Three Proofs of a Constructive Commuting Quantum Lovász Local Lemma

The recently proven Quantum Lovász Local Lemma generalises the well-known Lovász Local Lemma. It states that, if a collection of subspace constraints are “weakly dependent”, there necessarily exists a state satisfying all constraints. It implies e.g. that certain instances of the quantum k–QSAT satisfiability problem are necessarily satisfiable, or that many-body systems with “not too many” interactions are never frustrated. However, the QLLL only asserts existence; it says nothing about how to find the state. Inspired by Moser’s breakthrough classical results, we present a constructive version of the QLLL in the setting of commuting constraints, proving that a simple quantum algorithm converges efficiently to the required state. In fact, we provide three different proofs, all of which are independent of the original QLLL proof. So these results also provide independent, constructive proofs of the commuting QLLL itself, but strengthen it significantly by giving an efficient algorithm for finding the state whose existence is asserted by the QLLL.