| In the era of big data,relying on the rapid development of hardware and the powerful tools proposed for data mining,machine learning has an impact on various fields such as scientific research,finance,and living.However,in the face of the massive increase in data and the bottleneck of capacity of computation,machine learning algorithms will face great challenges.In recent years,with putting forward of Noisy Intermediate-Scale Quantum(NISQ),Quantum computing has taken a step forward in practical applications.For some specific problems such as solving linear equations,integer factorization,etc.,quantum computing technology exhibits the amazing computing power that classical computing cannot efficiently achieved.This computational model that leverages the phenomena of quantum mechanics to perform calculations and provides a potential resolution for the problems discussed above.Quantum machine learning is the fusion of machine learning techniques and quantum algorithms,which takes advantages from both sides.This research field explores how to use the quantum phenomenon of superposition and entanglement and integrate with classical machine learning algorithms to create new data analysis and processing tools which might promote the development of various fields.Although it has attracted more and more researchers’ attention in recent years,and related algorithms have been constantly proposed,the research on some key issues is still in its infancy.Therefore,this dissertation focuses on studying the quantum version of machine learning models.Based on the hybrid quantum-classical computing framework and quantum algorithms,we proposed quantum machine learning models for generating classical or quantum data and the dimensionality reduction task,which provide useful references for future research.The content of this dissertation is organized as following five parts.Focusing on the hybrid quantum-classical computing framework that has been widely used and implemented on NISQ equipment in recent years,this dissertation systematically analyzes and compares the performance of the framework applied in three typical machine learning scenarios.The quantum-classical hybrid computing framework is mainly composed of a variational quantum circuit and optimizer.Different circuit ansatz and optimizers affect the performance of quantum machine learning models.In this work,the performances of two types of quantum circuit ansatz and nine widely used optimizers are investigated to build a quantum machine learning model.Meanwhile,this dissertation also provide the analysis and comparison of the models applied in different learning tasks,including discrimination,generation,and optimization based on the convergence of the cost function curve,the time consumption of the model,and the distribution of the optimized value when the model reaches the maximum number of iterations.In the current hybrid framework,the variational quantum circuit is mainly constructed with a fixed structure and depth,which lacks flexibility.At the same time,a few research pay attention to comparing the performance of the quantum generation model based on different cost functions.Therefore,this article investigates how to use the variable-depth quantum circuits to build the quantum generation models.Moreover,we build the quantum generative models whose loss functions are related to four representative metrics for probability distribution,including KL divergence,JS divergence,total variation,and maximum mean discrepancy.The experiments demonstrate the performance of the models on classical discrete data and show the following conclusion.The quantum generative model with KL divergence as the cost function performs best and those with the variable-depth quantum circuit are more flexible in training.The Generative adversarial network(GAN)is a class of learning framework.It has received as much attention since its distinctive design in the machine learning community.The work of the Wasserstein generative adversarial network not only reveals the cause of the vanishing gradient and mode collapse but also provides an ingenious solution.Due to the property of quantum data,such as superposition and entanglement,traditional metrics are no longer applicable to quantum data.Therefore,inspired by the optimal transport theory,this dissertation proposes a quantum Wasserstein distance that can measure the distance between quantum data(quantum states)and also prove that it is a semi-metric in Hilbert space.Based on quantum Wasserstein distance and hybrid quantum-classical framework,a quantum Wasserstein generative adversarial network are proposed.This quantum generative model not only has good performance on different types of quantum data but also performs robustly under noisy environments.This research is a significant reference and provides the idea for the quantization of traditional algorithms.Besides,it is a powerful tool with potential applications in quantum computing and quantum information.Maximum mean discrepancy(MMD)is a distance based on embedding probability distribution in a reproducing kernel Hilbert space and finds a maximum distance between the mean embedding points.It has been used in various applications in machine learning.Inspired by MMD,this dissertation proposes a quantum version of the maximum mean discrepancy for dealing with quantum data(quantum state)and also prove that quantum maximum mean discrepancy is a metric in Hilbert space and satisfies the triangle inequality compared with quantum Wasserstein distance.Based on the hybrid quantum-classical framework,this dissertation gives the design and implemention of a quantum maximum mean discrepancy adversarial network and applies it to the crucial task in quantum computing such as quantum states preparation.Compared to the other quantum generative adversarial networks,this model has fewer parameters and is easier to train.Besides,the model also has many potential applications,including compressing,compiling,and transferring of quantum circuits.Laplacian eigenmap is a typical dimensionality reduction algorithm in machine learning.Inspired by Laplacian eigenmap and related quantum algorithm,this dissertation proposes a quantum dimensionality reduction algorithm: quantum Laplacian eigenmap.By regarding the target matrix as the density matrix of a quantum state,using quantum random access memory(QRAM)and applying Hermitian matrix multiplication technology,this dissertation provides a way to construct the target matrix under the quantum setting.Then the proposed method uses quantum phase estimation to find the largest k eigenvalues and eigenvectors of the target matrix and store them in the quantum state.After extracting the data from quantum states,the method can achieve the task of dimensionality reduction.This method can be widely used as a pre-processing step in quantum machine learning algorithms. |
