Quantum simulation 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 where ~ 1 and the solutions exhibit strong multiple-scale behavior due to a small parameter, 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 applications in many fundamental problems in quantum chemistry, including those from Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. However, the presence of the small parameter ~ in these applications compromises both the spatial resolution and the time integration accuracy. Therefore, to assess a Hamiltonian simulation algorithm in this regime, it is important to quantify the complexity in terms of both ~ and an error tolerance ε. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of ~ and the precision ε are obtained. It is found that the number of required qubits, m, scales only logarithmically with respect to ~. When the solution has bounded derivatives up to order `, the symmetric Trottering method has gate complexity (Formula presented), 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 ~. The gate complexity in this case is reduced to (Formula presented), with ` again indicating the smoothness of the solution.