ABSTRACT The fine physical details for the quantum computation can easily be provided by high-order conservative schemes. By compensating the high-order difference operator to the central difference operator δ, a time-dependent semi-discrete system, which conserves both charge and energy, is derived for the two-dimensional nonlinear Dirac equation (NLDE). Two kinds of fully discrete schemes are obtained by discretizing this semi-discrete system in time with the time-midpoint method and the time-splitting method. We prove theoretically that the former one conserves both charge and energy while the latter one only keeps the charge conservation. Some numerical experiments are given to verify the accuracy order, the stability and the conservative properties. In addition, the dynamics of the NLDE in one dimension and two dimensions are simulated.

中文翻译：

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非线性狄拉克方程的高阶保守格式

摘要 量子计算的精细物理细节可以很容易地由高阶保守方案提供。通过将高阶差分算子补偿到中心差分算子δ，推导出了二维非线性狄拉克方程（NLDE）的电荷和能量守恒的瞬态半离散系统。用时间中点法和时间分裂法对该半离散系统进行时间离散，得到两种全离散方案。我们从理论上证明，前者既守恒电荷又守恒能量，而后者只守恒电荷守恒。给出了一些数值实验来验证精度阶数、稳定性和保守性。此外，模拟了一维和二维的NLDE动力学。