Spin-$s$ $U(1)$-eigenstate preparation

We formulate a deterministic algorithm for preparing a general $U(1)$-eigenstate of a spin-$s$ chain of length $n$. These states consist of linear combinations of computational basis states $|\vec{m}\rangle$ of $n$ qudits, each with $(2s+1)$ levels and $s= 1/2, 1, 3/2, \ldots$, whose ditstrings $\vec{m}$ have a fixed digit sum. Exploiting a Gray code for bounded integer compositions, whose consecutive ditstrings obey the Gray property, the quantum state is prepared by applying corresponding ``Gray gates.'' We use this algorithm to prepare exact eigenstates of integrable spin-$s$ XXX Hamiltonians.