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

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 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 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\({varepsilon^-frac12 polylog( varepsilon^{-frac3{2ell}} hbar^-1\)}Big), with ell again indicating the smoothness of the solution.