  The expressibility of an ansatz used in a variational quantum algorithm is
defined as the uniformity with which it can explore the space of unitary
matrices. The expressibility of a particular ansatz has a well-defined upper
bound. In this work, we show that the expressibiliity also has a well-defined
lower bound in the hypothesis space. We provide an analytical expression for
the lower bound of the covering number, which is directly related to
expressibility. We also perform numerical simulations to to support our claim.
To numerically calculate the bond length of a diatomic molecule, we take
hydrogen ($H_2$) as a prototype system and calculate the error in the energy
for the equilibrium energy point for different ansatzes. We study the variation
of energy error with circuit depths and show that in each ansatz template, a
plateau exists for a range of circuit depths, which we call the set of
acceptable points, and the corresponding expressibility is known as the best
expressive region. We report that the width of this best expressive region in
the hypothesis space is inversely proportional to the average error. Our
analysis reveals that alongside trainability, the lower bound of expressibility
also plays a crucial role in selecting variational quantum ansatzes.
