Mathematical Problems in Engineering

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Volume 2021 |Article ID 6655455 | https://doi.org/10.1155/2021/6655455

Haibin Wang, Jiaojiao Zhao, Bosi Wang, Lian Tong, "A Quantum Approximate Optimization Algorithm with Metalearning for MaxCut Problem and Its Simulation via TensorFlow Quantum", Mathematical Problems in Engineering, vol. 2021, Article ID 6655455, 11 pages, 2021. https://doi.org/10.1155/2021/6655455

A Quantum Approximate Optimization Algorithm with Metalearning for MaxCut Problem and Its Simulation via TensorFlow Quantum

Academic Editor: Gonglin Yuan
Received22 Oct 2020
Revised20 Nov 2020
Accepted04 Mar 2021
Published25 Mar 2021

Abstract

A quantum approximate optimization algorithm (QAOA) is a polynomial-time approximate optimization algorithm used to solve combinatorial optimization problems. However, the existing QAOA algorithms have poor generalization performance in finding an optimal solution from a feasible solution set of combinatorial problems. In order to solve this problem, a quantum approximate optimization algorithm with metalearning for the MaxCut problem (MetaQAOA) is proposed. Specifically, a quantum neural network (QNN) is constructed in the form of the parameterized quantum circuit to detect different topological phases of matter, and a classical long short-term memory (LSTM) neural network is used as a black-box optimizer, which can quickly assist QNN to find the approximate optimal QAOA parameters. The experiment simulation via TensorFlow Quantum (TFQ) shows that MetaQAOA requires fewer iterations to reach the threshold of the loss function, and the threshold of the loss value after training is smaller than comparison methods. In addition, our algorithm can learn parameter update heuristics which can generalize to larger system sizes and still outperform other initialization strategies of this scale.

1. Introduction

With the rapid development of quantum computing, people pay more and more attention to using quantum computing to solve combinatorial optimization problems. The maximum cut problem (MaxCut) [1] is a typical NP-hard combinatorial optimization problem in graph theory. A promising approach to solve combinatorial optimization problems on near-term machines is the quantum approximate optimization algorithm (QAOA) [2]. This is a heuristic method, which can be considered as a time discretization of adiabatic quantum computing [3]. Like many quantum computing [46] methods, QAOA is a hybrid classical-quantum algorithm that combines quantum circuits and classical optimization of those circuits. The goal of QAOA is a function of the quantum state, which in turn is a function parameterized by one- and two-qubit gates, and its parameters can be continuously varied. In 2014, Edward Farhi and Jeffrey Goldstone jointly published the QAOA for the MaxCut problem (MaxCut QAOA) algorithm [7]. This algorithm is a polynomial-time approximate optimization algorithm for solving combinatorial optimization problems and has the potential to demonstrate quantum hegemony. However, this classical algorithm could require space doubly exponential in , where is the time of unitary transformation. Later, a classical computer is used as the outer-loop optimizer of quantum neural network to find the best parameters, which then were fed to the quantum computer with different sets of parameters. In 2018, Gavin proposed an algorithm [8] that used classical simulation with automatic differentiation [9] of parameters and stochastic gradient descent to explore the MaxCut QAOA problem. However, it is difficult to implement stochastic gradient descent directly on a quantum computer, since the requisite gradients are expensive to measure [10], requiring many observations for each gradient component. A variety of approaches have been used to optimize MaxCut QAOA circuits, including Nelder–Mead [11], gradient descent [12], limited-memory Broyden–Fletcher–Goldfarb–Shanno Bound (L-BFGS-B) [13], and Bayesian [14] methods. Among them, Nelder–Mead and L-BFGS-B are the local optimization methods, and gradient descent and Bayesian methods are the global optimization methods. However, these approaches are relatively poor in generalization. In other words, a classical neural network trained by these methods to optimize small quantum neural network (QNN) [15] cannot learn parameter update heuristics which can generalize to larger system sizes.

In order to solve these problems, we propose a new local optimization method, which is the quantum approximate optimization algorithm with metalearning for the MaxCut problem (MetaQAOA). This algorithm makes use of the quantum-classical metalearning approach composed of a group of metaoptimization techniques to learn how to modify the parameters of learning algorithms to further customize them for the MaxCut QAOA problems. This could be to ensure that the learning can be well generalized, to better fit the given data with fewer iterations, and to adapt a pretrained neural network model to a new task. Through the simulation experiment on TensorFlow Quantum (TFQ) [16], the convergence of the model shows that the total number of optimization iterations required to achieve a given accuracy has been significantly improved. The experiment simulation via TensorFlow Quantum (TFQ) shows that MetaQAOA requires fewer iterations to reach the threshold of the loss function, and the threshold of the loss value after training is smaller than the other local optimization methods, i.e., Nelder–Mead and L-BFGS-B. In addition, the path chosen by the long short-term memory (LSTM) optimizer on a larger MaxCut problem after being trained on many random smaller MaxCut problems demonstrates that even when there is less data, the neural network learns to generalize its parameter update heuristic to larger system sizes. This provides the possibility of training on small, classical, and simulatable MaxCut QAOA problem instances, in order to initialize larger, classically difficult to simulate MaxCut QAOA problem instances on quantum devices.

