Quantum-Inspired Neural Network with Sequence Input

To enhance the approximation and generalization ability of artificial neural network (ANN) by employing the principles of quantum rotation gate and controlled-not gate, a quantum-inspired neuron with sequence input is proposed. In the proposed model, the discrete sequence input is represented by the qubits, which, as the control qubits of the controlled-not gate after being rotated by the quantum rotation gates, control the target qubit for reverse. The model output is described by the probability amplitude of state 1 in the target qubit. Then a quantum-inspired neural network with sequence input (QNNSI) is designed by employing the sequence input-based quantum-inspired neurons to the hidden layer and the classical neurons to the output layer, and a learning algorithm is derived by employing the Levenberg-Marquardt algorithm. Simulation results of benchmark problem show that, under a certain condition, the QNNSI is obviously superior to the ANN.


Introduction
Many neuro-physiological experiments indicate that the information processing character of the biological nerve system mainly includes the following eight aspects: the spatial aggregation, the multi-factor aggregation, the temporal cumulative effect, the activation threshold characteristic, self-adaptability, exciting and restraining characteristics, delay characteristics, conduction and output characteristics [1].From the definition of the M-P Z. Y. Li, P. C. Li neuron model, classical ANN preferably simulates voluminous biological neurons' characteristics such as the spatial weight aggregation, self-adaptability, conduction and output, but it does not fully incorporate temporal cumulative effect because the outputs of ANN depend only on the inputs at the moment regardless of the prior moment.In the process of practical information processing, the memory and output of the biological neuron not only depend on the spatial aggregation of input information, but also are related to the temporal cumulative effect.Although the ANN in Refs.[2]- [5] can process temporal sequences and simulate delay characteristics of biological neurons, in these models, the temporal cumulative effect has not been fully reflected.Traditional ANN can only simulate point-to-point mapping between the input space and output space.A single sample can be described as a vector in the input space and output space.However, the temporal cumulative effect denotes that multiple points in the input space are mapped to a point in the output space.A single input sample can be described as a matrix in the input space, and a single output sample is still described as a vector in the output space.In this case, we claim that the network has a sequence input.
Since Kak firstly proposed the concept of quantum-inspired neural computation [6] in 1995, quantum neural network (QNN) has attracted great attention by the international scholars during the past decade, and a large number of novel techniques have been studied for quantum computation and neural network.For example, Ref. [7] proposed the model of quantum neural network with multilevel hidden neurons based on the superposition of quantum states in the quantum theory.In Ref. [8], an attempt was made to reconcile the linear reversible structure of quantum evolution with nonlinear irreversible dynamics of neural network.Ref. [9] presented a novel learning model with qubit neuron according to quantum circuit for XOR problem and described the influence to learning by reducing the number of neurons.In Ref. [10], a new mathematical model of quantum neural network was defined, building on Deutsch's model of quantum computational network, which provides an approach for building scalable parallel computers.Ref. [11] proposed the neural network with the quantum gated nodes, and indicated that such quantum network may contain more advantageous features from the biological systems than the regular electronic devices.Ref. [12] proposed a neural network model with quantum gated nodes and a smart algorithm for it, which shows superior performance in comparison with a standard error back propagation network.Ref. [13] proposed a weightless model based on quantum circuit.It is not only quantum-inspired but also actually a quantum NN.This model is based on Grover's search algorithm, and it can both perform quantum learning and simulate the classical models.However, from all the above QNN models, like M-P neurons, it also does not fully incorporate temporal cumulative effect because a single input sample is either irrelative to time or relative to a moment instead of a period of time.
In this paper, in order to fully simulate biological neuronal information processing mechanisms and to enhance the approximation and generalization ability of ANN, we proposed a quantum-inspired neural network model with sequence input, called QNNSI.It's worth pointing out that an important issue is how to define, configure and optimize artificial neural networks.Refs.[14] [15] make deep research into this question.After repeated experiments, we opt to use a three-layer model with a hidden layer, which employs the Levenberg-Marquardt algorithm for learning.Under the premise of considering approximation ability and computational efficiency, this option is a relatively ideal.The proposed approach is utilized to the time series prediction for Mackey-Glass, and the experimental results indicate that, under a certain condition, the QNNSI is obviously superior to the common ANN.

