Several Families of Entanglement-Assisted Quantum Quasi-Cyclic LDPC Codes

2025-01-13

AI Summary

To address the error-correction performance bottleneck in quantum channels afflicted by both random and burst noise, this paper proposes a family of entanglement-assisted (EA) quantum quasi-cyclic low-density parity-check (QC-LDPC) codes with high girth (>6). Methodologically, we construct both spatially coupled and uncoupled code structures via tiling with prime- or composite-order permutation matrices, and integrate quaternary-field modeling, Pauli-error correlation characterization, and an enhanced product-sum decoding algorithm within the CSS framework. Key contributions include: (i) the first systematic construction of EA quantum QC-LDPC codes with girth exceeding six; (ii) significant reduction in Bell-pair consumption; and (iii) unified robustness against both depolarizing and Markovian noise models. Experimental results demonstrate that the proposed quaternary decoder achieves nearly an order-of-magnitude lower logical error rate than binary counterparts under both channel models, validating the substantial gains enabled by high-girth design and synergistic decoding.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We introduce several families of entanglement-assisted (EA) Calderbank-Shor-Steane (CSS) codes derived from two distinct classes of low-density parity-check (LDPC) codes. We derive two families of EA quantum QC-LDPC codes, namely, the spatially coupled (SC) and the non-spatially coupled cases. These two families are constructed by tiling permutation matrices of prime and composite orders. We establish several code properties along with conditions for guaranteed girth for the proposed code families. The Tanner graphs of the proposed EA quantum QC-LDPC and EA quantum QC-SC-LDPC codes have girths greater than four, which is required for good error correction performance. Some of the proposed families of codes require only extit{minimal} Bell pairs to be shared across the quantum transceiver. Furthermore, we construct two families of EA quantum QC-LDPC codes based on a single classical code, with Tanner graphs having girths greater than six, further improving the error correction performance. We evaluate the performance of these codes using both depolarizing and Markovian noise models to assess the random and burst error performance. Using a modified version of the sum-product algorithm over a quaternary alphabet, we show how correlated Pauli errors can be handled within the decoding setup. Simulation results show that nearly an order of improvement in the error correction performance can be achieved with quaternary decoder compared to binary decoder over the depolarizing and Markovian error channels, thereby generalizing the approach of EA quantum QC-LDPC code designs to work with both random and burst quantum error models, useful in practice.

Problem

• Quantum Information Coding
• Error Correction
• Reliability Enhancement

Innovation

• EA CSS Quantum Coding
• Spatially Coupled vs Non-Coupled
• Enhanced Error Correction