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Mapping the XY Hamiltonian onto a Network of Coupled Lasers
arXiv - CS - Emerging Technologies Pub Date : 2020-09-10 , DOI: arxiv-2009.05091
Mostafa Honari-Latifpour, Mohammad-Ali Miri

In recent years there has been a growing interest in the physical implementation of classical spin models through networks of optical oscillators. However, a key missing step in this mapping is to formally prove that the dynamics of such a nonlinear dynamical system is toward minimizing a global cost function which is equivalent with the spin model Hamiltonian. Here, we introduce a minimal dynamical model for a network of dissipatively coupled optical oscillators and prove that the dynamics of such a system is governed by a Lyapunov function that serves as a cost function for the system. This cost function is in general a function of both phases and intensities of the oscillators and depends strongly on the pump parameter. In case of bipartite network topologies, the amplitudes of the oscillators become identical in the steady state and the cost function reduces to the XY Hamiltonian. In the general case for non-trivial network topologies, however, the cost function approaches the XY Hamiltonian only in the strong pump limit. We show that by adiabatically tuning the pump parameter, the network can largely avoid trapping into the local minima of the governing cost function and stabilize into the ground state of the associated XY Hamiltonian. These results show the great potential of laser networks for unconventional computing.

中文翻译:

将 XY 哈密顿量映射到耦合激光网络

近年来,人们对通过光学振荡器网络物理实现经典自旋模型越来越感兴趣。然而,这个映射中缺少的一个关键步骤是正式证明这种非线性动力系统的动力学是为了最小化与自旋模型哈密顿量等效的全局成本函数。在这里,我们为耗散耦合光学振荡器网络引入了一个最小动力学模型,并证明了这种系统的动力学是由作为系统成本函数的李雅普诺夫函数控制的。该成本函数通常是振荡器的相位和强度的函数,并且强烈依赖于泵参数。在双向网络拓扑的情况下,振荡器的振幅在稳态下变得相同,成本函数减少到 XY 哈密顿量。然而,在非平凡网络拓扑的一般情况下,成本函数仅在强泵限制中接近 XY 哈密顿量。我们表明,通过绝热调整泵参数,网络可以在很大程度上避免陷入控制成本函数的局部最小值并稳定到相关 XY 哈密顿量的基态。这些结果显示了激光网络在非常规计算方面的巨大潜力。网络可以在很大程度上避免陷入控制成本函数的局部最小值并稳定到相关的 XY 哈密顿量的基态。这些结果显示了激光网络在非常规计算方面的巨大潜力。网络可以在很大程度上避免陷入控制成本函数的局部最小值并稳定到相关的 XY 哈密顿量的基态。这些结果显示了激光网络在非常规计算方面的巨大潜力。
更新日期:2020-09-14
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