拓扑场论与瞬子计算
Topological field theory and computing with instantons
摘要：
Chern-Simons拓扑场论TFT是唯一已经在描述某些奇异的强关联电子系统和拓扑量子计算的相应概念中找到应用的TFT。在这里，我们表明，另一种类型的TFT，特别是规范场的无威滕型TFT被称为拓扑西格玛模型，描述了最近提出的数字存储器计算机（DMMs）工程的动态系统与点吸引子是解决特定任务的相应的逻辑电路的解决方案。这一结果源于最近的发现，任何随机微分方程具有拓扑超对称性，并实现了解决方案的搜索DMM收益通过瞬相。DMM中的某些TFT量子化器则揭示了在空间和时间上都存在一种瞬态的长程序，这与瞬子对拓扑超对称性的有效破坏有关。随之而来的非局域性和低维的瞬子是物理原因，为什么DMM可以有效地解决复杂的问题，尽管他们的非量子特性。我们用素分解的解推广了这些结果。
英文摘要：
Chern-Simons topological field theories TFTs are the only TFTs that have already found application in the description of some exotic strongly-correlated electron systems and the corresponding concept of topological quantum computing. Here, we show that TFTs of another type, specifically the gauge-field-less Witten-type TFTs known as topological sigma models, describe the recently proposed digital memcomputing machines (DMMs) - engineered dynamical systems with point attractors being the solutions of the corresponding logic circuit that solves a specific task. This result derives from the recent finding that any stochastic differential equation possesses a topological supersymmetry, and the realization that the solution search by a DMM proceeds via an instantonic phase. Certain TFT correlators in DMMs then reveal the presence of a transient long-range order both in space and time, associated with the effective breakdown of the topological supersymmetry by instantons. The ensuing non-locality and the low dimensionality of instantons are the physical reasons why DMMs can solve complex problems efficiently, despite their non-quantum character. We exemplify these results with the solution of prime factorization.
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