We present a quantum circuit that prepares an arbitrary $U(1)$-invariant state on a quantum computer, including the exact eigenstates of the spin-1/2 XXZ quantum spin chain with either open or closed boundary conditions. The algorithm is deterministic, does not require ancillary qubits, and does not require QR decompositions. The circuit prepares such an L-qubit state with M down-spins using LM-1 multi-controlled rotation gates and $2M(L-M)$ CNOT-gates.
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Raveh et al. (2024) studied this question.
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