![]() ![]() The Monte Carlo sampling of these diagrams will again contribute to the stochastic error, when a large m is needed. Even after such reductions, however, as the number of points in the time sequence m increases, the total number of diagrams still grows as a double factorial O(( m − 1)!!). This approach is further improved in by writing the evolution of the bold lines as an integro-differential equation, which only requires to sum over 'linked' diagrams, so that the computational cost can be further reduced. It introduces bold lines as partial resummations of bare dQMC, so that the total number of diagrams can be reduced, and the sign problem is hence suppressed. Recently, the inchworm Monte Carlo method was proposed to mitigate the numerical sign problem. ![]() However, such technique may encounter the notorious numerical sign problem, meaning that the number of Monte Carlo samples is required to grow at least exponentially (with respect to physical time) in order to keep the accuracy of the simulation. While these deterministic methods require some additional modeling of the open quantum system, the bare diagrammatic quantum Monte Carlo (dQMC) method applies Monte Carlo sampling to directly compute the summations and high-dimensional integrals in the Dyson series expansion of the quantum observable, and after applying Wick's theorem, this approach can be represented as the summation of all possible diagrams, each of which is determined by a finite time sequences and a partition of them into pairs. ![]() For example, the quasi-adiabatic propagator path integral method assumes finite memory length and so that the path integral can be numerically computed iteratively by assuming that the bath response function has a special form, the hierarchical equations of motion can be applied the method of multiconfiguration time dependent Hartree is developed based on ansatz of wave functions. In the more challenging case where memory effect has to be taken into account, a number of numerical methods have been proposed in the literature. In the weak coupling limit, such evolution can be approximated by the Markovian process described by the Lindblad equation, which simplifies the numerical simulation. The coupling between the system and the environment leads to non-Markovian evolution of the quantum state of the system. Open quantum systems, which characterize quantum systems coupled with environment, have been studied extensively for many decades, as it arises in many context including quantum optics, quantum computation, and dynamical mean field theory, just to list a few. ![]()
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