  The time-dependent one-dimensional nonlinear Schr\"odinger equation (NLSE) is
solved numerically by a hybrid pseudospectral-variational quantum algorithm
that connects a pseudospectral step for the Hamiltonian term with a variational
step for the nonlinear term. The Hamiltonian term is treated as an integrating
factor by forward and backward Fourier transformations, which are here carried
out classically. This split allows us to avoid higher-order time integration
schemes, to apply a first-order explicit time stepping for the remaining
nonlinear NLSE term in a variational algorithm block, and thus to avoid
numerical instabilities. We demonstrate that the analytical solution is
reproduced with a small root mean square error for a long time interval over
which a nonlinear soliton propagates significantly forward in space while
keeping its shape. We analyze the accuracy of the quantum algorithm and compare
it with classical approaches. Furthermore, we investigate the influence of
algorithm parameters on the accuracy of the results, including the temporal
step width and the depth of the quantum circuit.
