  In this work, we develop a highly efficient representation of functions and
differential operators based on Fourier analysis. Using this representation, we
create a variational hybrid quantum algorithm to solve static,
Schr\"odinger-type, Hamiltonian partial differential equations (PDEs), using
space-efficient variational circuits, including the symmetries of the problem,
and global and gradient-based optimizers. We use this algorithm to benchmark
the performance of the representation techniques by means of the computation of
the ground state in three PDEs, i.e., the one-dimensional quantum harmonic
oscillator, and the transmon and flux qubits, studying how they would perform
in ideal and near-term quantum computers. With the Fourier methods developed
here, we obtain low infidelities of order $10^{-4}-10^{-5}$ using only three to
four qubits, demonstrating the high compression of information in a quantum
computer. Practical fidelities are limited by the noise and the errors of the
evaluation of the cost function in real computers, but they can also be
improved through error mitigation techniques.
