  A frustration-free local Hamiltonian has the property that its ground state
minimises the energy of all local terms simultaneously. In general, even
deciding whether a Hamiltonian is frustration-free is a hard task, as it is
closely related to the QMA1-complete quantum satisfiability problem (QSAT) --
the quantum analogue of SAT, which is the archetypal NP-complete problem in
classical computer science. This connection shows that the frustration-free
property is not only relevant to physics but also to computer science. The
Quantum Lov\'asz Local Lemma (QLLL) provides a sufficient condition for
frustration-freeness. A natural question is whether there is an efficient way
to prepare a frustration-free state under the conditions of the QLLL. Previous
results showed that the answer is positive if all local terms commute. In this
work we improve on the previous constructive results by designing an algorithm
that works efficiently for non-commuting terms as well, assuming that the
system is "uniformly" gapped, by which we mean that the system and all its
subsystems have an inverse polynomial energy gap. Also, our analysis works
under the most general condition for the QLLL, known as Shearer's bound.
Similarly to the previous results, our algorithm has the charming feature that
it uses only local measurement operations corresponding to the local
Hamiltonian terms.
