Quantum process tomography of near-unitary maps

We study the problem of quantum process tomography given the prior information that the implemented map is near to a unitary map on a $d$-dimensional Hilbert space. In particular, we show that a perfect unitary map is completely characterized by a minimum of $d^2 + d$ measurement outcomes. This contrasts with the $d^4$ measurement outcomes required in general. To achieve this lower bound, one must probe the system with a particular set of $d$ states in a particular order. This order exploits unitarity but does not assume any other structure of the map. We further consider the more general case of noisy quantum maps, with a low level of noise. Our study indicates that transforming to the interaction picture, where the noiseless map is represented by a diagonal operator, can provide a useful tool to identify the noise structure. This, in turn, can lead to a substantial reduction in the numerical resources needed to estimate the noisy map.