Quantum simulation 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 parameterℏ, 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 ofℏand the precisionεare obtained. It is found that the number of required qubits,m, scales only logarithmically with respect toℏ. When the solution has bounded derivatives up to orderℓ, the symmetric Trotting method has gate complexityO((εℏ)−12polylog(ε−32ℓℏ−1−12ℓ)),provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented withpoly(m)operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently ofℏ. The gate complexity in this case is reduced toO(ε−12polylog(ε−32ℓℏ−1)),withℓagain indicating the smoothness of the solution.