Applications Of Machine Learning In Quantum Physics

With the successful applications of machine learning in image recognitions,deci-sion making and logical inference,machine learning has gained more and more interest in physics community.Machine learning algorithm is good at extracting the intrinsic laws in the data,and physics problems are logical and ruled by physics laws,there-fore it is straightforward to use machine learning to solve physics problems.In classi-cal physics,machine learning can learn the laws of Newton's mechanics.In quantum physics,machine learning can solve the ground-state wave functions of Schrodinger equations,identify the location of phase t.ransition,represent the quantum many-body system states,find out the order parameters of many-body systems and so on.Machine learning algorithms can be classified into supervised learning,un-supervised learning and reinforcement learning.Supervised learning is firstly training the machine by labelled data,such as training a neural network.After training,the neu-ral network is used for making predictions,meanwhile the input data may not in the training dataset.Un-supervised learning does not need laballed training dataset.On the goal of minimizing the loss function,the optimization is performed on the input data.After the optimization,the algorithm can divide the input data into several cat-egories.Reinforcement learning is training a learning agent to accomplish a mission by collecting the maximal rewards,after training the algorithm can accomplish the job.Reinforcement learning is suitable to control the instruments.Although machine learning algorithm is not originated from solving physics prob-lems.By understanding and properly applying machine learning,it can help physicists to understand physics in a new perspective and finding out new physics.As the de-velopment of quantum computing,it is possible to run machine learning on quantum computers,meanwhile achieving high accelerations.Therefore,machine learning can help solving physics problems,and physics can improve machine learning to higher ef-ficiencies.In the future,machine learning and physics will have mutual improvements.The thesis describe some examples that using machine learning to solve quantum physics problems,it is composed of two main parts.The first part describes how to use a deep neural network to generate the ground-state wave functions of Bose-Einstein Condensates.Because all the particle in the BEC has the same phase,the ground-state wave function of BEC is a distribution in the real space,while the wave function is obtainable by solving GP equations.We can build a neural network to learn the mapping between the GP equation and the ground-state wave functions.We considered two cases:(1)various coupling strengths with the same potential fields.(2)various potential fields with the same coupling strengths.After supervised learning,the deep neural network can generate the ground-state wave functions within high precisions.When inputting the Gaussian disorder with ?D=0.39,the network still generates the correct wave functions.Furthermore,when inputting different kinds of potentials,the neural network trained by the Gaussian disorder gives the correct wave functions.Thus the neural network has learned the process of solving Gross-Pitaevskii equations.The second part describes how to design a convolutional neural network as a vari-ational ansatz to solve the two-dimensional quantum spin models.Tensor network is a promising ansatz to solve quantum many-particle systems,however the computational complexity of tensor network is very high.Alternatively,a neural network can achieve the results that are comparable to tensor networks.Therefore,it is necessary to test the neural networks on several nontrivial models,such as the J1-J2 model.We built a convolutional neural network with a convolutional layer,a maxpooling layer and a transposed convolution layer.To avoid the local optimizations,we use the replica exchange molecular dynamics method.We firstly set several temperatures,then randomly initialize the parameters in the network,then we have several networks cor-responding to the temperatures,the network with the lowest temperature is the solution for ground-states.The ground energies achieved by the neural networks even lower than those achieved by the string-bond-states.Therefore it is possible to use convolutional neural networks to solve quantum m any-body problems in faster speeds.The third part describes a quantum version of the cocktail party problem(CPP).CPP is an important problem in signal processing.It is an algorithm to resolve the sources signals by using several detected signals,while each detected signal is a linear mixture of the source signals.CPP can be solved by the independent component analysis(ICA)algorithm.We have developped a quantum version of CPP,namely q-CPP.In q-CPP,the source is a pure state density matrix,while the pure states from each pair of sources are not orthogonal.Part of the total Hilbert space of each pure state is not detectable by detectors.Thus the density matrix detected by each detector is a mixed state.Based on the classical ICA,we defined a new loss function.We show that the optimization of the loss function can be mapped to solve the ground state of a spin-1/2 many-body Hamiltonian.The optimization is performed by either Newton's method and simulated annealing,we show that the pure states can be recovered within the filideties higher than 0.99 via both optimization methods.