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Quantum Lower Bound for Graph Collision Implies Lower Bound for Triangle Detection
[article]

2015
*
arXiv
*
pre-print

We show that an improvement to the best known

arXiv:1507.03885v1
fatcat:m7gwpzsbcnc5rbsbvjd4dlag3i
*quantum**lower**bound**for**GRAPH*-*COLLISION*problem*implies*an improvement to the best known*lower**bound**for**TRIANGLE*problem in the*quantum*query complexity model ...*For*both of these problems the known*lower**bounds*are trivial (Ω(√(n)) and Ω(n), respectively) and there is no known matching upper*bound*. ... Setting f = OR and g =*Graph*-*Collision*G gives the desired*bound*. As the next corollary shows, a better*lower**bound*on*Graph*-*Collision**implies*a better*lower**bound*on the*Triangle*problem. ...##
###
Quantum Lower Bound for Graph Collision Implies Lower Bound for Triangle Detection

2016
*
Baltic Journal of Modern Computing
*

We show that an improvement to the best known

doi:10.22364/bjmc.2016.4.4.10
fatcat:xwe77rfkbncmvckbfzc55zk4am
*quantum**lower**bound**for**GRAPH*-*COLLISION*problem*implies*an improvement to the best known*lower**bound**for**TRIANGLE*problem in the*quantum*query complexity model ...*For*both of these problems the known*lower**bounds*are trivial (Ω( √ n) and Ω(n), respectively) and there is no known matching upper*bound*. ... Setting f = OR and g =*GRAPH*-*COLLISION*G gives the desired*bound*. As the next corollary shows, a better*lower**bound*on*GRAPH*-*COLLISION**implies*a better*lower**bound*on the*TRIANGLE*problem. ...##
###
On the Power of Non-adaptive Learning Graphs

2013
*
2013 IEEE Conference on Computational Complexity
*

This also gives a

doi:10.1109/ccc.2013.14
dblp:conf/coco/BelovsR13
fatcat:wzeax7k42jfqnesi2nbnr65seq
*quantum*query*lower**bound**for*the*triangle*sum problem. ... The construction is based on orthogonal arrays and generalizes the*quantum*query*lower**bound**for*the k-sum problem derived recently by Belovs andŠpalek (Proceeding of 4th ACM ITCS, 323-328, 2012). ... Acknowledgements A.B. would like to thank Troy Lee, Robin Kothari and Rajat Mittal*for*sharing their ideas on the limitations of learning*graphs*. ...##
###
A quantum query algorithm for the graph collision problem
[article]

2012
*
arXiv
*
pre-print

We construct a new

arXiv:1204.1527v1
fatcat:o44qph5yrvdx7pagl35ivsirnu
*quantum*algorithm*for*the*graph**collision*problem; that is, the problem of deciding whether the set of marked vertices contains a pair of adjacent vertices in a known*graph*G. ... up to the sqrt(log n) factor on most*graphs*. ... Acknowledgments Dmitry Gavinsky is grateful to Ryan O'Donnell, Rocco Servedio, Srikanth Srinivasan and Li-Yang Tan*for*helpful discussions. ...##
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On the Power of Non-adaptive Learning Graphs

2014
*
Computational Complexity
*

This also gives a

doi:10.1007/s00037-014-0084-1
fatcat:3a2lurou7nhd5ilcvv6dxakd4y
*quantum*query*lower**bound**for*the*triangle*sum problem. ... The construction is based on orthogonal arrays and generalizes the*quantum*query*lower**bound**for*the k-sum problem derived recently by Belovs and Špalek (Proceeding of 4th ACM ITCS, 323-328, 2012). ... Acknowledgements A.B. would like to thank Troy Lee, Robin Kothari and Rajat Mittal*for*sharing their ideas on the limitations of learning*graphs*. ...##
###
On the Power of Non-Adaptive Learning Graphs
[article]

2012
*
arXiv
*
pre-print

This also gives a

arXiv:1210.3279v2
fatcat:dnoyeb5stfaa5ok575jjjo4dvi
*quantum*query*lower**bound**for*the*triangle*-sum problem. ... The construction is based on orthogonal arrays, and generalizes the*quantum*query*lower**bound**for*the k-sum problem derived recently in arXiv:1206.6528. ... Acknowledgments A.B. would like to thank Troy Lee, Robin Kothari and Rajat Mittal*for*sharing their ideas on the limitations of learning*graphs*. ...##
###
Adversary Lower Bound for the k-sum Problem
[article]

2012
*
arXiv
*
pre-print

We prove a tight

arXiv:1206.6528v2
fatcat:epzhatctw5hmhdrrekebgdcfsi
*quantum*query*lower**bound*Ω(n^k/(k+1))*for*the problem of deciding whether there exist k numbers among n that sum up to a prescribed number, provided that the alphabet size is sufficiently ... We are grateful to Kassem Kalach*for*informing about the applications of the k-sum problem in Merkle puzzles, and*for*reporting on some minor errors in the early version of the paper. ... Acknowledgments A.B. would like to thank Andris Ambainis, Troy Lee and Ansis Rosmanis*for*valuable discussions. ...##
###
New Results on Quantum Property Testing

