Quantum Neural Network States: A Brief Review of Methods and Applications

Abstract

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.

Entanglement classification via neural network quantum states

“…As mentioned above, NNSs provide a parameterisation for the wavefunction of quantum systems by means of RBM-like architectures [5], which have recently received considerable attention [7]. RBMs consist of a single visible and hidden layer of neurons, mediated by weighted inter-layer connections and with no intralayer links.…”

Entanglement area law for shallow and deep quantum neural network states

“…However, the eigenstates of the Hamiltonians of these natural systems often have an internal simplified structure, which makes many approximating or even exact methods possible. Neural network states are introduced as ansatz states of many-body quantum systems recently, and because of their good performance in solving some problems which can not be solved using the state-of-the-art method, many attentions are attracted [27,28,30,40]. Here, to explore the area-law entanglement of the neural network states, we first introduce the concept of quasi-product states.…”

“…For a K-local neural network, a corresponding quantum state can be given. Usually, there are two different ways to build quantum neural network states [40], the first approach, which is also the approach we choose to use in this work, is to introduce complex weights and biases into the neural network; the second approach is to represent the amplitude and phase of a wavefunction separately. We will prove that quantum states build from K-local neural networks obey the entanglement area-law, since there are all quasi-product states, and the entanglement area law of quasi-product states will be established later.…”

“…In practical applications, introducing complex weights and biases and complex activation functions may lead to some difficulties for training the neural network, to overcome this shortage, the amplitude and phase of quantum state are usually represented by two feed-forward neural network separately, see reference e.g.,[40]. But here for our purpose, we choose to use the complex neural network approach.…”

Quantum Machine Learning and Bioinspired Quantum Technologies

“…‐A prediction of the band gap which represents one of the basic properties of a crystalline material via machine learning calculations, by Alexander V. Balatsky and co‐workers ‐A Review Article on the progress in using artificial neural networks to build quantum many‐body states, by Zhih‐Ahn Jia et al …”