NDLI: A quantum Lovász local lemma

Content Provider	ACM Digital Library
Author	Ambainis, Andris Kempe, Julia Sattath, Or
Copyright Year	2012
Abstract	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 $\textit{k}-qsat$ problem (quantum analog of $\textit{k}-sat):$ For instance we show that any collection of rank-1 projectors, with the property that each qubit appears in at most $2^{k}/(e$ ċ $\textit{k})$ of them, has a joint satisfiable state. We then apply our results to the recently studied model of random $\textit{k}-qsat.$ Recent works have shown that the satisfiable region extends up to a density of 1 in the large $\textit{k}$ limit, where the density is the ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. [2009, 2010] we greatly extend the known satisfiable region for random $\textit{k}-qsat$ to a density of $Ω(2^{k}/k^{2}).$ 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.
Starting Page	1
Ending Page	24
Page Count	24
File Format	PDF
ISSN	00045411
e-ISSN	1557735X
DOI	10.1145/2371656.2371659
Journal	Journal of the ACM (JACM)
Volume Number	59
Issue Number	5
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2012-11-05
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Local lemma Probabilistic method Quantum computation Quanum SAT Random quantum SAT
Content Type	Text
Resource Type	Article
Subject	Hardware and Architecture Information Systems Control and Systems Engineering Artificial Intelligence Software

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

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