  We assess the use of variational quantum imaginary time evolution for solving
partial differential equations. Our results demonstrate that real-amplitude
ansaetze with full circular entangling layers lead to higher-fidelity solutions
compared to those with partial or linear entangling layers. To efficiently
encode impulse functions, we propose a graphical mapping technique for quantum
states that often requires only a single bit-flip of a parametric gate. As a
proof of concept, we simulate colloidal deposition on a planar wall by solving
the Smoluchowski equation including the Derjaguin-Landau-Verwey-Overbeek (DLVO)
potential energy. We find that over-parameterization is necessary to satisfy
certain boundary conditions and that higher-order time-stepping can effectively
reduce norm errors. Together, our work highlights the potential of variational
quantum simulation for solving partial differential equations using near-term
quantum devices.