2010
*
Foundations of Software Technology and Theoretical Computer Science
*

, based on Shor's algorithm and a modification of a classical

doi:10.4230/lipics.fsttcs.2010.145
dblp:conf/fsttcs/ChakrabortyFMW10
fatcat:vwmxfq2oxvcu7bp3gkzlqy5wvu
*lower**bound*by Lachish and Newman [27] . ... Based on this result, we also reduce the query complexity of*graph*isomorphism testers with*quantum*oracle access. ...*for*pointing out that his Fourier checking result in [1] was the first constant-vs-polynomial*quantum*speed-up in property testing. ...##
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Quantum Algorithm for Commutativity Testing of a Matrix Set
[article]

2005
*
arXiv
*
pre-print

We also present Ambainis's method of

arXiv:quant-ph/0509206v1
fatcat:d2vtqjggnrdgtbflzrke6rpmqa
*lower**bounding*technique (quant-ph/0002066) to obtain a*lower**bound**for*this problem. ... We give an O(k^4/5n^9/5) algorithm as well as a*lower**bound*of Omega(k^1/2n). We generalize the technique used in coming up with the upper*bound*to solve a broader range of similar problems. ... Acknowledgements The author would like to acknowledge Ashwin Nayak*for*supervion, Richard Cleve*for*reading this essay, Andris Ambainis and Frederic Magniez*for*consultation on*lower**bounds*and the differences ...##
###
New Results on Quantum Property Testing
[article]

2010
*
arXiv
*
pre-print

, based on Shor's algorithm and a modification of a classical

arXiv:1005.0523v3
fatcat:2ceczft5bvhmdfdvaujqhfktwq
*lower**bound*by Lachish and Newman lachish&newman:periodicity. ... Based on this result, we also reduce the query complexity of*graph*isomorphism testers with*quantum*oracle access. ... result in [1] was the first constant-vs-polynomial*quantum*speed-up in property testing. ...##
###
Applications of the Adversary Method in Quantum Query Algorithms
[article]

2014
*
arXiv
*
pre-print

Our results are as follows: * We develop a new technique

arXiv:1402.3858v1
fatcat:etcbmdej4bfxbhieqqfq6wd3lq
*for*the construction of*quantum*algorithms: learning*graphs*. * We use learning*graphs*to improve*quantum*query complexity of the*triangle**detection*... and the k-distinctness problems. * We prove tight*lower**bounds**for*the k-sum and the*triangle*sum problems. * We construct*quantum*algorithms*for*some subgraph-finding problems that are optimal in terms ...*Lower**Bounds**for**Quantum*Query Complexity In this chapter, we describe some known techniques*for*proving*lower**bounds*on*quantum*query complexity. ...##
###
Quantum Complexity of Testing Group Commutativity
[article]

2007
*
arXiv
*
pre-print

*For*the

*lower*

*bound*of Omega(k^2/3), we give a reduction from a special case of Element Distinctness to our problem. ... We construct a quite optimal

*quantum*algorithm

*for*this problem whose complexity is in O (k^2/3). The algorithm uses and highlights the power of the quantization method of Szegedy. ... Then Theorem 2 (due to Aaronson and Shi [AS04] and Kutin [Kut05] , together with Ambainis [Amb05] )

*implies*the

*lower*

*bound*. ...

##
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Adversary Lower Bounds for the Collision and the Set Equality Problems
[article]

2017
*
arXiv
*
pre-print

We prove tight Ω(n^1/3)

arXiv:1310.5185v4
fatcat:iy732z4akvdxjk324sjxfn5us4
*lower**bounds*on the*quantum*query complexity of the*Collision*and the Set Equality problems, provided that the size of the alphabet is large enough. ... In particular, this gives nearly tight*lower**bounds**for*the k-Sum and the*Triangle*-Sum problems. ... Note that H Dual Learning*Graph*Perspective Our*lower**bounds**for*the*Collision*and the Set Equality problems are intrinsicly based on the dual learning*graph**for*these problems developed in [9] . ...##
###
Improved Quantum Query Algorithms for Triangle Finding and Associativity Testing
[chapter]

2013
*
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
*

We show that the

doi:10.1137/1.9781611973105.107
dblp:conf/soda/LeeMS13
fatcat:k65v7a5l2rcv5o446odv3n57jq
*quantum*query complexity of*detecting*if an n-vertex*graph*contains a*triangle*is O(n 9/7 ). This improves the previous best algorithm of Belovs [2] making O(n 35/27 ) queries. ... Our algorithms are designed using the learning*graph*framework of Belovs. We give a family of algorithms*for**detecting*constant-sized subgraphs, which can possibly be directed and colored. ... Acknowledgements We would like to thank Aleksandrs Belovs*for*discussions and comments on an earlier draft of this work. ...##
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Improved Quantum Query Algorithms for Triangle Finding and Associativity Testing
[article]

2012
*
arXiv
*
pre-print

We show that the

arXiv:1210.1014v1
fatcat:57t2dmwmm5bbfo3ejdvahgcak4
*quantum*query complexity of*detecting*if an n-vertex*graph*contains a*triangle*is O(n^9/7). This improves the previous best algorithm of Belovs making O(n^35/27) queries. ... Our algorithms are designed using the learning*graph*framework of Belovs. We give a family of algorithms*for**detecting*constant-sized subgraphs, which can possibly be directed and colored. ... Acknowledgements We would like to thank Aleksandrs Belovs*for*discussions and comments on an earlier draft of this work. ...
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