Lower bound of the expressibility of ansatzes for Variational Quantum Algorithms

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, i.e., its covering number. The expressibility of a particular ansatz has a well-defined upper bound [1]. In this work, we show that the expressibility 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. Further, we provide numerical analysis to support our claim. By calculating the bond length of hydrogen molecule ($H_2$) using different ansatzes in a variational quantum eigensolver (VQE) setting, we study the variation of equilibrium energy error with circuit depths. We 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 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