Quantum Equation-of-Motion Method with Single, Double, and Triple Excitations

The quantum equation-of-motion (qEOM) method with single and double excitations (qEOM-SD) has been proposed to study electronically excited states, but it fails to handle states dominated by double excitations. In this work, we reformulate the qEOM method within the effective Hamiltonian framework that satisfies the killer condition, and then present an efficient implementation incorporating single, double, and triple excitations. To reduce computational complexity, we employ point-group symmetry and perturbation theory to screen triple excitations, effectively reducing the scaling from N_o⁶N_v⁶ to N_o⁵N_v⁵, where N_o and N_v are the numbers of occupied and virtual spin orbitals, respectively. Furthermore, we account for the effect of neglected triple excitations by introducing a perturbative correction to the excitation energy. We apply this method to challenging cases where the qEOM-SD method exhibits significant errors, such as the 2 ¹Δ state of CH⁺ and the 2 ¹Σ state of HF. Our new method achieves energy errors below 0.18 eV while incorporating less than 8.2% of triple excitations. Additionally, we extend the operator screening technique to the quantum subspace expansion method for the efficient inclusion of selected triple excitations.