论文标题
半经典schrödinger方程的Trotter公式的均匀观察误差界限
Uniform observable error bounds of Trotter formulae for the semiclassical Schrödinger equation
论文作者
论文摘要
在最坏的情况下，汉密尔顿进化的模拟时间必须为$ o（\ | h \ | t）$，在量子计算中被称为$ o（\ | h \ \ | t）$，在量子计算中被称为快速定理，在最坏的情况下，这基本上表明，由于汉密尔顿进化的仿真时间必须比物理时间更大。我们在半经典的schrödinger方程的背景下证明了一类可观察的计算成本可能要比最新的界限要低得多。在半经典制度（有效的普朗克常数$ h \ ll 1 $）中，汉密尔顿的运营商标准为$ O（h^{ - 1}）$。我们表明，可观察到的进化的猪猪步骤的数量可以为$ o（1）$，也就是说，在量子标度上模拟Schrödinger方程的某些可观察到的东西只需要与经典量表相当的模拟时间。在错误分析方面，我们将可观察到的误差界[Lasser-Lubich 2020]提高到了均匀的$ H $可观察到的误差界。据我们所知，这是半经典schrödinger方程的第一个均匀观察误差，而无需牺牲数值方法的收敛顺序。基于半经典的微积分和离散的微局部分析，我们的结果展示了利用多尺度特性的潜在改进，例如有效planck常数的较小性，基础动力学的较小性，并阐明了跨越规模的量子动力学模拟。
Known as no fast-forwarding theorem in quantum computing, the simulation time for the Hamiltonian evolution needs to be $O(\|H\| t)$ in the worst case, which essentially states that one can not go across the multiple scales as the simulation time for the Hamiltonian evolution needs to be strictly greater than the physical time. We demonstrated in the context of the semiclassical Schrödinger equation that the computational cost for a class of observables can be much lower than the state-of-the-art bounds. In the semiclassical regime (the effective Planck constant $h \ll 1$), the operator norm of the Hamiltonian is $O(h^{-1})$. We show that the number of Trotter steps used for the observable evolution can be $O(1)$, that is, to simulate some observables of the Schrödinger equation on a quantum scale only takes the simulation time comparable to the classical scale. In terms of error analysis, we improve the additive observable error bounds [Lasser-Lubich 2020] to uniform-in-$h$ observable error bounds. This is, to our knowledge, the first uniform observable error bound for semiclassical Schrödinger equation without sacrificing the convergence order of the numerical method. Based on semiclassical calculus and discrete microlocal analysis, our result showcases the potential improvements taking advantage of multiscale properties, such as the smallness of the effective Planck constant, of the underlying dynamics and sheds light on going across the scale for quantum dynamics simulation.
