This paper introduces protocols for preparing many-body quantum states using quantum circuits and local operations and classical communication (LOCC). The key idea is that relaxing the requirement of exact preparation allows for significant resource savings. The authors demonstrate that states like the W and Dicke states can be prepared with circuit depth and ancilla count independent of system size. They also present an efficient method for implementing non-Clifford unitary operations and discuss applications to eigenstates of spin models. The preparation of Dicke states is achieved by measuring the number of excitations, which allows for efficient state preparation with controlled infidelity. The authors show that by using amplitude amplification, the preparation time can be reduced, leading to a deterministic protocol with lower resource requirements. The results are applied to prepare eigenstates of the XX Hamiltonian and interacting spin chains, including the Richardson-Gaudin model. The paper also discusses the broader implications of these results for quantum state preparation in current quantum devices, highlighting the potential for efficient protocols that allow for controlled infidelities and probabilities of failure. The work raises theoretical questions about the application of these methods to more general interacting Hamiltonians and the classification of quantum phases of matter.This paper introduces protocols for preparing many-body quantum states using quantum circuits and local operations and classical communication (LOCC). The key idea is that relaxing the requirement of exact preparation allows for significant resource savings. The authors demonstrate that states like the W and Dicke states can be prepared with circuit depth and ancilla count independent of system size. They also present an efficient method for implementing non-Clifford unitary operations and discuss applications to eigenstates of spin models. The preparation of Dicke states is achieved by measuring the number of excitations, which allows for efficient state preparation with controlled infidelity. The authors show that by using amplitude amplification, the preparation time can be reduced, leading to a deterministic protocol with lower resource requirements. The results are applied to prepare eigenstates of the XX Hamiltonian and interacting spin chains, including the Richardson-Gaudin model. The paper also discusses the broader implications of these results for quantum state preparation in current quantum devices, highlighting the potential for efficient protocols that allow for controlled infidelities and probabilities of failure. The work raises theoretical questions about the application of these methods to more general interacting Hamiltonians and the classification of quantum phases of matter.