The method of increments and frozen natural orbital (MI-FNO) framework is introduced to help expedite the application of noisy, intermediate-scale quantum~(NISQ) devices for quantum chemistry simulations. The MI-FNO framework provides a systematic reduction of the occupied and virtual orbital spaces for quantum chemistry simulations. The correlation energies of the resulting increments from the MI-FNO reduction can then be solved by various algorithms, including quantum algorithms such as the phase estimation algorithm and the variational quantum eigensolver (VQE). The unitary coupled-cluster singles and doubles VQE framework is used to obtain correlation energies for the case of small molecules (i.e., BeH$_2$, CH$_4$, NH$_3$, H$_2$O, and HF) using the cc-pVDZ basis set. The quantum resource requirements are estimated for a constrained geometry complex (CGC) catalyst that is utilized in industrial settings for the polymerization of $alpha$-olefins. We show that the MI-FNO approach provides a significant reduction in the qubit requirements relative to the full system simulations. We propose that the MI-FNO framework can create scalable examples of quantum chemistry problems that are appropriate for assessing the progress of NISQ devices.
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