Efficient quantum simulation of unitary gates based on atomic ensembles

We develop a technique to perform efficient unitary time evolution of a general spin Hamiltonian based on the system with identical qubits, as naturally realized by permutation symmetric atomic ensembles. We employ Cartan decomposition method to the associated Hamiltonian Lie algebra and construct the symmetry adapted generators to implement unitary operator of the ensemble Hamiltonian. This enables the compact circuit representations of the system dynamics in terms of experimentally accessible collective spin operations and exploits the Dicke basis structure of the ensembles as an effective qudit representation. We benchmark Cartan based evolution against both exact time evolution and the Suzuki Trotter product formula. Our results show that the Cartan protocol accurately reproduces the operator dynamics, spin expectation values, and fluctuation observables with zero error that are substantially smaller than those obtained through Trotterization while maintaining an optimized circuit structure. In addition, scaling analysis with the ensemble size demonstrates that the Cartan error remains independent of the atom number while the corresponding Trotter error increases with the system size, highlighting the improved efficiency of the Cartan approach for a larger system size.