  Well-controlled quantum devices with their increasing system size face a new
roadblock hindering further development of quantum technologies: The effort of
quantum tomography---the characterization of processes and states within a
quantum device---scales unfavorably to the point that state-of-the-art systems
can no longer be treated. Quantum compressed sensing mitigates this problem by
reconstructing the state from an incomplete set of observables. In this work,
we present an experimental implementation of compressed tomography of a seven
qubit system---the largest-scale realization to date---and we introduce new
numerical methods in order to scale the reconstruction to this dimension.
Originally, compressed sensing has been advocated for density matrices with few
non-zero eigenvalues. Here, we argue that the low-rank estimates provided by
compressed sensing can be appropriate even in the general case. The reason is
that statistical noise often allows only for the leading eigenvectors to be
reliably reconstructed: We find that the remaining eigenvectors behave in a way
consistent with a random matrix model that carries no information about the
true state. We report a reconstruction of quantum states from a topological
color code of seven qubits, prepared in a trapped ion architecture, based on
tomographically incomplete data involving 127 Pauli basis measurement settings
only, repeated 100 times each.