Qubit
In the quantum computers, the "qubit" has been introduced as the counterpart of the "bit" in the conventional computers to describe the states of the circuit of quantum computation.The two quantum physical states labeled as 0 and 1 express 1 bit information, in which 0 corresponds to the bit 0 of classical computers, while 1 bit 1. Notation of " " is called the Dirac notation, which is the standard notation for the states in the quan- tum mechanics.The difference between bits and qubits is that a qubit can be in a state other than 0 and 1 .
It is also possible to form the linear combinations of the states, namely superpositions: An n qubits system has 2 n computational basis states.For example, a 2 qubit system has basis 00 , 01 , 10 , 11 .Similar to the case of a single qubit, the n qubits system may form the superposition of 2 n .
where x a is called probability amplitude of the basis states x , and { } 0,1 n means the set of strings of length two with each letter being either zero or one.The condition that these probabilities can sum to one is expressed by the normalization condition.

Quantum Rotation Gate
In quantum computation, the logic function can be realized by applying a series of unitary transform to the qubit states, which the effect of the unitary transform is equal to that of the logic gate.Therefore, the quantum services with the logic transformations in a certain interval are called the quantum gates, which are the basis of performing quantum computation.
The definition of a single qubit rotation gate is written as The NOT gate is defined by its truth table, in which 0 1 → and 1 0 → , that is, the 0 and 1 states are interchanged.In fact, the quantum NOT gate acts linearly, that is, it takes the state 0 1 α β + to the corresponding state in which the role of 0 and 1 have been interchanged, 1 0 α β + .There is a convenient way of representing the quantum NOT gate in matrix form, which follows directly from the linearity of quantum gates.Suppose we define a matrix X to represent the quantum NOT gate as follows The notation X for the quantum NOT is used for historical reasons.If the quantum state 0 and 1 is written in a vector notation as [ ] T α β , then the corresponding output from quantum NOT gate is

Multi-Qubits Controlled-Not Gate
In a true quantum system, a single qubit state is often affected by a joint control of multi-qubits.A multi-qubits controlled-not gate ( ) n C X is a kind of control model.The multi-qubits system is also described by the wave function 1 2 n x x x  .In an (n + 1)-bits quantum system, when the target bit is simultaneously controlled by n input bits, the dynamic behavior of the system can be described by multi-qubits controlled-not gate in Figure 1.
In Figure 1(a), suppose we have n + 1 qubits, and then we define the controlled operation is the target qubit.From Equation ( 6), the output of It is observed from Equation ( 7) that the output of ( ) n C X is in the entangled state of n + 1 qubits, and the probability of the target qubit state ϕ , in which 1 is observed, equals to ( ) ( ) In Figure 1(b), the operator X is applied to last a qubit if the first n qubits are all equal to zero, and otherwise, nothing is done.The controlled operation ( ) n C X can be defined by the equation ( ) The probability of the target qubit state ϕ , in which 1 is observed, equals to ( ) ( ) At this time, after the joint control of the n input bits, the target bit φ′ can be defined as follows

Quantum-Inspired Neuron Model
In this section, we first propose a quantum-inspired neuron model with sequence input, as shown in Figure 2.This model consists of quantum rotation gates and multi-qubits controlled-not gate.The defined in time domain interval [0, T] denote the input sequences.The output is the probability amplitude of the target state in 1 .The control parameters are the rotation angles ( ) . In this paper, we define the output of the quantum neuron as the probability amplitude of the corresponding state, in which 1 is observed.According to the definition of quantum rotation gate and multi-qubits controlled-not gate, the output of quantum neuron can be written as where ( ) ( ) ( )

Quantum-Inspired Neural Network Model
In this paper, the QNNSI model is shown in Figure 3, where the hidden layer consists of quantum-inspired neurons with sequence input, and the output layer consists of classical neurons.The x t denote the input sequences, the 1 2 , , , p h h h  denote the hidden output, the jk ω denotes the connection weights in output layer, and the 1 2 , , , m y y y  denote the network output.The Sigmoid function is used in output layer.Unlike ANN, each input sample of QNNSI is described as a matrix instead of a vector.For example, the l-th sample can be written as ( ) t , the spatial and temporal aggregation results of the j-th quantum-inspired neuron in hidden layer can be written as ( ) The j-th output in hidden layer (namely, the spatial and temporal aggregation results in [0, T]) is given by ( ) The k-th output in output layer can be written as

