We demonstrate a practical application of quantum computing by using it to investigate the linear H4 molecule as a simple model for singlet fission. We use the Peeters-Devreese-Soldatov energy functional to calculate the necessary energetics based on the moments of the Hamiltonian estimated on the quantum computer. To reduce the number of required measurements, we use several independent strategies: 1) reduction of the size of the relevant Hilbert space by tapering off qubits; 2) measurement optimization via rotations to eigenbases shared by groups of qubit-wise commuting Pauli strings; and 3) parallel execution of multiple state preparation and measurement operations using all 20 qubits available on the Quantinuum H1-1 quantum hardware. Our results meet the energetic requirements for singlet fission, are in excellent agreement with exact transition energies (for the chosen one-particle basis), and outperform classical methods considered computationally feasible for singlet fission candidates.