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I am trying to reproduce this paper: https://arxiv.org/pdf/2211.02302.pdf. However, I am having problems in getting the singlet first excited state (S1). The authors in the paper, they use the SSVQE algorithm with a penalty term in the cost function to obtain the S1 of the H2 molecule. I am using Qulacs (as in the paper) and the same parameters for the cost function. However, when using the SSVQE with the penalty term, I get all the time the ground state and the triplet first excited state (S0 and T1), instead of getting S0 and S1 as in the paper. Can someone help me with that? I attach my cost function:

def cost(theta_list):

    mu = 10**5
    s0=0
    s1=0
    gamma = 0.6

    state0 = QuantumState(n_qubit) 
    state1 = QuantumState(n_qubit); state1.set_computational_basis(1) 

    cost_function = get_exp(state0, theta_list) + gamma *get_exp(state1, theta_list) + mu*abs(get_exp_s2(state0, theta_list)-s0*(s0+1) + (get_exp_s2(state1, theta_list)-s1*(s1+1)))**2 

    return cost_function

I have also compared my observables (H and S²) with the ones in the paper and they are exactly the same, so I am guessing the problem is in how I implement the cost function with the penalty term... Thank you a lot for the help!

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