Quantum Circuits assisted by LOCC: Transformations and Phases of Matter.

We introduce deterministic state-transformation protocols between many-body quantum states which can be implemented by low-depth Quantum Circuits (QC) followed by Local Operations and Classical Communication (LOCC). We show that this gives rise to a classification of phases in which topologically-ordered states or other paradigmatic entangled states become trivial. An implication of this fact is that the toric code can be prepared with a finite (low) depth QC and LOCC. We also investigate how the set of unitary operations is enhanced by LOCC in this scenario, allowing one to perform certain large-depth QC in terms of low-depth ones.