  Quantum algorithms can potentially overcome the boundary of computationally
hard problems. One of the cornerstones in modern optics is the beam propagation
algorithm, facilitating the calculation of how waves with a particular
dispersion relation propagate in time and space. This algorithm solves the wave
propagation equation by Fourier transformation, multiplication with a transfer
function, and subsequent back transformation. This transfer function is
determined from the respective dispersion relation, which can often be expanded
as a polynomial. In the case of paraxial wave propagation in free space or
picosecond pulse propagation, this expansion can be truncated after the
quadratic term. The classical solution to the wave propagation requires
$\mathcal{O}(N log N)$ computation steps, where $N$ is the number of points
into which the wave function is discretized. Here, we show that the propagation
can be performed as a quantum algorithm with $\mathcal{O}((log{}N)^2)$
single-controlled phase gates, indicating exponentially reduced computational
complexity. We herein demonstrate this quantum beam propagation method (QBPM)
and perform such propagation in both one- and two-dimensional systems for the
double-slit experiment and Gaussian beam propagation. We highlight the
importance of the selection of suitable observables to retain the quantum
advantage in the face of the statistical nature of the quantum measurement
process, which leads to sampling errors that do not exist in classical
solutions.
