Three proofs of a constructive commuting quantum Lovász local lemma

Toby S Cubitt, Martin Schwarz, Frank Verstraete, Or Sattath, Itai Arad

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


The recently proven Quantum Lov´asz Local Lemma generalises the well-known Lov´asz 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.
Original languageEnglish GB
Title of host publicationQuantum Information Processing workshop, Montreal, Canada
StatePublished - 2012


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