Ptychography of pure quantum states

Ptychography is an imaging technique in which a localized illumination scans overlapping regions of an object and generates a set of diffraction intensities used to computationally reconstruct its complex-valued transmission function. We propose a quantum analogue of this technique designed to reconstruct d-dimensional pure states. A set of n rank-r projectors "scans" overlapping parts of an input state and the moduli of the d Fourier amplitudes of each part are measured. These nd outcomes are fed into an iterative phase retrieval algorithm that estimates the state. Using d up to 100 and r around d/2, we performed numerical simulations for single systems in an economic $n=4$ and a costly $n=d$ scenario, as well as for multiqubit systems $n=6log d$. This numeric study included realistic amounts of depolarization and poissonian noise, and all scenarios yielded, in general, reconstructions with infidelities below 10^-2. The method is shown, therefore, to be resilient to noise and, for any d, requires a simple and fast postprocessing algorithm. We show that the algorithm is equivalent to an alternating gradient search, which ensures that it does not suffer from local-minima stagnation. Unlike traditional approaches to state reconstruction, the ptychographic scheme uses a single measurement basis; the diversity and redundancy in the measured data---key for its success---are provided by the overlapping projections. We illustrate the simplicity of this scheme with the paradigmatic multiport interferometer.