  The entanglement-induced barren plateau phenomenon is an exponential
vanishing of the parameter gradients with system size that limits the use of
variational quantum algorithms(VQA). Recently, it was observed that a landscape
transition from a barren plateau to no barren plateau occurs if the volume-law
growth of entanglement is suppressed by adding measurements with
post-selection. This suppression appears to coincide with a measurement-induced
phase transition (MIPT) that measurements are known to cause in monitored
circuits. From an information theory perspective, we argue that these are
different transitions. We back this hypothesis with a numerical study of the
cost landscape of such hybrid variational quantum circuits with extensive
results on the behavior of cost-gradient variances with and without
post-selection, direct visualizations of optimization runs for specific local
quantum circuits, and a mutual information measure we introduce and compare
with entanglement measures used in the study of MIPT. Specifically, our results
show there are two transitions, a measurement-induced landscape transition
(MILT) that seems universal across different VQA ansatzes and appears at a
lower probability of measurements, and the MIPT that appears at a higher
probability of measurements and appears at an ansatz specific location.
Finally, to reap the benefits of MILT for optimization, our numerical
simulations suggest the necessity of post-selecting measurement outcomes.
