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This research presents a deterministic algorithm for preparing general $U(1)$-eigenstates in spin-$s$ chains using Gray codes to construct specific quantum gates. The method enables the exact preparation of eigenstates for integrable XXX Hamiltonians, as well as AKLT and Dicke states within qudit systems. By leveraging bounded integer compositions, this approach offers a structured pathway for state engineering in higher-dimensional quantum architectures without relying on probabilistic techniques.
Spin-$s$ $U(1)$-eigenstate preparation
arXiv:2601.14513v3 Announce Type: replace 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 $|\vec{m}\rangle$ 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. We also consider the preparation of AKLT states and spin-$s$ Dicke states.
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 $|\vec{m}\rangle$ 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. 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)
arXiv:2601.14513 [quant-ph]
(or arXiv:2601.14513v3 [quant-ph] for this version)
https://doi.org/10.1002/andp.70239 DOI(s) linking to related resources
From: Rafael I. Nepomechie [view email] [v1] Tue, 20 Jan 2026 22:07:41 UTC (18 KB) [v2] Sat, 27 Jun 2026 14:00:18 UTC (19 KB) [v3] Tue, 4 Aug 2026 12:48:43 UTC (19 KB)
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