A Quantum Lovasz Local Lemma
Andris Ambainis, Julia Kempe, Or Sattath

TL;DR
This paper extends the Lovasz Local Lemma to a geometric setting involving subspaces and relative dimension, providing new bounds for the existence of satisfying states in quantum satisfiability problems, especially for random k-QSAT.
Contribution
It introduces a geometric version of the Lovasz Local Lemma applicable to quantum problems, enabling bounds on the dimension of intersections of subspaces and advancing understanding of satisfiability in quantum systems.
Findings
Extended LLL to subspaces and relative dimension
Bound the dimension of intersections under independence conditions
Significantly increased the known satisfiable density for random k-QSAT
Abstract
The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions. Our result immediately applies to the k-QSAT problem: For instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most of them, has a joint satisfiable state. We then apply our results to the recently studied model of random k-QSAT. Recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the…
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