Two types of neural network representations of quantum mixed states

Abstract

Quantum information and artificial intelligence are the two most cutting-edge research fields in recent years, which have made a lot of progress in changing the traditional science, realizing the cross fusion of the two fields has become a hot topic of scientists. Although scholars have made many explorations in this field and used them to simulate the steady state and the dynamics of open quantum manybody systems, little attention has been paid to the problem of accurate representation of neural networks. In this paper, we focus on neural network representations of quantum mixed states. We first propose neural network quantum mixed virtual states (NNQMVS) and neural network quantum mixed states (NNQMS) with general input observables using two neural network architectures, respectively, and then explore their properties, the related conclusions of NNQMVS and NNQMS under tensor product operation and local unitary operation are obtained. To quantify the approximation degree normalized NNQMVS and NNQMS of a given mixed state, we define the best approximation degree using normalized NNQMVS and NNQMS, and obtain the necessary and sufficient conditions for the representability of a general mixed state using normalized NNQMVS and NNQMS. Moreover, we explore the types of mixed states that can be represented by these two neural network architectures and show their accurate neural network representations.