Quantum simulation in the semi-classical regime

Quantum 6, 739 (2022).

https://doi.org/10.22331/q-2022-06-17-739

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 $hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a 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 $hbar$ 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 $mathcal{O}Big({ (varepsilon hbar)^{-frac12} mathrm{polylog}(varepsilon^{-frac{3}{2ell}} hbar^{-1-frac{1}{2ell}})}Big),$ 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 $mathcal{O}Big({varepsilon^{-frac12} mathrm{polylog}( varepsilon^{-frac3{2ell}} hbar^{-1} )}Big),$ with $ell$ again indicating the smoothness of the solution.