Variational quantum simulation: a case study for understanding warm
  starts
Ricard Puig i Valls Marc Drudis Supanut Thanasilp Zoë Holmes
The barren plateau phenomenon, characterized by loss gradients that vanish
exponentially with system size, poses a challenge to scaling variational
quantum algorithms. Here we explore the potential of warm starts, whereby one
initializes closer to a solution in the hope of enjoying larger loss variances.
Focusing on an iterative variational method for learning shorter-depth circuits
for quantum real and imaginary time evolution we conduct a case study to
elucidate the potential and limitations of warm starts. We start by proving
that the iterative variational algorithm will exhibit substantial (at worst
vanishing polynomially in system size) gradients in a small region around the
initializations at each time-step. Convexity guarantees for these regions are
then established, suggesting trainability for polynomial size time-steps.
However, our study highlights scenarios where a good minimum shifts outside the
region with trainability guarantees. Our analysis leaves open the question
whether such minima jumps necessitate optimization across barren plateau
landscapes or whether there exist gradient flows, i.e., fertile valleys away
from the plateau with substantial gradients, that allow for training.