The rest of the paper is organized as follows: Section 2 describes the preliminaries about the classical LSTM neural network, metalearning, and the quantum approximate optimization algorithm for the MaxCut QAOA problem; Section 3 presents the design focus of MetaQAOA, including the description of the MaxCut problem, the process of algorithm, and the method of optimizing parameter; in Section 4, experiments are performed in the quantum cloud computing platform to verify the feasibility of the proposed MetaQAOA; Section 5 summarizes what we find in this work and the prospects for future research studies.

2. Preliminaries

2.1. Long Short-Term Memory (LSTM)

In general, LSTM refers to long and short-term memory artificial neural network, which is a time loop neural network, and is used to solve the problem of long-term dependence. There are special units called memory blocks (shown in Figure 1) in the hidden layer of LSTM. In addition to special multiplicative units called gates to control the flow of information, the memory blocks contain memory cells with self-connections storing the temporal state of the network. Each memory block contained an input gate, an output gate, and a forget gate [17]. The input gate controls the flow of input activations into the memory cell. The output gate controls the output flow of cell activations into the other part of the network. Because the forget gate can scale the cell before adding it as input to the cell through the self-connection of the cell, the cell’s memory is forgotten or reset adaptively. In addition, the current LSTM architecture (as shown in Figure 2) contains peephole connections from its internal cells to the gates in the same cell to understand the precise timing of the outputs [18].

The LSTM calculates the activation of the network cell iteratively from to by making use of the following equations to calculate the mapping from the input sequence to the output sequence :where represents weight matrices, diagonal weight matrices for peephole connections are , , and , bias vector is (the input gate bias vector is ), the logistic sigmoid function is , and the input gate, output gate, forget gate, and cell activation vectors are , , , and . They are all the same size as the cell output activation vector , represents the element-wise product of the vectors, the input and output activation functions are and , and the network output activation function is . In addition, LSTM neural networks are already deep architectures in the sense that they can be considered as a feed-forward neural network unrolled over time, where each layer shares the same parameters, which is shown in Figure 2.

2.2. Metalearning

Metalearning is to study how to design a machine learning model with less training set, so that it can learn well and quickly [19]. One specific example is a neural network model which learns how to optimize other neural network models with parameterized function. Metalearners can not only train machine learning models but also train and optimize general functions. In specific regions where models are used to optimize other models, early studies explored guided strategy search, which has been replaced by LSTMs [20]. The LSTM is a kind of recurrent neural network, which is used to reduce the vanishing or exploding gradients in other recurrent neural network structures [21, 22]. An LSTM is composed of a cell state, a hidden state, and a gate, all of which are called LSTM cell. In each time step, the cell state is changed based on the hidden state, gate, and data information entered into the LSTM cell. The change of the hidden state depends on the gate and the input, and then, the cell state and hidden state are passed to the LSTM cell in the next time step. A complete processing method for LSTM is given in [23]. An LSTM is instrumental in learning long-term dependencies, such as those in optimization.

Metalearners have been used in the rapid general optimization of the model, and there are few training examples [24]: given the random initial parameters, we seek to quickly converge to the “optimal” general parameters defined by some measures. This same problem characteristic also appears in QAOA, and some common distribution across problems [25] may be followed by good parameters. A metalearner could be used to find general good parameters and fine-tuning left to some other optimizer [26].

2.3. The Quantum Approximate Optimization Algorithm for the MaxCut Problem

The MaxCut problem is a typical combinatorial optimization problem and an NP-hard problem. A promising approach to solve combinatorial optimization problems on near-term quantum machines is QAOA. The steps of the quantum approximate optimization algorithm for the MaxCut problem are as shown in Algorithm 1.(1)An initial state is prepared, which is a uniformly distributed quantum superposition state.(2)The mixer Hamiltonian and the cost Hamiltonian are selected appropriately. The selected cost and mixer Hamiltonian need meet this condition that the mixer Hamiltonian is noncommuting with the cost Hamiltonian, i.e., .(3)The unitary applied to this initial state is about the exponential form of mixer Hamiltonian and cost Hamiltonian and can be expressed in the form of a parameterized quantum circuit (PQC).(4)We can get the final state after unitary transformation, as shown in Algorithm 1.

