Preparing angular momentum eigenstates using engineered quantum walks

Coupled angular-momentum eigenstates are widely used in atomic and nuclear physics calculations and are building blocks for spin networks and the Schur transform. To combine two angular momenta ${\mathbf{J}}_{1}$ and ${\mathbf{J}}_{2}$, forming eigenstates of their total angular momentum $\mathbf{J}={\mathbf{J}}_{1}+{\mathbf{J}}_{2}$, we develop a quantum-walk scheme that does not require inputting $O({j}^{3})$ nonzero Clebsch--Gordan (CG) coefficients classically. In fact, our scheme may be regarded as a unitary method for computing CG coefficients on quantum computers with a typical complexity of $O(j)$ and a worst-case complexity of $O({j}^{3})$. Equivalently, our scheme provides decompositions of the dense CG unitary into sparser unitary operations. Our scheme prepares angular-momentum eigenstates using a sequence of Hamiltonians to move an initial state deterministically to desired final states, which are usually highly entangled states in the computational basis. In contrast with usual quantum walks, whose Hamiltonians are prescribed, we engineer the Hamiltonians in $\text{su}(2)\ifmmode\times\else\texttimes\fi{}\text{su}(2)$, which are inspired by, but different from, Hamiltonians that govern magnetic resonances and dipole interactions. To achieve a deterministic preparation of both ket and bra states, we use projection and destructive interference to double pinch the quantum walks, such that each step is a unit-probability population transfer within a two-level system. We test our state preparation scheme on classical computers, reproducing tables of CG coefficients. We also implement small test problems on current quantum hardware.