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.

这篇论文介绍了一种方法，用于在不均匀分布的不规则网格上实现 ISING 哈密顿量的均衡。该方法采用（多边缘）QC-LDPC 码和玻尔兹曼机，通过系统地扩展系统、用环替换电荷，并通过环移表示距离，将电荷系统映射到一个空间中，将不规则网格转化为均匀配置，适用于 Torical 和 Circular Hyperboloid 拓扑结构。该论文还讨论了与 QC-LDPC 码、多边缘 QC-LDPC 码和玻尔兹曼机相关的基本定义和符号，并探讨了用于评估分区函数的图上代码概率模型中的边际化问题，包括精确和近似估计技术。提供了关于 Torical 和 Circular Hyperboloid 下玻尔兹曼机达到平衡态的严格证明，为我们方法的应用铺平了道路。我们的方法在有限几何 QC-LDPC 码中的实际应用进行了研究，特别是在材料科学中。论文进一步探讨了该方法在自然语言处理转换器深度神经网络领域的有效性，检验了广义重复积累码、空间耦合码和笼状图 QC-LDPC 码。我们的拓扑感知硬件高效拟循环码均衡方法的多功能和有影响力的特点在多个科学领域得到了展示，而无需使用特定的部分划分。