  We present a detailed numerical study of an alternative approach, named
Quantum Non-Demolition Measurement (QNDM), to efficiently estimate the
gradients or the Hessians of a quantum observable. This is a key step and a
resource-demanding task when we want to minimize the cost function associated
with a quantum observable. In our detailed analysis, we account for all the
resources needed to implement the QNDM approach with a fixed accuracy and
compare them to the current state-of-the-art method. We find that the QNDM
approach is more efficient, i.e. it needs fewer resources, in evaluating the
derivatives of a cost function.These advantages are already clear in small
dimensional systems and are likely to increase for practical implementations
and more realistic situations. Since the vast majority of the Variational
Quantum Algorithms can be formulated in the discussed framework, our results
can have significant implications in quantum optimization algorithms and make
the QNDM approach a valuable alternative to implement Variational Quantum
Algorithms on near-term quantum computers.
