Elsevier

Neural Networks

Volume 71, November 2015, Pages 11-26
Neural Networks

Complex Rotation Quantum Dynamic Neural Networks (CRQDNN) using Complex Quantum Neuron (CQN): Applications to time series prediction

https://doi.org/10.1016/j.neunet.2015.07.013Get rights and content

Highlights

  • A novel hybrid networks model Complex Rotation Quantum Dynamic Neural Networks (CRQDNN) is proposed.

  • Deep quantum entanglement is realized by incorporating Complex Quantum Neurons (QRN).

  • The embedded IIR filter structure enables the dynamic properties to treat with time series input.

  • The application studies of chaotic time series prediction and electronic prognostics are investigated.

Abstract

Quantum Neural Networks (QNN) models have attracted great attention since it innovates a new neural computing manner based on quantum entanglement. However, the existing QNN models are mainly based on the real quantum operations, and the potential of quantum entanglement is not fully exploited. In this paper, we proposes a novel quantum neuron model called Complex Quantum Neuron (CQN) that realizes a deep quantum entanglement. Also, a novel hybrid networks model Complex Rotation Quantum Dynamic Neural Networks (CRQDNN) is proposed based on Complex Quantum Neuron (CQN). CRQDNN is a three layer model with both CQN and classical neurons. An infinite impulse response (IIR) filter is embedded in the Networks model to enable the memory function to process time series inputs. The Levenberg–Marquardt (LM) algorithm is used for fast parameter learning. The networks model is developed to conduct time series predictions. Two application studies are done in this paper, including the chaotic time series prediction and electronic remaining useful life (RUL) prediction.

Introduction

Time series prediction refers to the study of the present and past behavior of the system for the prediction of the future (Widiputra, Pears, & Kasabov, 2011). Artificial Neural Networks (ANN) are universally employed in time series prediction since they have the properties such as self-organizing, data-driven, self-study, self-adaptive and associated memory (Zhang, Eddy Patuwo, & Y Hu, 1998).

Traditional static ANN models do not incorporate the temporal cumulative effect because a single input sample is either irrelative to time or relative to a moment instead of a period of time. The time-varying effect is supposed to be incorporated in neural networks models (Kim, 1997). To better deal with the time series inputs, from about 1990s, great attention has been paid to the development of Dynamic Neural Networks (DNN) due to their capabilities in modeling nonlinear dynamical systems (Lou and Cui, 2007, Meyer-Bäse et al., 1996).

Recently, Quantum Neural Networks (QNN) models have attracted great attention worldwide because of its novel computational characteristics. Quantum entanglement is involved in the quantum networks modeling, which is responsible for the associations between input and output patterns in the proposed architecture (Svitek, 2008). The quantum entanglement mechanism can be well brought in neural networks modeling that provides more adaptability for parameter learning of the networks. Many researchers confirm that the entanglement mechanism is necessary to realize high-level quantum computing (Jozsa and Linden, 2003, Menneer, 1999), including quantum neural computing (QNC).

In existing QNN models, QNC is generally based on the real assumption, i.e. the quantum probability amplitudes are all real. Notwithstanding, the degree of quantum entanglement is limited if QNC is only defined in the real number domains. An extension from the real number domain to the complex number domain not only complies with more of the real physical world, but also can lead to a higher level of quantum entanglement by combined interaction of the real parts and the imaginary parts. In this way, the data information is encoded into an enhanced level of uncertainty, and the neural networks have a higher flexibility in parameter learning.

To improve the approximation and generalization ability of ANN by utilizing the mechanism of deep quantum entanglement within complex number domain, in this paper, a novel hybrid network model Complex Rotation Quantum Dynamic Neural Networks (CRQDNN) is proposed. The CRQDNN model is established on the definition of a new quantum neuron model based on the newly defined Quantum Complex Rotation Gate (QCRG). Since there are two freedom degrees considering the combination of the real and imaginary parts, a CQN has two output ports with each representing an individual portion of the deep quantum entanglement.

In this paper, the structure of the CRQDNN model is proposed, and the input/output mathematical relationships are derived. We select the Levenberg–Marquardt (LM) networks learning algorithm to ensure a fast convergence and a high probability of finding the global minimum. Two application studies are done in this paper, including the chaotic time series prediction and electronic remaining useful life (RUL) prediction. The experimental results show that CRQDNN is superior to traditional QNN model in the metrics of prediction accuracy.

The remainder of this paper is organized as follows. Section  2 sorts out the related works in previous researches. Section  3 introduces the background knowledge of qubits and quantum gates. Section  4 proposes the novel quantum neuron model CQR. Section  5 proposes the networks structure of CRQDNN with its input/output relationships. Section  6 conducts the application study in chaotic time series prediction. Section  7 conducts the application study in electronic RUL prediction. Finally, Section  8 concludes the whole paper.

Section snippets

DNN

DNN with different time-scales can model the dynamics of the short-term memory of neural activity levels (Lou and Cui, 2007, Meyer-Bäse et al., 1996). Recurrent Neural Networks (RNN) are a basic kind of DNN with feedback paths introducing dynamics into the model. Recurrent Neural Networks (RNN) are a closed loop system, with feedback paths introducing dynamics into the model. Unlike feed-forward neural networks, RNN can use their internal memory to process arbitrary sequences of inputs (Mandic

Qubits and quantum gates

In this section, we introduce the basic knowledge of QC that lays the foundation of the subsequent discussions, including the definition of the qubits and the quantum gates.

Quantum neuron model

The quantum neuron model would be the most important part of the quantum neural networks model. Kouda et al. (2002) and Matsui et al. (2005) propose the single layer quantum perceptron model for the networks formulation, which is widely used as the quantum neuron model in the literature. The basic form of quantum neuron model proposed by Kouda et al. is shown in Fig. 2. This quantum neuron model only processes the real input quantized data |xh (h=1,,n), and utilizes the real quantum rotation

Networks structure of CRQDNN

In this section we propose the CRQDNN structure combining both CQNs and classical neurons. The mathematical relationships in the CRQDNN models are also exploited.

Sunspot time series prediction application

In this section, we present the first application study towards chaotic time series prediction using CRDQNN. The famous Sunspot time series (SIDC, 2015) as the real-world experimental data is used as the experimental data.

Electronic prognostics application

Addressing on the requirement of early fault prediction with high accuracy in EPHM, we focus on the issue of electronic prognostics, and show that the CRQDNN model can fulfill the RUL estimation task. In this section, we will show that CRQDNN has the ability to track the major deterioration trends of the developing fault, and it is also robust towards noise compared with methods based on traditional networks models.

Conclusions

To enhance the approximation and generalization ability of ANN by utilizing the mechanism of deep quantum entanglement within complex number domain, a novel hybrid networks model Complex Rotation Quantum Dynamic Neural Networks (CRQDNN) is proposed in this paper. CRQDNN is a three layer model with a quantum layer which employs the Quantum Complex Rotation Gate (QCRG) and quantum CNOT gate, and a hidden layer which employs the IIR filter as the memory units. Compared with previous real QNN

Acknowledgments

We express sincere appreciation to the editor and reviewers for their efforts to improve this paper. We are very thankful for the help from No. 10 Research Institute, Electronics Technology Group of China. This work is supported by Science and Technology on Reliability and Environmental Engineering Laboratory, Beihang University. This work is also sponsored by fund projects 61316705, 51319040301 and Z132014B002.

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