On quantum algorithms for the Schr\"odinger equation in the semi-classical regime

• URL: http://arxiv.org/abs/2112.13279v3
• Date: Sat, 4 Jun 2022 03:12:09 GMT
• Title: On quantum algorithms for the Schr\"odinger equation in the semi-classical regime
• Authors: Shi Jin, Xiantao Li, Nana Liu
• Abstract summary: We consider Schr"odinger's equation in the semi-classical regime. Such a Schr"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics.
• Score: 27.175719898694073
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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