热力学矩阵指数与热力学并行性
Thermodynamic Matrix Exponentials and Thermodynamic Parallelism
AI总结：
提出一种利用耦合振荡器电路热平衡对实矩阵求指数的热力学算法，实现与维度成线性关系的渐近加速，并引入热力学并行性概念解释该优势。
摘要：
热力学计算利用物理系统中的涨落和耗散来高效求解各种数学问题。例如，最近研究表明，某些线性代数问题可以通过热力学方式求解，从而带来随矩阵维度扩展的渐近加速。这种“热力学优势”的起源尚未得到完全解释，且尚不清楚还有哪些其他问题可能从中受益。在此，我们提出了一种新的热力学算法用于对实矩阵求指数，并应用于模拟线性动力系统。我们描述了一个包含耦合振荡器的简单电路，其热平衡可以实现我们的算法。我们还表明，该算法也提供了与维度成线性关系的渐近加速。最后，我们引入了热力学并行性的概念来解释这种加速，指出热力学噪声提供了一种资源，能够实现计算的有效并行化，并且我们假设这是一种更广泛地解释热力学优势的机制。
英文摘要：
Thermodynamic computing exploits fluctuations and dissipation in physical systems to efficiently solve various mathematical problems. For example, it was recently shown that certain linear algebra problems can be solved thermodynamically, leading to an asymptotic speedup scaling with the matrix dimension. The origin of this "thermodynamic advantage" has not yet been fully explained, and it is not clear what other problems might benefit from it. Here we provide a new thermodynamic algorithm for exponentiating a real matrix, with applications in simulating linear dynamical systems. We describe a simple electrical circuit involving coupled oscillators, whose thermal equilibration can implement our algorithm. We also show that this algorithm also provides an asymptotic speedup that is linear in the dimension. Finally, we introduce the concept of thermodynamic parallelism to explain this speedup, stating that thermodynamic noise provides a resource leading to effective parallelization of computations, and we hypothesize this as a mechanism to explain thermodynamic advantage more generally.
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