  Several proposals aiming to demonstrate quantum advantage on near-term
quantum computers rely on the optimization of variational circuits. These
approaches include, for example, variational quantum eigensolvers and many-body
quantum simulators and their realization with limited computational techniques
critically depends on the development of efficient optimization techniques. In
this paper, we introduce a new optimization strategy for dense quantum
circuits, leveraging tensor network optimization principles. Our approach
focuses on optimizing one gate at a time by fully characterizing the dependency
of the cost function on the gate through environment tensor tomography,
obtained via noisy measurements on a quantum device. We compute the minimal
number of measurements needed to perform a full tensor tomography and relate
this number to unitary 2-design. We then describe a general framework for
landscape tomography based on linear regression and compare two different
implementations based on shadow tomography and Clifford tableaux, respectively.
Finally, we compare our strategy with both gradient-free optimization and
gradient-based optimization based on the parameter-shift rule, highlighting
potential benefits of our algorithm for the development of quantum algorithms
in noisy devices.
