离散耦合Hopfield模型的分析
Analysis of the Hopfield Model with Discrete Coupling
摘要：
对高速伊辛计算专用硬件的需求不断增长，这促使人们需要确定精度如何取决于物理资源有限的硬件实现。例如，在诸如现场可编程门阵列的数字硬件中，随着表示耦合强度的位数的减少，可以增加集成伊辛自旋的密度和计算速度，同时计算精度变得更低。为了优化精度-效率的权衡，我们必须根据代表耦合强度的位数来估计伊辛计算机的性能变化。在这项研究中，我们解决这个问题，专注于离散耦合的Hopfield模型。Hopfield模型是一个经典的伊辛计算模型。以往的研究都是用统计力学的方法分析几个非线性函数（如符号）对Hopfield模型的耦合强度的影响，但没有详细分析耦合强度离散化的影响。本文利用复制方法导出了离散耦合Hopfield模型的序参量方程，阐明了表征耦合强度的位数与临界存储容量之间的关系。本文利用Sompolinsky（1986）提出的一般非线性耦合Hopfield模型的复制方法，对具有多比特离散耦合强度的模型进行了分析，并新颖地导出了一般非线性耦合模型的de Almeida-numerless线。
英文摘要：
Growing demand for high-speed Ising-computing-specific hardware has prompted a need for determining how the accuracy depends on a hardware implementation with physically limited resources. For instance, in digital hardware such as field-programmable gate arrays, as the number of bits representing the coupling strength is reduced, the density of integrated Ising spins and the speed of computing can be increased while the calculation accuracy becomes lower. To optimize the accuracy-efficiency trade-off, we have to estimate the change in performance of the Ising computing machine depending on the number of bits representing the coupling strength. In this study, we tackle this issue by focusing on the Hopfield model with discrete coupling. The Hopfield model is a canonical Ising computing model. Previous studies have analyzed the effect of a few nonlinear functions (e.g. sign) for mapping the coupling strength on the Hopfield model with statistical mechanics methods, but not the effect of discretization of the coupling strength in detail. Here, we derived the order parameter equations of the Hopfield model with discrete coupling by using the replica method and clarified the relationship between the number of bits representing the coupling strength and the critical memory capacity. In this paper, we used the replica method for the Hopfield model with general nonlinear coupling (Sompolinsky (1986)) to analyze the model with a multi-bit discrete coupling strength, and we novelly derived the de Almeida-Thouless line of the model with general nonlinear coupling.
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