Preparing Angular Momentum Eigenstates on Quantum Computers

Coupled angular momentum eigenstates |j,m, j₁, m₁, j₂, m₂> are widely used in atomic and nuclear physics calculations. In order to accelerate such calculations on a quantum computer, we investigate how to combine two angular momenta J₁ and J₂ to form total angular momentum J=J₁+J₂ eigenstates faster than standard Clebsch–Gordan/Racah methods. Starting from the ground state, simulated magnetic resonance gates U_b prepare states of fixed j. Then, simulated dipole transition gates U_d change the j value and prepare a superposition of |j,m> states. The success probability of preparing a specific |j,m> state can be controlled using parameters of the U_b and U_d gates. Since all eigenstates are needed for the target calculations, we use an ancilla register to record the prepared |j,m> eigenstate. Two entangling gates U_m and U_j transfer the m and j values from the computational basis to the ancilla register, which can be partially readout or retained during subsequent calculations. We experimentally demonstrate our state preparation scheme for the j₁=j₂=1/2 case using four levels in a superconducting transmon qudit. The complexity scaling of this state preparation scheme to higher j values is discussed and compared with classical algorithms.