Elsevier

Neurocomputing

Volume 246, 12 July 2017, Pages 3-11
Neurocomputing

Chaos in a quantum neuron: An open system approach

https://doi.org/10.1016/j.neucom.2016.06.081Get rights and content

Abstract

Researches in natural neuron dynamics have shown that phase transition and chaos provide optimal behaviour for information processing. In artificial neural models that behavior is expected also to maximize the neuron learnability. By employing an open quantum systems approach, we propose a new quantum information extraction method in the quantum RAM node dynamics where complex values are iterated. We experimentally show bifurcation and chaos emergence by varying their parameters.

Introduction

Obtaining the analytical description of a given system does not necessarily imply understanding its parametric dependency, its temporal behavior, the existence chaos and fractality, bifurcation and other dynamics characteristics [1]. The dynamics of natural (chemical, biological, physics) or artificial (artificial neural networks, computer networks, data centers) systems need to be understood quantitatively and qualitatively for them to be useful. A dynamics that is well understood can be used to produce an expected behavior, e.g. either serving as input signal generator or working as specific task module in a complex system. For example, text encryption system may involve a chaotic module [2]. Fractals can help either to generate or to segment textures [3], [4]. Biological neurons can learn fast in chaotic regime, bifurcation or phase change [5], [6], [7].

Biological neurons have received the attention of dynamical systems researchers since the brain is a self-feedback system. The studies about the parameters, bifurcations and phase change of the neurons have helped scientists to understand how information processing can be maximized in the brain activities [8], [9], [10], [11]. In the artificial world, there is further evidence pointing relationships amongst nonlinearity, chaotic environments and artificial neuron networks learnability [10], [12]. It is argued that neuron populations are predisposed to instability and bifurcation that depend on external input and internal parameters [13], [14].

Previous studies in the machine learning have dealt with simulation and detecting specific behaviors in neural models [15], [16], [17], [18], [19], [20], [21]. As biological neurons have shown propensity to learn more efficiently under certain dynamics conditions, those kinds of studies have increased. In this work, we analyze the dynamics of a quantum artificial weightless neuron model named qRAM in [22].

Existing quantum artificial neurons in the literature come basically in two guises [23], [24], [25], [26]. Some are just inspired in quantum mechanics, violating a principle or another. Others are actual quantum models. For a discussion see [27]. Quantum neurons have been used to solve pattern recognition problems and machine learning tasks [25], [28], [29], [30].

Parameters need to be adjusted in the qRAM model and they influence the neuron characteristics. The parameter choice can induce certain set of behaviors in qRAM. In this work we define an extraction method that searches for a chaotic qRAM parametric setting. Classically there are evidences that neural models work at maximum efficiency in a chaotic configuration, for specific tasks in machine learning and pattern recognition [8], [9], [10], [11], [13], [14].

In [31], we show that the neuron operation may entangle the qubits representing the neuron input, output and parameters. The neuron entangled configuration prohibits access to the qRAM output result separating from the other qubits. The output can be estimated by the real extraction method (REM) proposed in [31]. The estimated output is fed back in the input. The REM estimates its qubits amplitudes norms instead of its complex values amplitudes.

Other works investigate chaos in quantum systems but the states are in a predetermined format to increase their entanglement [32], [33], [34]. The REM experiments have shown that restricting to real values the qRAM system dynamics is not chaotic.

Based on REM and the open quantum systems approach [35], this paper presents a new quantum qubit extraction method that produces complex values which we name quadratic extraction method (QEM). We use QEM to find a qRAM configuration that generates chaos and bifurcations in the neurons dynamics. The advantage of QEM over REM is the capacity to generate a qRAM chaotic configuration and this can be visualized through an orbit diagram. The system parameters are the neuron selectors. The system initial condition is the initial neuron input. The qRAM chaotic configuration found in this paper should help artificial neural computing designers to fit neurons parameters to provoke specific chaotic behavior or oscillatory dynamics. The experimental results show a qualitatively changing of dynamical critical points during its parameter variation.

This paper is organized as follows. Section 2 presents basic concepts of quantum computing. Section 3 describes the working of quantum weightless neuron nodes. Section 4 explains the REM dynamics and in Section 5 the proposed method QEM dynamics is presented. Section 6 shows the experiments of the QEM dynamics chaotic configuration where it is possible to see orbit diagrams showing bifurcations and chaos. The conclusion is presented in Section 7.

Section snippets

Quantum computing

A quantum bit, qubit, is a complex bidimensional unit vector. In spite of fact that 0 and 1 bits can be represented by any orthogonal base of C2, the mostly used one is the canonical (or computational) basis defined as the pair of vectors |0=[1,0]T and |1=[0,1]T. A qubit |ψ⟩ can be written as shown in Eq. 1, where α and β are complex numbers and |α|2+|β|2=1. |ψ=α|0+β|1

In quantum computing it is not possible to make a copy of an unknown state [36]. Composite quantum systems are formed using

Quantum weightless networks

The quantum RAM based neuron qRAM [43] was defined as the quantization of the RAM node which is the neural unit of the weightless neural networks first proposed in [44] and reviewed in [45].

