Quantum process tomography of unitary and near-unitary maps

We study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a d-level system is completely characterized by a minimal set of d(2)+d elements associated with a collection of POVMs, in contrast to the d(4)-d(2) elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with particular sets of d pure states. We further compare the performance of different estimators inspired by compressed sensing, to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Convex optimization inspired by the techniques of compressed sensing can therefore be used both as indicators of error models and to validate the use of the prior assumptions.