We also test the performance of adaptive algorithms under different convergence criteria as defined by Eqs. (7) and (8). The convergence threshold of ε m = 10 -m is set and is denoted by m. For instance, (AR-)ADAPT(m) indicates that the algorithm will stop when the gradient norm of all operators is less than ε m , and here m = 1, 2 and 3 are checked. (AR-)AES-VQE(m) indicates that the calculation will stop when the energy drop of all operators is less than ε m , m = 4 and 5 are checked in this work. Fig. shows the simulated results of three small molecules by using (AR-)ADAPT-VQE(m) and (AR-)AES-VQE(m) with different value of m. The result comparisons between five-pairs of AAs are made, but to make the figures more concise and clear, we only display three of them, i.e. (AR-)ADAPT( ), (AR-)ADAPT( ) and (AR-)AES (5). The results with all values of m are shown in Section F of SI.