Solving the time-dependent Schr\"odinger equation is an important application area for quantum algorithms. We consider Schr\"odinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schr\"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $\hbar$ and the precision $\varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hbar$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $\mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $\hbar$. The gate complexity in this case is reduced to $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.
翻译:解析基于时间的 Schr\ odinger 方程式是量子算法的一个重要应用区域 。 我们考虑在半古典体系中Schr\ “ odinger 方程式 ” 。 在这里, 解决方案显示由于一个小参数$\ hbar$而具有强烈的多重行为。 也就是说, 量子状态的动态和引致的可观察性可以在不同的空间和时间尺度上发生。 这种“ schr\” 方程式会发现许多应用程序, 包括在 Born- Oppenheimer 分子动态和 Ehrenfest 动态中。 本文考虑了古典计算机中伪光谱$$( PS) 方法的量类比值。 以$\ hbar$\ hbar$为单位的门数 $\\\\\\\ mar\\\ r\\\\ m\ m\ m\ listal=lational rexcial_\\\\ r\\\\\\\\\\\ ral remax ral ral r= r@ r= r= maxxxxxxxxxxxxxxm=xxxxxxxxxxxxxxxxxxxx======= =========== =========================================xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx