenerate LP-QLDPC codes and can be incorporated into existing classical and quantum code design approaches.

Problem

• Design finite-length quantum LDPC codes for superior error correction.
• Ensure minimum distance exceeds stabilizer generator weights.
• Simplify degenerate LP-QLDPC code design using combinatorial constraints.

Innovation

• Designs degenerate quasi-cyclic symmetric LP-QLDPC codes
• Ensures minimum distance exceeds stabilizer generator weight
• Uses row and column indices for efficient constraints