This paper presents a method for achieving equilibrium in the ISING Hamiltonian when confronted with unevenly distributed charges on an irregular grid. Employing (Multi-Edge) QC-LDPC codes and the Boltzmann machine, our approach involves dimensionally expanding the system, substituting charges with circulants, and representing distances through circulant shifts. This results in a systematic mapping of the charge system onto a space, transforming the irregular grid into a uniform configuration, applicable to Torical and Circular Hyperboloid Topologies. The paper covers fundamental definitions and notations related to QC-LDPC Codes, Multi-Edge QC-LDPC codes, and the Boltzmann machine. It explores the marginalization problem in code on the graph probabilistic models for evaluating the partition function, encompassing exact and approximate estimation techniques. Rigorous proof is provided for the attainability of equilibrium states for the Boltzmann machine under Torical and Circular Hyperboloid, paving the way for the application of our methodology. Practical applications of our approach are investigated in Finite Geometry QC-LDPC Codes, specifically in Material Science. The paper further explores its effectiveness in the realm of Natural Language Processing Transformer Deep Neural Networks, examining Generalized Repeat Accumulate Codes, Spatially-Coupled and Cage-Graph QC-LDPC Codes. The versatile and impactful nature of our topology-aware hardware-efficient quasi-cycle codes equilibrium method is showcased across diverse scientific domains without the use of specific section delineations.

翻译：本文提出了一种在不规则网格上电荷分布不均匀时实现ISING哈密顿量平衡的方法。通过采用（多边缘）QC-LDPC码和玻尔兹曼机，我们的方法涉及对系统进行维度扩展，用循环矩阵替代电荷，并通过循环移位表示距离。这实现了将电荷系统系统性地映射到一个空间，将不规则网格转化为均匀构型，适用于环面和圆形双曲面拓扑。论文涵盖了与QC-LDPC码、多边缘QC-LDPC码及玻尔兹曼机相关的基本定义和符号。它探讨了图概率模型编码中的边缘化问题以评估配分函数，包括精确和近似估计技术。严格证明了在环面和圆形双曲面拓扑下玻尔兹曼机平衡态的可达性，为我们的方法论应用铺平了道路。我们的方法在有限几何QC-LDPC码中的实际应用被研究，特别是在材料科学领域。论文进一步探讨了其在自然语言处理Transformer深度神经网络领域的有效性，研究了广义重复累积码、空间耦合和笼图QC-LDPC码。无需特定章节划分，我们展示了这种拓扑感知的硬件高效准循环码平衡方法在多个科学领域的通用性和影响力。