TY - GEN

T1 - A quantum Lovász local lemma

AU - Ambainis, Andris

AU - Kempe, Julia

AU - Sattath, Or

PY - 2010

Y1 - 2010

N2 - The Lovász 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 2k/(e·k) 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 ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. we greatly extend the known satisfiable region for random k-QSAT to a density of Ω(2k/k2). Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.

AB - The Lovász 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 2k/(e·k) 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 ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. we greatly extend the known satisfiable region for random k-QSAT to a density of Ω(2k/k2). Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.

KW - local lemma

KW - probabilistic method

KW - quantum computation

KW - quanum SAT

KW - random quantum sat

UR - http://www.scopus.com/inward/record.url?scp=77954735628&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954735628&partnerID=8YFLogxK

U2 - 10.1145/1806689.1806712

DO - 10.1145/1806689.1806712

M3 - Conference contribution

AN - SCOPUS:77954735628

SN - 9781605588179

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 151

EP - 160

BT - STOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing

T2 - 42nd ACM Symposium on Theory of Computing, STOC 2010

Y2 - 5 June 2010 through 8 June 2010

ER -