Simplified projection on total spin zero for state preparation on quantum computers

We introduce a simple algorithm for projecting on $J=0$ states of a many-body system by performing a series of rotations to remove states with angular momentum projections greater than zero. Existing methods rely on unitary evolution with the two-body operator ${J}^{2}$, which when expressed in the computational basis contains many complicated Pauli strings requiring Trotterization and leading to very deep quantum circuits. Our approach performs the necessary projections using the one-body operators ${J}_{x}$ and ${J}_{z}$. By leveraging the method of Cartan decomposition, the unitary transformations that perform the projection can be parametrized as a product of a small number of two-qubit rotations, with angles determined by an efficient classical optimization. Given the reduced complexity in terms of gates, this approach can be used to prepare approximate ground states of even-even nuclei by projecting onto the $J=0$ component of deformed Hartree-Fock states. We estimate the resource requirements in terms of the universal gate set ${H,S,\mathrm{CNOT},T}$ and briefly discuss a variant of the algorithm that projects onto $J=1/2$ states of a system with an odd number of fermions.