  Digital quantum simulation (DQS) of continuous-variable quantum systems in
the position basis requires efficient implementation of diagonal unitaries
approximating the time evolution operator generated by the potential energy
function. In this work, we provide efficient implementations suitable for
potential functions approximable by piecewise polynomials, with either uniform
or adaptively chosen subdomains. For a fixed precision of approximation, we
show how adaptive grids can significantly reduce the total gate count at the
cost of introducing a small number of ancillary qubits. We demonstrate the
circuit construction with both physically motivated and artificially designed
potential functions, and discuss their generalizations to higher dimensions.
