On quantum algorithms for the Schrödinger equation in the semi-classical regime

Abstract

Solving the time-dependent Schrödinger equation is an important application area for quantum algorithms. We consider Schrödinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hslash$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schrödinger 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 $\hslash$ and the precision $\epsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hslash$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}((\epsilon \hslash {)}^{-\frac{1}{2}}\mathrm{polylog}({\epsilon }^{-\frac{3}{2\ell }}{\hslash }^{-1-\frac{1}{2\ell }})),$ 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 $\hslash$. The gate complexity in this case is reduced to $\mathcal{O}({\epsilon }^{-\frac{1}{2}}\mathrm{polylog}({\epsilon }^{-\frac{3}{2\ell }}{\hslash }^{-1})),$ with $\ell$ again indicating the smoothness of the solution.

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Cited by in corpus (5)

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