Preparing angular-momentum eigenstates by quantum walk on su(2)×su(2) algebra

To enable quantum computation of atomic and nuclear physics problems, we develop an efficient state preparation scheme for combining two angular momenta J^­₁ and J₂ to form eigenstates of their total angular momentum J=J₁+J₂. Compared to a brute force approach, which performs the basis change using O(j³) nonzero Clebsch–Gordan (CG) coefficients, our scheme prepares the |j,m>eigenstate using quantum walk with O(j) steps generated by su(2)×su(2) algebra. To prepare states deterministically, we choose a series of evolution Hamiltonians, each is sparse and confines the quantum walk to a two-dimensional Hilbert space. We encode the ground state as the top state |j₁+j₂, j₁+j₂>. When m ≥ 0, our scheme walks the state from |j'+1, m'+1>to |j',m'>until j'=j. Then, keeping j constant, another series of walks moves |j,m'+1>to |j,m'>until m'=m. Analogously, when m<0, a series of steps reaches the desired final state from the bottom state |j₁+j₂, -j₁-j₂>after swapping it with the ground state. We test our state preparation scheme on classical computers, reproducing known CG coefficients, and implement small test problems on current quantum hardware. The state preparation scheme paves the way towards quantum calculation of full matrix elements.