Nontrivial Attractors of the Perturbed Nonlinear Schrödinger Equation: Applications to Associative Memory and Pattern Recognition

Abstract

Computers are dynamical systems that carry out information processing through their change of state with time. For instance, neural networks such as Hopfield's associative memory are dissipative dynamical systems in a finite dimensional configuration space with attractors that represent stored patterns. In particular, dissipative dynamical systems with an infinite dimensional configuration space are of broad interest and have the possibility to store and restore complex and strongly distorted data structures. In this work, the nonlinear Schrödinger equation (NLSE) with a dissipative perturbation which creates a frictional force acting on soliton is considered. It is shown that the control of the perturbative term allows one to decrease the velocity of soliton to zero and conserve a positive value of its amplitude. The existence of such a frictional force would suggest that the perturbation makes the zero‐velocity solitonic solution of non‐zero amplitude into an attractor for all evolution trajectories whose initial conditions are moving solitons. This allows a way to store information in the infinite dimensional dynamical system governed by NLSE using principles which are completely analogous to that of Hopfield's associative memory. This approach can be realized with solitons in Bose–Einstein condensates and nonlinear optical systems.

Solitonic Fixed Point Attractors in the Complex Ginzburg–Landau Equation for Associative Memories

“…It is shown that the control of dissipative perturbation allows us to handle the attractor of system similarly to Ref. [34], such that it is possible to store and process information. This approach can be realized with solitons in Bose-Einstein condensates and nonlinear optical systems.…”

“…In this paper, we consider the complex Ginzburg-Landau equation (CGLE) [35,36] and show that it has similar properties as seen in Ref. [34] that can be exploited towards associative memory and pattern recognition. The CGLE is of interest since it is a model of experimentally accessible systems such as nonlinear optics, which can form the basis of experimental realization of the general approach.…”

“…The use of BECs to solve classical optimization problems [21] and perform quantum algorithms [22][23][24][25][26][27][28][29] have also been investigated. Recently, the nonlinear Schrodinger equation (NLSE), which can be realized in BEC, with a simplified dissipative perturbation which creates a frictional force acting on soliton [30][31][32][33] was considered in an application to associative memory and pattern recognition [34]. It was shown that the control of the perturbative term allows one to decrease the velocity of soliton to zero and conserve a positive value of its amplitude.…”

Quantum Machine Learning and Bioinspired Quantum Technologies

“…Among the collection of articles, we have works more oriented to quantum machine learning, including ‐The use of the first quantum Ansätze for the statistical relational learning on knowledge graphs using parametric quantum circuits, by Yunpu Ma et al ‐The analysis of the nonlinear Schrödinger equation to store information analogously as with a Hopfield's associative memory, by Alexey N. Pyrkov and co‐workers ‐Experimentally carrying out a quantum autoencoder based on quantum adders with the Rigetti cloud quantum computer, by Yongcheng Ding et al …”

“…Among the collection of articles, we have works more oriented to quantum machine learning, including -The use of the first quantum Ansätze for the statistical relational learning on knowledge graphs using parametric quantum circuits, by Yunpu Ma et al [1] -The analysis of the nonlinear Schrödinger equation to store information analogously as with a Hopfield's associative memory, by Alexey N. Pyrkov and co-workers. [2] -Experimentally carrying out a quantum autoencoder based on quantum adders with the Rigetti cloud quantum computer, by Yongcheng Ding et al [3] -A quantum experiment to reconstruct an unknown photonic quantum state with a limited amount of copies, in the context of reinforcement learning, by Shang Yu et al [4] -A prediction of the band gap which represents one of the basic properties of a crystalline material via machine learning calculations, by Alexander V. Balatsky and co-workers. [5] -A Review Article on the progress in using artificial neural networks to build quantum many-body states, by Zhih-Ahn Jia et al [6] In the quantum biomimetic field, we have -Contributions related to quantum synchronization via quantum machine learning, in the parallel works by Francisco A. Cárdenas-López et al [7] and Gabriel Garau Estarellas et al [8] -An article motivating the debate on possible quantum effects in conscious minds, by Göran Wendin.…”