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A New Quantum Lower Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs
[article]

2006
*
arXiv
*
pre-print

Finally, we present

arXiv:quant-ph/0511200v2
fatcat:udxouud47nd53ji3v2jmjpg27q
*a**quantum*algorithm for evaluating solutions*to*systems of linear inequalities,*and*use our*direct**product**theorems**to*show that the*time*-*space**tradeoff*of this algorithm is close*to*... We also use the polynomial*method**to*prove*a**direct**product**theorem*for 1-sided error algorithms for k threshold functions*with**a*stronger*bound*on the success probability. ...*Lower**bound*Here we use our*direct**product**theorems**to**lower*-*bound*the quantity T 2 S for T -query, S-*space**quantum*algorithms for systems of linear inequalities. ...##
###
A New Quantum Lower Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs

2007
*
Algorithmica
*

Finally, we present

doi:10.1007/s00453-007-9022-9
fatcat:5zy2kbfg6nhgfjoexvi3uxqwzi
*a**quantum*algorithm for evaluating solutions*to*systems of linear inequalities,*and*use our*direct**product**theorems**to*show that the*time*-*space**tradeoff*of this algorithm is close*to*... We also use the polynomial*method**to*prove*a**direct**product**theorem*for 1-sided error algorithms for k threshold functions*with**a*stronger*bound*on the success probability. ...*Lower**bound*Here we use our*direct**product**theorems**to**lower*-*bound*the quantity T 2 S for T -query, S-*space**quantum*algorithms for systems of linear inequalities. ...##
###
Quantum and Classical Strong Direct Product Theorems and Optimal Time‐Space Tradeoffs

2007
*
SIAM journal on computing (Print)
*

Our

doi:10.1137/05063235x
fatcat:yxa4y3g4snhrlhnxa3jjjwztd4
*direct**product**theorems*imply*a**time*-*space**tradeoff*T 2 S = Ω'N 3´f or sorting N items on*a**quantum*computer, which is optimal up*to*polylog factors. ...*A*strong*direct**product**theorem*says that if we want*to*compute k independent instances of*a*function, using less than k*times*the resources needed for one instance, then our overall success probability ... about [BPSW05] ,*and*the anonymous referees for comments that improved the presentation of the paper. ...##
###
Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs
[article]

2004
*
arXiv
*
pre-print

Our

arXiv:quant-ph/0402123v2
fatcat:zjptysi4afchxotkemro3wgake
*direct**product**theorems*imply*a**time*-*space**tradeoff*T^2*S=Omega(N^3) for sorting N items on*a**quantum*computer, which is optimal up*to*polylog factors. ...*A*strong*direct**product**theorem*says that if we want*to*compute k independent instances of*a*function, using less than k*times*the resources needed for one instance, then our overall success probability ... Acknowledgments We thank Scott Aaronson for email discussions about the evolving results in his [Aar04] that motivated some of our proofs,*and*Harry Buhrman for useful discussions. ...##
###
Quantum and Classical Communication-Space Tradeoffs from Rectangle Bounds
[chapter]

2004
*
Lecture Notes in Computer Science
*

We derive

doi:10.1007/978-3-540-30538-5_32
fatcat:y3gj4isd7ra2xazlrfo5x4t5ge
*lower**bounds*for*tradeoffs*between the communication C*and**space*S for communicating circuits. The first such*bound*applies*to**quantum*circuits. ...*direct**product*result for the one-sided rectangle*lower**bound*on randomized communication complexity. ... All these results are collected in the following*Direct**Product*Results As in [KSW04] we use*direct**product*type results*to*obtain*quantum*communication-*space**tradeoff**lower**bounds*for functions*with*...##
###
Quantum and Classical Communication-Space Tradeoffs from Rectangle Bounds
[article]

