  Compressive sensing is a sensing protocol that facilitates reconstruction of
large signals from relatively few measurements by exploiting known structures
of signals of interest, typically manifested as signal sparsity. Compressive
sensing's vast repertoire of applications in areas such as communications and
image reconstruction stems from the traditional approach of utilizing
non-linear optimization to exploit the sparsity assumption by selecting the
lowest-weight (i.e. maximum sparsity) signal consistent with all acquired
measurements. Recent efforts in the literature consider instead a data-driven
approach, training tensor networks to learn the structure of signals of
interest. The trained tensor network is updated to "project" its state onto one
consistent with the measurements taken, and is then sampled site by site to
"guess" the original signal. In this paper, we take advantage of this computing
protocol by formulating an alternative "quantum" protocol, in which the state
of the tensor network is a quantum state over a set of entangled qubits.
Accordingly, we present the associated algorithms and quantum circuits required
to implement the training, projection, and sampling steps on a quantum
computer. We supplement our theoretical results by simulating the proposed
circuits with a small, qualitative model of LIDAR imaging of earth forests. Our
results indicate that a quantum, data-driven approach to compressive sensing,
may have significant promise as quantum technology continues to make new leaps.