In its simplest form a RAM node stores in its memory one bit addressed by an input bit string. The corresponding qRAM represents the stored bit with selectors. To simulate the changing of the stored value in the RAM memory one changes the selector values. In spite of the simplicity of the RAM-based nodes,

Real extraction method

For iterating a system, we need to extract the system output and feed it back in the system input. An extraction method of the output qubit was proposed in [31], named REM, which extracts the amplitudes norms to build the output qubit. For completeness the REM method is explained in this section. In the next section, another qRAM dynamics method named QEM is proposed considering the summation of the squared amplitudes to build the output qubit. QEM generates complex numbers during its iteration

Quadratic extraction method

The real extraction method restricts the dynamics to deal with only real values, since only the module of the amplitude values are used. Here, the REM method is extended to recover the value of the output register through the summation of the squared amplitudes and not the summation of squared module of the amplitudes. With only the module of the squared amplitudes values, the dynamics is not chaotic and has no bifurcation in the orbit diagram. We know that non-linearity is a necessary

Experiments

In this section, experiments where chaos emerges are reported when the QEM dynamics are employed. That was not yet found in the REM dynamics. The orbit diagrams for the QEM dynamics exhibits chaos. Three steps of QEM dynamics are investigated: (a) first, the amplitudes are squared through the circuit as explained in Section 5; (b) after that, the qubit is filtered by the Of operator in such away that the output qubit amplitudes are passed to build |0⟩ and |1⟩ states; (c) the extracted output

Conclusion

In this work, we present a new quantum qubit extraction method using the open quantum systems approach [35], which we named QEM, to evaluate a qRAM dynamics. QEM allows complex values iterations and a qRAM chaotic dynamics configuration is encountered in its orbit diagram, unlike the REM method. The circuit proposed in this work can be used to fit parameters during the neuron learning procedure with guarantee that chaos and bifurcation are present in that method.

The proposed QEM is presented in

Acknowledgments

This work is supported by research grants from CNPq under grant numbers 302526/2011-0, 442668/2014-7 and 484754/2012-2, by CAPES under grant number 1022/2014 and by FACEPE under grant number APQ-1188-1.03/10. CNPq, CAPES and FACEPE are Brazilian research agencies.

Fernando M. de Paula Neto received the Bachelor degree cum laude (2014) in Computer Engineering and the Master degree (2016) in Computer Science at Universidade Federal de Pernambuco (Brazil). He is currently working toward the Ph.D. degree in Computer Science at the Universidade Federal de Pernambuco, Brazil. His current research interests include Quantum Computing, Dynamical Systems and Chaos, Neural Networks, Hybrid Intelligent Systems and Machine Learning.

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    Fernando M. de Paula Neto received the Bachelor degree cum laude (2014) in Computer Engineering and the Master degree (2016) in Computer Science at Universidade Federal de Pernambuco (Brazil). He is currently working toward the Ph.D. degree in Computer Science at the Universidade Federal de Pernambuco, Brazil. His current research interests include Quantum Computing, Dynamical Systems and Chaos, Neural Networks, Hybrid Intelligent Systems and Machine Learning.

    Wilson R. de Oliveira has completed his B.Sc degree (1982), M.Sc. degree (1985) and Ph.D. degree (2004) in Computer Science at Universidade Federal de Pernambuco (Brazil) specialising in Computing Theory of Artificial Neural Networks and Automata Theory. He was on sabbatical leave at the School of Computer Science, University of Birmingham, UK (2008) working on Matroid Theory and Discrete Differential Geometry. He has been working recently on Quantum Weightless Neural Networks, Combinatorial Hopf Algebras, Matroid Representations and Non-linearity and Chaos in Physical Systems. He has published over 50 articles in scientific journals and conferences. He joined the “Departmento de Estatística e Informàtica” at the “Universidade Federal Rural de Pernambuco” in 2000 where he is now an Associate Professor.

    Teresa Ludermir received the Ph.D. degree in Artificial Neural Networks in 1990 from Imperial College, University of London, UK. From 1991 to 1992, she was a lecturer at Kings College London. She joined the Center of Informatics at Federal University of Pernambuco, Brazil, in September 1992, where she is currently a Professor and head of the Computational Intelligence Group. She has published over 300 articles in scientific journals and conferences, three books in Neural Networks and organized two of the Brazilian Symposium on Neural Networks. Her research interests include weightless Neural Networks, hybrid neural systems and applications of Neural Networks.

    Adenilton J. da Silva received the Licenciate degree (cum laude) in mathematics from Universidade Federal Rural de Pernambuco, Brazil and received the Ph.D. degree in Computer Science from Universidade Federal de Pernambuco, Brazil. He joined the Departamento de Estatística e Informàtica at the Universidade Federal Rural de Pernambuco in 2014 where is now Assistant Professor. His current research interests include quantum computation, artificial neural networks, machine learning and hybrid neural systems.

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