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

Abstract

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 $|\vec{m}⟩$ of $n$ qudits, each with $(2s+1)$ levels and $s=1/2,1,3/2,\dots$, 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. We also consider the preparation of AKLT states and spin-$s$ Dicke states.

13 pages, 3 figures, implementations of circuits in cirq are available on GitHub; v2: minor changes, version accepted in Annalen der Physik; v3: added missing term in Eq. (2.8)