Weingarten Calculus And Application In Random Tensor Networks

Tensor network is an important tool,which is widely applied in quantum physics,quantum chemistry,machine learning and other fields.Different from the classical simulation methods,such as Lanczos method,tensor network can do well in handling the quantum lattice system with any scale.Since the tensor network satisfies the area law of entanglement entropy,it can be used to describe the ground state and low-energy excited state of local Hamiltonian.The quantum lattice system constructed by tensor network can directly reflect the neighborhood relationship between the constituent elements of the quantum lattice,and describe the degree of freedom of each constituent element.In this thesis,we will focus on the projected entanglement pair states(PEPS),which is a high-dimensional generalization of matrix product states.In this thesis,the quantum states represented by tensor networks with simple topological structures and PEPS are investigated,and their canonical typicality is discussed.Our main results are the following:Firstly,this thesis studies three kinds of random tensor networks with simple topology structure.For three-node tensor network and four-node tensor network,each node is given appropriate dimension and randomization,and the expectation of their corresponding reduced density matrixes are estimated by Weingarten calculus.By means of the measure concentration phenomenon of the related Lipschitz functions,their regularity and typicality are discussed.Secondly,this thesis focuses on a certain PEPS model via the above approach.More precisely,two kinds of truncations are investigated as follows,one is that the PEPS is truncated in a strip region,another case is that the PEPS is truncated on a central region.Under the assumption of appropriate boundary conditions and randomization,the canonical typicality of the corresponding quantum states is discussed.