Various methods have been explored to prepare the spin-adapted ground state—the lowest energy state within the Hilbert space constrained by externally specified values of the total spin magnitude and the spin-$z$ component. In such problem settings, nonvariational methods incorporate penalty terms into the original Hamiltonian to enforce the desired constraints. Such additional terms would require a quartic number of gates in terms of the number $n_{\text{spin}}$ of spins when naively implemented. To circumvent this computationally intensive scaling, this paper proposes a nonvariational procedure to obtain the spin-adapted ground states that requires only quadratic scaling. The proposed method consists of two stages: the first stage is to prepare a spin-magnitude adapted state and the second stage is postprocessing for the desired $S_z$. By separating into two stages, the procedure achieves the desired spin-adapted ground state while reducing the number of penalty terms from $O\left(n_{\text{spin}}^4\right)$ to $O\left(n_{\text{spin}}^2\right)$. We conducted numerical experiments for spin-1/2 Heisenberg ring models and manganese trimer systems. The results confirmed the effectiveness of our method, demonstrating a significant reduction in gate complexity and validating its practical usefulness.

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• Quantum algorithms & computation
• Quantum algorithms for chemical calculations
• Quantum computation
• Quantum simulation