The rapid progress in quantum computing has opened up new possibilities for tackling complex scientific problems. Variational quantum eigensolver (VQE) holds the potential to solve quantum chemistry problems and achieve quantum advantages. However, the measurement step within the VQE framework presents challenges. It can introduce noise and errors while estimating the objective function with a limited measurement budget. Such error can slow down or prevent the convergence of VQE. To reduce measurement error, many repeated measurements are needed to average out the noise in the objective function. By consolidating Hamiltonian terms into cliques, simultaneous measurements can be performed, reducing the overall measurement shot count. However, limited prior knowledge of each clique, such as noise level of measurement, poses a challenge. This work introduces two shot assignment strategies based on estimating the standard deviation of measurements to improve the convergence of VQE and reduce the required number of shots. These strategies specifically target two distinct scenarios: overallocated and underallocated shots. The efficacy of the optimized shot assignment strategy is demonstrated through numerical experiments conducted on a H$_2$ molecule. This research contributes to the advancement of VQE as a practical tool for solving quantum chemistry problems, paving the way for future applications in complex scientific simulations on quantum computers.
