We introduce a completely different method to calculate the evolution of a spin interacting with a sufficient large spin bath,especially suitable for treating the central spin model in a quantum dot(QD).With only an approximation on the envelope of central spin,the symmetry can be exploited to reduce a huge Hilbert space which cannot be calculated with computers to many small ones which can be solved exactly.This method can be used to calculate spin-bath evolution for a spin bath containing many(say,1000)spins,without a perturbative limit such as strong magnetic field condition,and works for long-time regime with sufficient accuracy.As the spin-bath evolution can be calculated for a wide range of time and magnetic field,an optimal dynamic of spin flip-flop can be found,and more sophisticated approaches to achieve extremely high polarization of nuclear spins in a QD could be developed.
Stochastic optimization has established itself as a major method to handle uncertainty in various optimization problems by modeling the uncertainty by a probability distribution over possible realizations.Traditionally,the main focus in stochastic optimization has been various stochastic mathematical programming(such as linear programming,convex programming).In recent years,there has been a surge of interest in stochastic combinatorial optimization problems from the theoretical computer science community.In this article,we survey some of the recent results on various stochastic versions of classical combinatorial optimization problems.Since most problems in this domain are NP-hard(or#P-hard,or even PSPACE-hard),we focus on the results which provide polynomial time approximation algorithms with provable approximation guarantees.Our discussions are centered around a few representative problems,such as stochastic knapsack,stochastic matching,multi-armed bandit etc.We use these examples to introduce several popular stochastic models,such as the fixed-set model,2-stage stochastic optimization model,stochastic adaptive probing model etc,as well as some useful techniques for designing approximation algorithms for stochastic combinatorial optimization problems,including the linear programming relaxation approach,boosted sampling,content resolution schemes,Poisson approximation etc.We also provide some open research questions along the way.Our purpose is to provide readers a quick glimpse to the models,problems,and techniques in this area,and hopefully inspire new contributions.
