  The quantum analogue of ptychography, a powerful coherent diffractive imaging
technique, is a simple method for reconstructing $d$-dimensional pure states.
It relies on measuring partially overlapping parts of the input state in a
single orthonormal basis and feeding the outcomes to an iterative
phase-retrieval algorithm for postprocessing. We provide a proof of concept
demonstration of this method by determining pure states given by superpositions
of $d$ transverse spatial modes of an optical field. A set of $n$ rank-$r$
projectors, diagonal in the spatial mode basis, is used to generate $n$
partially overlapping parts of the input and each part is projectively measured
in the Fourier transformed basis. For $d$ up to 32, we successfully
reconstructed hundreds of random states using $n=5$ and $n=d$ rank-$\lceil
d/2\rceil$ projectors. The extension of quantum ptychography for other types of
photonic spatial modes is outlined.
