Quantum simulation in the semi-classical regime

Solving the time-dependent Schr\"odinger equation is an important application area for quantum algorithms. We consider Schr\"odinger's in semi-classical regime. Here solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, sense that dynamics of states and induced observables can occur on different spatial temporal scales. Such finds many applications, including Born-Oppenheimer molecular Ehrenfest dynamics. This paper considers analogues pseudo-spectral (PS) methods classical computers. Estimates gate counts terms $\hbar$ precision $\varepsilon$ are obtained. It found number required qubits, $m$, scales only logarithmically with respect $\hbar$. When solution has bounded derivatives up order $\ell$, symmetric Trotting method complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided diagonal unitary operators be implemented $\mathrm{poly}(m)$ operations. physical desired outcomes, however, step size time integration chosen independently The this case reduced $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ $\ell$ again indicating smoothness solution.