Input: initial state Hamiltonian .
Output: final state .
(1);
(2)random ;
(3);
(4);
(5)fortodo
(6);
(7)
(8)end
(9)return

A variety of approaches have been used to optimize PQC, including the local optimization methods (Nelder–Mead [11] and L-BFGS-B [13]) and the global optimization methods (gradient descent [12] and Bayesian [14]). However, these local optimization approaches are relatively poor in generation. In order to enable the classical neural networks trained by these methods to optimize small QNNs, it is possible to learn parameter update heuristics that can be extended to larger system sizes. In this case, we propose a new local optimization method, which is a quantum approximate optimization algorithm with metalearning for the MaxCut problem.

3. The Quantum Approximate Optimization Algorithm with Metalearning for the MaxCut Problem

The main purpose of the quantum optimization algorithm with metalearning for the MaxCut problem proposed in this paper is to train a classical long short-term memory (LSTM) network as an outer-loop optimizer to directly optimize the parameters in the QNN. In this section, we mainly introduce the process and training strategy of the MetaQAOA.

3.1. MaxCut Problem

The MaxCut problem is to find a split of the vertex set so that the maximum number of edge weights is divided, i.e., one vertex of the edge is in and the other vertex is in . This is a typical combination problem. Given a graph with nodes and edges , find a subset to maximize the number of edges between and . This problem can be attributed to finding the ground state of an Ising model.

The simplest method is the enumeration method, which is the only method that can find the global optimal solution (no approximation). However, this method is more complex, especially for the case of numerous vertices. Because when MaxCut only divides the vertices into two groups, there are a total of methods. Therefore, the MaxCut problem is an NP-hard problem.

3.2. The Quantum Approximate Optimization Algorithm with Metalearning for the MaxCut Problem

The quantum approximate optimization algorithm with metalearning for the MaxCut problem (MetaQAOA) we proposed will be to train an optimizer network, i.e., metalearner, with metalearning to learn parameter update heuristics for QNN constructed in the form of the parameterized quantum circuit. The MetaQAOA is a local optimizer, which has a notion of location in the solution space. It can search for candidate solutions from this location. Besides, MetaQAOA is usually fast and is susceptible to finding local minima. Training an outer-loop optimizer on many instances of MaxCut problem to enhance the efficiency of solving unseen instances is the idea of metalearning for optimization. This may be to ensure that the learning is well generalized and that the model can better fit the given data with fewer iterations. The algorithm we proposed mainly improves the update method of parameters in parameterized quantum circuits, and the steps of the proposed algorithm are shown in Algorithm 2.

  Input: initial state , Hamiltonian .
  Output: final state .
(1);
(2)random ;
(3);
(4); ;
(5)fortodo
(6)
(7)
(8)
(9)
(10)end
(11)return ;

The main steps of our algorithm are as follows. We first prepare a uniformly distributed quantum superposition state as an initial state :where represents the number of nodes in the graph for the MaxCut problem and is a superposition state.

Then, we select the mixer Hamiltonian and the cost Hamiltonian as follows:where is the sum of Pauli on each qubit, is diagonal in the computational basis, represent Pauli operator, respectively, is the edge of the graph , and are the two nodes connected by the edge.

The reason why we choose this mixer Hamiltonian is that each item is a noncommuting cost Hamiltonian, i.e., , and it can be exponentiated with minimal gate depth. One traditional family of Hamiltonian used in QAOA is the Ising models, and thus, we convert the cost Hamiltonian into Ising Hamiltonian , as shown in Algorithm 2. Therefore, equation (4) can be written as follows:

Thirdly, the unitary applied onto this initial state is given bywhere is the number of times the unitary transformation is executed and the difference is that the parameters are different at each time step. These parameters at each time step are as follows:

This unitary transformation with parameters expressed in the form of a general parameterized quantum circuit is shown in Figure 3. The initial state of the circuit is and Hadamard gate is performed on each qubit to prepare entangled state. Although these unitaries do not necessarily act on all qubits, we arrange them as “blocks” here, where “block” operations may be repeated times with different parameters in the circuit. Finally, the expected value is obtained by measurement.

A QNN is constructed in the form of a parameterized quantum circuit. A single time step of MetaQAOA is shown in Figure 4. An iteration starts from the LSTM sending the set of candidate parameters to the QNN. Then, the QNN executes a parameterized circuit , and it is responsible for generating the state . The purpose of measuring this state is to extract relevant information, i.e., the expectation value of the cost Hamiltonian. The classical subroutine suggests parameters based on the values provided by the quantum computer and sends the new parameters back to the quantum device.

This process is repeated times until the given goal is reached, as shown in Algorithm 2. In other words, the quantum-classical optimization loop can be unrolled into a single temporal quantum-classical computational graph over time, as shown in Figure 5. More precisely, at the optimization iteration, the LSTM receives the previous QNN query’s estimated cost function expectation , where is the estimate of and the parameters represent input, as shown in Algorithm 2. At this time step, LSTM also receives information stored in its internal hidden state from the previous time step. The LSTM itself has trainable parameters , so the parameterized mapping it applies iswhere is the information stored in the internal hidden state, represent the parameters of QNN, is the expectation value of the cost function at time step, and is the trainable of the LSTM. Besides, represents a new hidden state and are the parameters of the QNN generated by LSTM at time step.

Once a new set of QNN parameters is proposed, the LSTM will send it to the QNN for evaluation, and then the loop will continue.

Finally, the final state iswhich is our parameterized output, where is a variational parameter to be optimized by the LSTM and is the number of times the unitary transformation is executed. In addition, is an initial state and are the mixer Hamiltonian and Ising Hamiltonian, respectively.

3.3. Training Strategy

In this section, we will provide more detailed information on how the MetaQAOA trains an LSTM to optimize the parameters of QNN. The goal of quantum-classical metalearning is to train LSTM to learn an effective parameter update heuristics for a set of cost functions of interest, i.e., to find an optimizer that efficiently optimizes the special distribution of optimization area on average. The better optimizer is an optimizer that finds the best approximate local minimum of the cost function in as few function queries as possible. The whole optimal qualification is decided by the current problem category and the application area of interest.

In order to learn this general optimization strategy, the LSTM is trained by using a random instance, which has a fairly general distribution of functions, that is, functions are sampled from Gaussian processes. Since we focus on the known QNN optimization landscapes which are different from the classical Gaussian process optimization landscapes, we aim to train neural optimizers for MaxCut QAOA problems and QNN ansatze. In order to explore the effectiveness of our method, we will train the LSTM for random QNN instances in the target problem category, i.e., MaxCut QAOA, and test the trained network on (larger) previously unseen instances in the target problem category. The number of variational parameters in these ansatzes does not depend on the size of the system, but only on the number of alternating operators. This means we can use this training strategy to train our LSTM on smaller QNN instance sizes for MaxCut QAOA problems and then test for generalization on larger QNN instance sizes.

4. Experiment Simulation

4.1. Experimental Environment and Dataset

In order to train and test the LSTM optimizer on MaxCut QAOA problems, our experiments are conducted on the CPU and GPU. The TensorFlow Quantum (TFQ) [16], an open-source quantum cloud computing platform for the rapid prototyping of hybrid quantum-classical models for classical or quantum data, is introduced to realize the simulation experiments. Their parameters are shown in Table 1. Besides, the random problem instances are generated in the following form. Firstly, we fix an integer , and then an integer uniformly from the range is sampled randomly. Finally, we can generate a random graph and the probability of connection between nodes is , where is the number of neighbors in each node and represents the number of nodes. In order to illustrate it more vividly, we generate a random 3-regular graph with 10 nodes and the probability of being neighbors between nodes is , as shown in Figure 6. To generate training data, we uniformly sample , yielding QNN system sizes of at most 12 qubits. To generate our testing data, we also uniformly sample . Because our experiment is to verify that the trained network can be generalized to larger system and problem instances, the condition, i.e., , has to be met.


EnvironmentsParameters

CPUIntel (R) Xeon (R) Gold 5218
Hard diskSamsung PM881 512 GB
GPUNVIDIA GeForce RTX 2080 Ti
Simulation platformTensorFlow Quantum 2.3.0
Operating systemWindows 10

In our experiment, 500 problem instances sampled from this training dataset and 250 problem instances as validation dataset are selected to train the LSTM. In addition, we select 100 problem instances sampled from this testing set as testing data. The batch size of each training is 64, and there are a total of 1500 epochs for iterative training. In addition, the Adam optimizer provided by the open-source library is introduced for the network model, and the learning rate is set as 0.001. We choose a short time horizon to minimize the complexity of the training. For reference, 5 iterations is a significantly less number of quantum-classical optimization iterations than what is typically seen in previous works on QNN optimization.

