Speeding Up Quantum Simulation With Coarse-grained Tight Binding Model And Optimized Basis

In quantum simulation,it is always a challenge to calculate multi-component complex systems at different scales,especially when there are bound states extending over long distances in the system.Considering a quantum well system as an example,close to the quantum well the wave function fluctuates a lot,which require atomic-accuracy simulations,but far from the quantum well the wave function has less fluctuations and does not need an atomistic description.The quantum state may extend over long distances.In order to avoid the finite size effect,we need to use a very large computational cell.Although the classical tight-binding algorithm can achieve atomistic precision,the computational burden increases rapidly with the size of the system.In order to solve the problem we propose two types of algorithms that optimize the expansion basis under the tight-binding framework.Their brief contents are as follows:(1)We divide the entire system uniformly or non-uniformly into blocks consisting of several unit cells.Each blocks relates to particular expansion basis and gather the expansion basis of all blocks we could get the whole expansion basis of the system.According to the local details of blocks we could find proper specific optimized basis for each blocks and find ways to reduce the number of expansion basis of the whole system.The dimension of the Hamiltonian of the whole system will be reduced accordingly,so that we can save a lot of time from the diagonalization of reduced Hamiltonian.(2)We utilize the fact that the two types of expansion coefficients of a same quantum state under the primitive cell basis and the supercell basis could convert with each other through a relation based on the Bloch theorem.Taking account the Brillouin zone folding mechanism we convert several primitive cell states derived by diagonalization to a set of states that fulfill the form of eigen states of the supercell.We could also find ways to manipulate these states and reduce its size to increase computational speed with the derived states as new expansion basis.The effective Hamiltonian can still describe the system well under the new expansion basis.In this thesis we simulate the GaAs/AlAs quantum well system and other band structure related things as an example to show that our proposed algorithms have a similar prediction ability with classical algorithms and also in advantage of the calculation speed.Applying the methods,we successfully reproduce the shift of band edge energies with an increasing well width,as well as density of states and dielectric function of the quantum well.The algorithms are also promising in the calculation of effective mass and evanescent states.The algorithm in this thesis is written and implemented in C++and have parallel computing ability.The algorithms can still maintain the speed improvement compared with the traditional method under parallel computing.