Quantum simulation in the semi-classical regime
Shi Jin, Xiantao Li, Nana Liu · Quantum · 2022
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 ℏ , 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 ℏ and the precision ε are obtained. It is found that the number of required qubits, m , scales only logarithmically with respect to ℏ . When the solution has bounded derivatives up to order ℓ , the symmetric Trotting method has gate complexity O ( ( ε ℏ ) − 1 2 p o l y l o g ( ε − 3 2 ℓ ℏ − 1 − 1 2 ℓ ) ) , provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with p o l y ( m ) operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of ℏ . The gate complexity in this case is reduced to O ( ε − 1 2 p o l y l o g ( ε − 3 2 ℓ ℏ − 1 ) ) , with ℓ again indicating the smoothness of the solution.