We detect the quantum phase transition of a quantum many-body system by mapping the observed results of the quantum state onto a neural network. In the present study, we utilized the simplest case of a quantum many-body system, namely a one-dimensional chain of Ising spins with the transverse Ising model. We prepared several spin configurations, which were obtained using repeated observations of the model for a particular strength of the transverse field, as input data for the neural network. Although the proposed method can be employed using experimental observations of quantum many-body systems, we tested our technique with spin configurations generated by a quantum Monte Carlo simulation without initial relaxation. The neural network successfully classified the strength of transverse field only from the spin configurations, leading to consistent estimations of the critical point of our modelIn recent years, the technique of deep learning has become well established and has shown outstanding performance in various fields. For instance, GoogLeNets is a deep neural network (DNN) that won a prize at the ImageNet Large Scale Visual Recognition Competition in 2014 (ILSVRC14). 1) AlphaGo is a DNN designed by Deep Mind that is trained using a novel combination of supervised learning from human expert games and reinforcement learning from games of self play. 2) AlphaGo has successfully beat many human Go players.In general, DNNs can be used extract unclear features of a given input dataset or to identify hidden relationships between input and output. DNNs have been applied to various problems in physics. [3][4][5] In addition, a theoretical assessment of a DNN has been performed using several tools in the theoretical physics. 6) In the previous study performed by Tanaka and Tomita,7) they have utilized convolutional neural networks (CNN) for detecting the phase transition of a classical two-dimensional Ising model on a square lattice using only the spin configurations obtained by the Markov-chain Monte Carlo method. The sharp change of weights in the resulting CNN signified the phase transition and successfully computed a * arai@smapip.is.tohoku.ac.jp
We derive macroscopically deterministic flow equations with regard to the order parameters of the ferromagnetic p-spin model with infinite-range interactions. The p-spin model has a first-order phase transition for p > 2. In the case of p ≥ 5 ,the p-spin model with anti-ferromagnetic XX interaction has a second-order phase transition in a certain region. In this case, however, the model becomes a non-stoqustic Hamiltonian, resulting in a negative sign problem. To simulate the p-spin model with anti-ferromagnetic XX interaction, we utilize the adaptive quantum Monte Carlo method. By using this method, we can regard the effect of the anti-ferromagnetic XX interaction as fluctuations of the transverse magnetic field. A previous study (J.Inoue, J. Phys.Conf. Ser.233, 012010, 2010) derived deterministic flow equations of the order parameters in the quantum Monte Carlo method. In this study, we derive macroscopically deterministic flow equations for the magnetization and transverse magnetization from the master equation in the adaptive quantum Monte Carlo method. Under the Suzuki-Trotter decomposition, we consider the Glauber-type stochastic process. We solve these differential equations by using the Runge-Kutta method and verify that these results are consistent with the saddle-point solution of mean-field theory. Finally, we analyze the stability of the equilibrium solutions obtained by the differential equations.
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