A method for obtaining the WKB wave function describing the particle tunneling outside of a two-dimensional potential well is suggested. The Cartesian coordinates
(x,y) are chosen in such a way that the
x axis has the direction of the probability flux at large distances from the well. The WKB wave function is then obtained by simultaneous expansion of the wave function in the coordinate
y and the parameter determining the curvature of the escape path. It is argued, both physically and mathematically, that these two expansions are mutually consistent. It is shown that the method provides systematic approximation to the outgoing probability flux. Both the technical and conceptual advantages of this approach in comparison with the usual approach based on the solution of classical equations of motion are pointed out. The method is applied to the problem of the coupled anharmonic oscillators and verified through the dispersion relations.
The scattered wave formalism is developed for a quantum subsystem interacting with the external environment through open boundaries. The total wave function is divided into incident and scattered components and Markovian outgoing wave boundary conditions are applied to the scattered wave function. This formalism significantly reduces the computational effort relative to other methods which rely on Green functions and memory kernels. The method is applied to one-dimensional barrier scattering and to a three-dimensional model for the field effect transistor.
Quantum chemistry has been identified as one of the most promising areas where quantum computing can have a tremendous impact. For current noisy intermediate-scale quantum (NISQ) devices, one of the best available methods to prepare approximate wave functions on quantum computers is the adaptive derivative-assembled pseudo-Trotter Ansatz variational quantum eigensolver (ADAPT-VQE). However, ADAPT-VQE suffers from a significant measurement overhead when estimating the importance of operators in the wave function. In this work, we propose fermionic adaptive sampling theory VQE (FAST-VQE), a method for selecting operators based on importance metrics solely derived from the populations of Slater determinants in the wave function. Thus, our method mitigates measurement overheads for ADAPT-VQE as it is only dependent on the populations of Slater determinants which can simply be determined by measurements in the computational basis. We introduce two heuristic importance metrics, one based on selected configuration interaction with perturbation theory and one based on approximate gradients. In state vector and finite shot simulations, FAST-VQE using the heuristic metric based on approximate gradients converges at the same rate or faster than ADAPT-VQE and requires dramatically fewer shots.