Universal Approximation Quantum Neural Network Algorithm And Applications

A Quantum Neural Network(QNN)is an artificial neural network model designed based on the fundamental principles of quantum mechanics.In classical machine learning theory,the universal approximation of artificial neural networks is guaranteed by the Universal Approximation Theorem(UAT),which states that neural networks can approximate any continuous function.This is an essential guarantee for ensuring that neural networks can handle complex learning problems.For classical neural networks,the universal approximation originates from the non-linear transformations provided by activation functions and the trainable parameters.However,achieving universal approximation in quantum neural networks is far from trivial.On the one hand,due to the intrinsic linearity of quantum systems,it is difficult to directly implement non-linear transformations.On the other hand,designing a quantum circuit ansatz that can theoretically generate any unitary transformation remains an open problem.Furthermore,even if the universality of quantum neural networks can be guaranteed theoretically,there still exists several challenges for quantum neural networks to solve real-world learning problems,such as how to handle learning problems based on the time-series data and how to efficiently generate adversarial attacks.To address these problems,this dissertation conducted research from two aspects: quantum neural network design and quantum neural network applications.The specific research results and innovations are as follows:1.To address the problem how to efficiently implement the nonlinear transformation of quantum neural networks,this dissertation presents a duplication-free quantum neural network(DQNN)model which combines the variational quantum computing framework with classical nonlinear activation functions.The model first applies multiple variational circuits on a single small-scale quantum register and measures the expectation values of several local observables.Then the measurement results are processed through classical activation functions and the output can be finally generated.In this way,the DQNN can effectively solve the given learning problem and its universality can be strictly proven.In addition,compablack to existing quantum neural network models,DQNN can effectively blackuce the number of qubits and circuit complexity requiblack by quantum neural networks in solving learning problems.In the meanwhile,DQNN can achieve better learning performance and has better robustness to defend quantum noise in quantum circuits.Experimental results show that,compablack to classical neural networks,DQNN requires fewer parameters when solving typical learning problems,which provides an opportunity for further exploration of the potential quantum advantages of quantum neural networks.2.To address the problem of designing quantum circuits capable of generating arbitrary unitary transformations,this dissertation combines the variational quantum circuit ansatz with the matrix product state(MPS)representation of quantum states and proposes a sequentially generated(SG)ansatz that can generate arbitary MPS with a fixed bond dimension.In theory,the SG ansatz has the capability to generate arbitrary MPS,and any quantum state can be represented by MPS.Therefore,the SG ansatz theoretically has the ability to generate arbitrary unitary transformations.By applying the SG ansatz to the DQNN introduced in Result 1,the constructed quantum neural network model can effectively solve typical machine learning problems.Furthermore,MPS and its extended form,string-bond state(SBS),can be applied to the representation of ground states of typical one-dimensional and high-dimensional quantum many-body systems.This dissertation extends the SG ansatz structure to enable it to generate SBS and further applies the extended ansatz to the task of ground state energy calculations in typical chemical and quantum many-body models.Simulation results show that,compablack to several typical variational quantum circuit ansatz,the SG ansatz achieves higher accuracy with lower circuit complexity in calculating the ground state energy of typical quantum systems.Specially,for two-dimensional and high-dimensional quantum systems,the SG ansatz also provides good approximate solutions.This research not only provides new insights for the circuit structure design of quantum neural networks but also offers an opportunity for solving complex quantum many-body problems.3.Time-series data is ubiquitous in natural language processing tasks.How to effectively process time-series data is a key challenge for quantum neural networks.To address this challenge,this dissertation adopts the duplication-free quantum neural network as the subroutine and further designs a duplication-free quantum long short-term(DQLSTM)neural network model for time-series data processing.Specially,DQLSTM utilizes the superposition principle of quantum states such that it can store and process exponentially long classical information with a polynomial-qubit quantum register.With the nonlinearity generation method as proposed in result 1,DQLSTM can efficiently blackuce the quantum resources requiblack to deal with natural language processing tasks.Numerical simulations demonstrate that,compablack with existed quantum long short-term neural network models,DQLSTM can use a smaller quantum register to achieve better performance in solving typical natural language processing tasks.Furthermore,compablack with the classical long short-term neural network with a similar parameter scale,the DQLSTM can achieve a better performance.4.Quantum adversarial attacks refer to the infinitesimal perturbations that can cause quantum neural networks to misclassify.In addressing the problem of efficiently generating quantum adversarial attacks,this dissertation proposes a quantum adversarial attack generation algorithm(Quantum Fool algorithm)based on a simplified DQNN model,which utilizes the gradient information of the input data with respect to the DQNN.This algorithm generates small perturbations through iterative accumulation,enabling the rapid generation of quantum adversarial attacks that cause misclassification in quantum neural networks.A comparison with typical quantum adversarial attack generation algorithms reveals that the quantum obfuscation algorithm generates smaller perturbations to the original data.Researching efficient methods for generating quantum adversarial attacks not only helps understand the vulnerability of quantum neural networks in solving classification problems but also provides new insights for enhancing the robustness of quantum neural networks.All above results on the universality and applications of quantum neural networks demonstrate that the quantum neural networks have the equivalent effectiveness as the classical neural networks and have the ability to solve typical classical and quantum learning problems.These results not only pave the way for a deeper understanding of the inner mechanisms of quantum neural networks,but also provide a possibility for exploring the practical applications of quantum neural networks,and contribute to further investigating the potential quantum advantages based on quantum neural networks.