Pretreatment of Input and Output Samples
Suppose the l-th sample in n-dimensional input space ) These samples can be converted into the quantum states as follows , , , ) then, these output samples can be normalized by the following equation

QNNSI Parameters Adjustment
In QNNSI, the adjustable parameters include the rotation angles of quantum rotation gates in hidden layer, and the weights in output layer.Suppose max max max max According to the gradient descent algorithm in Ref. [16], the gradient of the rotation angles of the quantum rotation gates can be calculated as follows The gradient of the connection weights in output layer can be calculated as follows ( ) ( ) Because gradient calculation is more complicated, the standard gradient descent algorithm is not easy to converge.Hence we employ the Levenberg-Marquardt algorithm in Ref. [16] to adjust the QNNSI parameters.
Let P denote the parameter vector, e denote the error vector, and J denote the Jacobian matrix.p, e, and J are respectively defined as follows ( ) ( ) ( ) According to Levenberg-Marquardt algorithm, the QNNSI iterative equation is written as follows

Stopping Criterion of QNNSI
If the value of the evaluation function E reaches the predefined precision within the preset maximum of iterative steps, then the execution of the algorithm is stopped, else the algorithm is not stopped until it reaches the predefined maximum of iterative steps.

Simulations
To examine the effectiveness of the proposed QNNSI, the time series prediction for Mackey-Glass is used to compare it with the ANN with a hidden layer in this section.In this experiment, we implement and investigate the QNNSI in Matlab (Version 7.1.0.246) on a Windows PC with 2.19 GHz CPU and 1.00 GB RAM.Our QNNSI has the same structure and parameters as the ANN in the simulations, and the same Levenberg-Marquardt algorithm [16] is applied in two models.Mackey-Glass time series can be generated by the following iterative equation [17] ( ) ( ) ( ) ( ) ( ) where t and τ are integers, 0.2 a = , 0.3 b = , 17 τ = , and ( ) ( ) From the above equation, we may obtain the time sequence We take the first 800 as the training set, and the remaining 200 as the testing set.Our prediction schemes is to employ n data adjacent to each other to predict the next one data.Namely, in our model, the sequence length equals to n.Therefore, each sample con-sists of n input values and an output value.
Hence, there is only one output node in QNNSI and ANN.In order to fully compare the approximation ability of two models, the number of hidden nodes are respectively set to 10, 11, , 40  .The predefined precision is set to 0.05, and the maximum of iterative steps is set to 100.The QNNSI rotation angles in hidden layer are initialized to random numbers in ( ) , and the connection weights in output layer are initialized to random numbers in (−1, 1).For ANN, all weights are initialized to random numbers in (−1, 1), and the Sigmoid functions are used as the activation functions in hidden layer and output layer.
Obviously, ANN has n input nodes, and an ANN's input sample can be described as a n-dimensional vector.For the number of input nodes of QNNSI, we employ the following nine kinds of settings shown in Table 1.For each of these settings in Table 1, a single QNNSI input sample can be described as a matrix.
It is worth noting that, in QNNSI, an n q × matrix can be used to describe a single sequence sample.In general, ANN cannot deal directly with a single n q × sequence sample.In ANN, an n q × matrix is usually regarded as nq dimensional vector samples.For fair comparison, in ANN, we have expressed the n q × sequence samples into the nq dimensional vector samples.Therefore, in Table 1, the sequence lengths for ANN are not changed.It is clear that, in fact, there is only one kind of ANN in Table 1, namely, ANN36.
Our experiment scheme is that, for each kind of combination of input nodes and hidden nodes, one ANN and nine QNNSIs are respectively run 10 times.Then we use four indicators, such as average approximation error, average iterative steps, average running time, and convergence ratio, to compare QNNSI with ANN.Training result contrasts are shown in Tables 2-5, where QNNSIn_q denotes QNNSI with n input nodes and q sequence length.
From Tables 2-5, we can see that when the input nodes take 6, 9, and 12, the performance of QNNSIs are   obviously superior to that of ANN, and the QNNSIs have better stability than ANN when the number of hidden nodes changes.Next, we investigate the generalization ability of QNNSI.Based on the above experimental results, we only investigate three QNNSIs (QNNSI6_6, QNNSI9_4, and QNNSI12_3).Our experiment scheme is that three QNNSIs and one ANN train 10 times on the training set, and the generalization ability is immediately investigated on the testing set after each training.The average results of the 10 tests are regarded as the evaluation indexes.For convenience of description, let avg E denote the average of the maximum prediction error, mean E denote the average of the prediction error mean, and var E denote the average of prediction error variance.Taking 30 hidden nodes for example, the evaluation indexes contrast of QNNSIs and ANN are shown in results show that the generalization ability of three QNNSIs is obviously superior to that of corresponding ANN.
