时间演化矩阵乘积算符方法及其在量子开放系统中的应用

Time-evolving matrix product operator method and its applications in open quantum system

量子开放系统在量子科学的发展中占据着举足轻重的地位,因此对其数值计算方法进行研究具有重要的意义.对于量子开放系统,在20世纪90年代发明的准绝热传播子路径积分方法是为数不多的精确的数值计算方法,但其计算复杂度随着系统的大小和关联时间长度呈指数增长,因此在实际计算中它所能计算的物理模型比较受限.近年来,张量网络的研究和应用有了长足的进展.使用张量网络来表达该方法可以使其计算复杂度变成多项式增长,极大地提高了计算效率.由此发展出的新方法则被称为时间演化矩阵乘积算符方法,是一种高效的、数值精确的、并且非马尔可夫的计算方法,在量子开放系统的研究中有着广泛的应用前景.本文首先综述了准绝热传播子路径积分方法,接着介绍了矩阵乘积态的基本思路,然后利用矩阵乘积态来表述准绝热传播子路径积分方法,从而对时间演化矩阵乘积算符方法进行介绍;最后综述了该方法在量子开放系统中的应用,并以自旋-玻色子系统中的关联函数和热流计算为例对该方法进行了展示.

Open quantum systems play an important role in developing quantum sciences,and therefore the study of corresponding numerical method is of great significance.For the open quantum systems,the quasi-adiabatic propagator path integral invented in 1990s is one of the few numerically exact methods.However,its computational complexity scales exponentially with system size and correlation length,and therefore its application is limited in practical calculation.In recent years,the study and application of tensor network have made rapid progress.Representing the path integral by tensor network makes the computational complexity increase polynomially,thus greatly improving the computational efficiency.Such a new method is called time-evolving matrix product operator.At the very beginning,the reduced density matrix is represented as a matrix product state.Then the time evolution of the system can be achieved by iteratively applying matrix product operators to the matrix product state.The iterative process is amenable to the standard matrix product states compression algorithm,which keeps the computational cost on a polynomial scale.The time-evolving matrix product operator is an efficient,numerically exact and fully non-Markovian method,which has a broad application prospect in the study of quantum open systems.For instance,it is already used in the study of the thermalization,heat statistic,heat transfer and optimal control of the quantum open systems,and conversely it can be also used to investigate the effect of the system on the environment.In addition,the TEMPO method is naturally related to the process tensor,and can be used to calculate the correlation function of the system efficiently.In this article we review this method and its applications.We give a brief introduction of the path integral formalism of Caldeira-Leggett model.According to the path integral formalism,we demonstrate the usage of quasi-adiabatic propagator path integral method.we give the basic idea of matrix product states,and we show how to recast quasi-adiabatic propagator path integral method into time-evolving matrix product operators method by employing the concept of matrix product states and matrix product operators,and give a review of its applications.In addition,we use the calculation results of physical quantities,correlation functions and heat currents in the spin-boson model to illustrate the applications of the time-evolving matrix product operator method.