The Quantum Satisfiability problem generalizes the Boolean satisfiability
problem to the quantum setting by replacing classical clauses with local
projectors. The Quantum Lov\'asz Local Lemma gives a sufficient condition for a
Quantum Satisfiability problem to be satisfiable [AKS12], by generalizing the
classical Lov\'asz Local Lemma.
The next natural question that arises is: can a satisfying quantum state be
efficiently found, when these conditions hold? In this work we present such an
algorithm, with the additional requirement that all the projectors commute. The
proof follows the information theoretic proof given by Moser's breakthrough
result in the classical setting [Mos09].
Similar results were independently published in [CS11,CSV13]