4.2. Circuit Design

We only show the circuits of MaxCut QAOA unitary with qubits and qubits at the first time step, i.e., . In our experiment, we have carried out 5 time steps. Expect that the parameters of the different time steps, i.e., , , are distinct, the unitary circuits with are the same as those with .

The unitary circuits are shown in Figure 7, where Figure 7(a) shows the circuit of MaxCut QAOA unitary with and and Figure 7(b) shows the circuit of MaxCut QAOA unitary with and . These circuits are all parameterized quantum circuits, in which qubits connected by ZZ gates are neighbours in the generated 3-regular graph and gamma_0 and eta_0 represent the MaxCut QAOA parameters.

4.3. Results Analysis
4.3.1. Comparison of Training Performance

In order to evaluate the training performance of the algorithm proposed in this paper, some local optimizers, i.e., Nelder–Mead [11], L-BFGS-B [13], and our MetaQAOA, are compared. Nelder–Mead and L-BFGS-B are gradient-free and gradient-based approaches, respectively, which are standard local optimizers. L-BFGS-B is a local optimizer and has access to the gradients. Out of all optimizers chosen, it is the closest to the MetaQAOA in terms of information available to the optimizer and computational burden (i.e., the cost of computing the gradients). Nelder–Mead was chosen as it provides a widely recognized benchmark. Our MetaQAOA requires fewer iterations to reach the threshold of the loss function.

In this paper, we use the observed improvement at each time step as the loss function to measure the quality of the neural network model. This loss function is as follows:which is summed over the history of the optimization and the observed improvement at time step is derived from the difference between the recommended value, , and the best value obtained over the past history of the optimization until the time step, . If there is no improvement at a given time step, then the contribution to the loss is zero. However, temporarily increasing the cost function and then significantly improving the historical best level will be rewarded rather than penalized.

Our experiment aims at the QAOA for MaxCut problem of random 3-regular graphs generated by n = 10 qubits, yielding QNN system sizes of at 10 qubits. The three methods are compared from the relationship between the training epoch and the value of the loss function as shown in Figure 8. From this figure, we can get that, compared with the other two methods, the metalearner requires fewer iterations to reach the threshold of the loss function, the threshold of loss value after iteration is smaller, and the closest comparable competitor to the metalearner among these methods is L-BFGS-B. This shows that our proposed algorithm has better training performance.

4.3.2. Generalization Performance

Different from the other two methods, our proposed algorithm is generalizable. To illustrate the generalization performance, four kinds of experiments are simulated via TFQ. In this paper, we choose for the first kind of experiment, for the second kind of experiment, for the third kind of experiment, and for the last kind of experiment.

By inputting 1000 randomly selected parameter combinations into the optimized neural network, the distribution of QNN cost function results can be obtained which form four circular regions as shown in Figure 9. In addition, the darker the colour of regions, the better the result of QNN’s cost function. We make the updated value of the parameter . In Figure 9, the horizontal axis is used to represent the value of gamma , and the vertical axis is used to represent the value of eta . In order to clearly show the change of parameters, we set . The red dotted line indicates the change trend of the two parameters through 5 time steps, and the red star represents the value of the parameter after the last update. As we can see from Figure 9, the LSTM immediately begins guessing near the basin of attraction and continues to explore around the region looking to improve the estimate, so we can draw a conclusion that the neural network learns to generalize its heuristic to larger system sizes.

5. Conclusions

According to the shortcomings of weak generalization of the current methods of optimizing MaxCut QAOA circuits, we propose a quantum approximate optimization algorithm with metalearning for MaxCut problem (MetaQAOA) via TFQ platform. Our algorithm trains a classical LSTM network as an outer-loop optimizer to directly optimize the parameters in the QNN constructed by the parameterized quantum circuit. The simulation experiment results via TFQ show that MetaQAOA requires fewer iterations to reach the threshold of the loss function, and the threshold of loss value after training is smaller than the other local optimization methods, i.e., Nelder–Mead and L-BFGS-B. In addition, our method can learn parameter update heuristics that generalize to larger system sizes and still outperform other initialization strategies of this scale. But the metalearning optimizer neural networks in our MetaQAOA cannot scale to arbitrary problems and number of parameters, and research that builds upon it might be addressed in future studies.

Data Availability

The data used to support this study are included within this article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The support of all the members of the quantum research group of NUIST is especially acknowledged, and their professional discussions and suggestions have provided us with a lot of help. This work was supported by the Natural Science Foundation of China under Grant nos. 62071240 and 61802002 and in part by the Natural Science Foundation of Jiangsu Higher Education Institutions of China under Grant no. 19KJB520028, and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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