The paper introduces protocols for preparing many-body quantum states using quantum circuits and local operations with classical communication (LOCC). The authors show that by relaxing the requirement of exact preparation, significant resource savings can be achieved. Specifically, the $W$ and Dicke states can be prepared with circuit depth and ancilla number per site that are independent of the system size. They also introduce an efficient scheme for implementing certain non-local, non-Clifford unitary operators. The paper discusses how similar ideas can be applied to prepare eigenstates of well-known spin models, both free and interacting. The main results include efficient protocols for preparing Dicke states and the $W$ state, as well as improved schemes using amplitude amplification. The authors provide detailed proofs and technical computations to support their findings.The paper introduces protocols for preparing many-body quantum states using quantum circuits and local operations with classical communication (LOCC). The authors show that by relaxing the requirement of exact preparation, significant resource savings can be achieved. Specifically, the $W$ and Dicke states can be prepared with circuit depth and ancilla number per site that are independent of the system size. They also introduce an efficient scheme for implementing certain non-local, non-Clifford unitary operators. The paper discusses how similar ideas can be applied to prepare eigenstates of well-known spin models, both free and interacting. The main results include efficient protocols for preparing Dicke states and the $W$ state, as well as improved schemes using amplitude amplification. The authors provide detailed proofs and technical computations to support their findings.