  Variational quantum algorithms (VQAs) are a broad class of algorithms with
many applications in science and industry. Applying a VQA to a problem involves
optimizing a parameterized quantum circuit by maximizing or minimizing a cost
function. A particular challenge associated with VQAs is understanding the
properties of associated cost functions. Having the landscapes of VQA cost
functions can greatly assist in developing and testing new variational quantum
algorithms, but they are extremely expensive to compute. Reconstructing the
landscape of a VQA using existing techniques requires a large number of cost
function evaluations, especially when the dimension or the resolution of the
landscape is high. To address this challenge, we propose a low-rank
tensor-completion-based approach for local landscape reconstruction. By
leveraging compact low-rank representations of tensors, our technique can
overcome the curse of dimensionality and handle high-resolution landscapes. We
demonstrate the power of landscapes in VQA development by showcasing practical
applications of analyzing penalty terms for constrained optimization problems
and examining the probability landscapes of certain basis states.