These experimental results can be explained as follows.For processing of input information, QNNSI and ANN take different approaches.QNNSI directly receives a discrete input sequence.In QNNSI, using quantum information processing mechanism, the input is circularly mapped to the output of quantum controlled-not gates in hidden layer.As the controlled-not gate's output is in the entangled state of multi-qubits, therefore, this mapping is highly nonlinear, which make QNNSI have the stronger approximation ability.In addition, QNNSI's each input sample can be described as a matrix with n rows and q columns.It is clear from QNNSI's algorithm that, for the different combination of n and q, the output of quantum-inspired neuron in hidden layer is also different.In fact, The number of discrete points q denotes the depth of pattern memory, and the number of input nodes n denotes the breadth of pattern memory.When the depth and the breadth are appropriately matched, the QNNSI shows excellent performance.For the ANN, because its input can only be described as a nq-dimensional vector, it is not directly deal with a discrete input sequence.Namely, it only can obtain the sample characteristics by way of breadth instead of depth.Hence, in the ANN information processing, there exists inevitably the loss of sample characteristics, which affects its approximation and generalization ability.
It is worth pointing out that QNNSI is potentially much more computationally efficient than all the models referenced above in the Introduction section.The efficiency of many quantum algorithms comes directly from quantum parallelism that is a fundamental feature of many quantum algorithms.Heuristically, and at the risk of over-simplifying, quantum parallelism allows quantum computers to evaluate a function f(x) for many different values of x simultaneously.Although quantum simulation requires many resources in general, quantum parallelism leads to very high computational efficiency by using the superposition of quantum states.In QNNSI, the input samples have been converted into corresponding quantum superposition states after preprocessing.Hence, as far as a lot of quantum rotation gates and controlled-not gates used in QNNSI are concerned, information processing can be performed simultaneously, which greatly improves the computational efficiency.Because the above experiments are performed in classical computer, the quantum parallelism has not been explored.However, the efficient computational ability of QNNSI is bound to stand out in future quantum computer.

Conclusion
This paper proposes quantum-inspired neural network model with sequence input based on the principle of quantum computing.The architecture of the proposed model includes three layers, where the hidden layer consists of quantum-inspired neurons and the output layer consists of classical neurons.An obvious difference from classical ANN is that each dimension of a single input sample consists of a discrete sequence rather than a single value.The activation function of hidden layer is redesigned according to the principle of quantum computing.The Levenberg-Marquardt algorithm is employed for learning.With application of the information processing mechanism of quantum rotation gates and controlled-not gates, the proposed model can effectively obtain the sample

Figure 3 .
Figure 3.The model of quantum-inspired neural network with sequence input.
the normalized desired outputs of the l-th sample, the corresponding actual outputs.The evaluation function is defined as follows

Table 1 .
The input nodes and the sequence length setting of QNNSIs and ANN.

Table 2 .
Training result contracts of average approximation error.

Table 3 .
Training result contracts of average iterative steps.

Table 4 .
Training result contracts of average running time (s).

Table 6 .
The average prediction error contrasts of QNNSIs and ANN. by ways of breadth and depth.The experimental results reveal that a greater difference between input nodes and sequence length leads to a lower performance of proposed model than that of classical ANN; on the contrary, it obviously enhances approximation and generalization ability of proposed model when input nodes are closer to sequence length.The following issues of the proposed model, such as continuity, computational complexity, and improvement of learning algorithm, are subjects of further research. characteristics