Distributed finite-time stabilization of entangled quantum states on tree-like hypergraphs

Preparation of pure states on networks of quantum systems by controlled dissipative dynamics offers important advantages with respect to circuit-based schemes. Unlike in continuous-time scenarios, when discrete-time dynamics are considered, dead-beat stabilization becomes possible in principle. Here, we focus on pure states that can be stabilized by distributed, unsupervised dynamics in finite time on a network of quantum systems subject to realistic quasi-locality constraints. In particular, we define a class of quasi-locality notions, that we name "tree-like hypergraphs," and show that the states that are robustly stabilizable in finite time are then unique ground states of a frustration-free, commuting quasi-local Hamiltonian. A structural characterization of such states is also provided, building on a simple yet relevant example.