We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rd log^2 d) measurement settings, compared to standard methods that require d^2 settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low-rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed. We present both theoretical bounds and numerical simulations.
@article{arxiv.0909.3304,
title = {Quantum state tomography via compressed sensing},
author = {David Gross and Yi-Kai Liu and Steven T. Flammia and Stephen Becker and Jens Eisert},
journal= {arXiv preprint arXiv:0909.3304},
year = {2015}
}