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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. 

Localization is one of the most interesting topics related to the promising millimeter wave (mmWave) technology. In this paper, we investigate joint channel estimation and localization for a cooperative mmWave system with several receivers. Due to the strong line-of-sight path common to mmWave channels, one can localize the position of the user by exploiting the signal's angle-of-arrival (AoA). Leveraging a variational Bayesian approach, we obtain soft information about the AoA for each receiver. We then use the soft AoA information and geometrical constraints to localize the position of the user and further improve the channel estimation performance. Numerical results show that the proposed algorithm has centimeter-level localization accuracy for an outdoor scene. In addition, the proposed algorithm provides 1-3 dB of gain for channel estimation by exploiting the correlation among the receiver channels depending on the availability of prior information about the path loss model.