One of the main challenges of quantum many‐body physics is the exponential growth in the dimensionality of the Hilbert space with system size. This growth makes solving the Schrödinger equation of the system extremely difficult. Nonetheless, many physical systems have a simplified internal structure that typically makes the parameters needed to characterize their ground states exponentially smaller. Many numerical methods then become available to capture the physics of the system. Among modern numerical techniques, neural networks, which show great power in approximating functions and extracting features of big data, are now attracting much interest. In this work, the progress in using artificial neural networks to build quantum many‐body states is reviewed. The Boltzmann machine representation is taken as a prototypical example to illustrate various aspects of the neural network states. The classical neural networks are also briefly reviewed, and it is illustrated how to use neural networks to represent quantum states and density operators. Some physical properties of the neural network states are discussed. For applications, the progress in many‐body calculations based on neural network states, the neural network state approach to tomography, and the classical simulation of quantum computing based on Boltzmann machine states are briefly reviewed.

Machine learning representations of many-body quantum states have recently been introduced as an ansatz to describe the ground states and unitary evolutions of many-body quantum systems. We investigate one of the most important representations, restricted Boltzmann machine (RBM), in stabilizer formalism. A general method to construct RBM representations for stabilizer code states is given and exact RBM representations for several types of stabilizer groups with the number of hidden neurons equal or less than the number of visible neurons are presented. The result indicates that the representation is extremely efficient. Then we analyze the surface code with boundaries, defects, domain walls and twists in full detail and find that almost all the models can be efficiently represented via RBM ansatz: the RBM parameters of perfect case, boundary case, and defect case are constructed analytically using the method we provide in stabilizer formalism; and the domain wall and twist case is studied numerically. Besides, the case for Kitaev's D(Z d ) model, which is a generalized model of surface code, is also investigated.