Approximating many-body quantum states with quantum circuits and measurements

We introduce protocols to prepare many-body quantum states with quantum circuits assisted by local operations and classical communication. First, we show that by lifting the requirement of exact preparation, one can substantially save resources. In particular, the so-called W and, more generally, Dicke states require a circuit depth and number of ancillas that are independent of the system size. We also show how one can save resources in the preparation of eigenstates of well-known spin models, both free and interacting. As a biproduct of our work, we introduce an efficient scheme to implement certain non-local, non-Clifford unitary operators.