Turing-Completeness and Undecidability in Coupled Nonlinear Optical Resonators

TL;DR

This paper proves that networks of coupled nonlinear optical resonators are Turing-complete, revealing their immense computational power and the undecidability of certain problems, which advances understanding of ultrafast all-optical computing.

Contribution

It establishes the Turing-completeness of coupled nonlinear optical resonators and identifies the low hardware complexity threshold for this property.

Findings

Coupled nonlinear optical resonators are Turing-complete.

Several problems related to these resonators are formally undecidable.

The minimum hardware complexity for Turing-completeness is surprisingly low.

Abstract

Networks of coupled nonlinear optical resonators have emerged as an important class of systems in ultrafast optical science, enabling richer and more complex nonlinear dynamics compared to their single-resonator or travelling-wave counterparts. In recent years, these coupled nonlinear optical resonators have been applied as application-specific hardware accelerators for computing applications including combinatorial optimization and artificial intelligence. In this work, we rigorously prove a fundamental result showing that coupled nonlinear optical resonators are Turing-complete computers, which endows them with much greater computational power than previously thought. Furthermore, we show that the minimum threshold of hardware complexity needed for Turing-completeness is surprisingly low, which has profound physical consequences. In particular, we show that several problems of interest in the study of coupled nonlinear optical resonators are formally undecidable. These theoretical findings can serve as the foundation for better understanding the promise of next-generation, ultrafast all-optical computers.