Expressibility and Trainability Analysis of Hardware-Efficient Ansatz Variants in Variational Quantum Eigensolver with a Linear Mixing Model

This study analyzes the expressibility characteristics and gradient distribution in the variational quantum eigensolver (VQE) algorithm using nine variations of hardware-efficient ansatz (HEA) in the hydrogen molecular system. The evaluation was conducted by measuring Kullback-Leibler divergence (DKL) value as an indicator of expressibility ansatz and the gradient variance as an indicator of the barren plateau phenomenon. The system was optimized using the L-BFGS algorithm, and each ansatz was tested to identify the extent to its parametric structure affected optimization performance and the potential for barren plateaus. The analysis results show differences among the nine ansatzes. Variations such as SRy, TRy, RyRx, and RyRz tend to produce high expressibility but are followed by a drastic decrease in gradient variance, thus showing symptoms of a barren plateau. Conversely, more complex ansatzes involving the mixing of rotation bases, such as HRy, Ry, and XRy provide a more stable gradient distribution and more consistent optimization performance. These findings are consistent with the proposed linear mixing expressibility-trainability model, states that increased expressibility is not always directly proportional to the ease of the optimization process; homogenization of the state space actually reduces the directional information needed by the optimizer. These results provide an important basis for selecting more effective ansatzes in future implementations of variational quantum algorithms.