2004
*
arXiv
*
pre-print

We then turn

arXiv:quant-ph/0412088v1
fatcat:l5kyxfgztrel3e5wk3maiey3nq
*to*randomized*bounded*error protocols,*and*derive the*bound*C=\Omega(n^3/S^2) for Boolean matrix multiplication, utilizing*a**new**direct**product*result for the one-sided rectangle*lower**bound*... We derive*lower**bounds*for*tradeoffs*between the communication C*and**space*S for communicating circuits. The first such*bound*applies*to**quantum*circuits. ...*Direct**Product*Results As in [KSW04] we use*direct**product*type results*to*obtain*quantum*communication-*space**tradeoff**lower**bounds*for functions*with*many outputs. ...##
###
A new quantum lower bound method,

2006
*
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06
*

Finally, we present

doi:10.1145/1132516.1132604
dblp:conf/stoc/AmbainisSW06
fatcat:4sc24jzya5gy7kywpckfqkdaka
*a**quantum*algorithm for evaluating solutions*to*systems of linear inequalities,*and*use our*direct**product**theorems**to*show that the*time*-*space**tradeoff*of this algorithm is close*to*... We also use the polynomial*method**to*prove*a**direct**product**theorem*for 1-sided error algorithms for k threshold functions*with**a*stronger*bound*on the success probability. ...*Lower**bound*Here we use our*direct**product**theorems**to**lower*-*bound*the quantity T 2 S for T -query, S-*space**quantum*algorithms for systems of linear inequalities. ...##
###
Time-space tradeoffs for two-way finite automata
[article]

2016
*
arXiv
*
pre-print

We prove: (1)

arXiv:1507.01346v2
fatcat:rgel6tnjpnc45opahfeb4ghe2y
*a**time*-*space**tradeoff*upper*bound*for recognition of the languages L_EQ(n) on two-way probabilistic finite automata (2PFA): TS= O(n n), whereas*a**time*-*space**tradeoff**lower**bound*on two-way ... deterministic finite automata is Ω(n^2), (2)*a**time*-*space**tradeoff*upper*bound*for recognition of the languages L_INT(n) on two-way finite automata*with**quantum**and*classical states (2QCFA): TS= O(n^3 ... Using communication complexity*lower**bound*proof*method*[21], we can get the*lower**bound*for*time*-*space**tradeoff*for 2DFA.*Theorem*2. ...##
###
A Strong Direct Product Theorem for Disjointness
[article]

2010
*
arXiv
*
pre-print

This also implies

arXiv:0908.2940v3
fatcat:peg2x6wjjffgfcuswgihxyvlsu
*a**new**lower**bound*for Disjointness in*a*restricted 3-player NOF protocol,*and*optimal communication-*space**tradeoffs*for Boolean matrix*product*. ...*A*strong*direct**product**theorem*states that if we want*to*compute k independent instances of*a*function, using less than k*times*the resources needed for one instance, then the overall success probability ... Acknowledgments I thank Rahul Jain*and*Shengyu Zhang for insightful discussions. The idea of*a*"smooth" rectangle*bound*originated in discussion between us. ...##
###
Remote Preparation of Quantum States

2005
*
IEEE Transactions on Information Theory
*

The paper includes an extensive discussion of our results, including the impact of the choice of model on the resources, the topic of obliviousness,

doi:10.1109/tit.2004.839476
fatcat:nishrbydifdb7pztykahq4ooka
*and*an*application**to*private*quantum*channels*and**quantum*... Our main result is*a*general*method*of remote state preparation for arbitrary states of many qubits, at*a*cost of 1 bit of classical communication*and*1 bit of entanglement per qubit sent. ... Readers of [30] will notice the similarity of the proofs of the*lower**bounds*in*Theorems*9*and*11. ...##
###
Quantum Verification of Matrix Products
[article]

2005
*
arXiv
*
pre-print

We present

arXiv:quant-ph/0409035v2
fatcat:wg56esv63vho3m6qayj37jztgu
*a**quantum*algorithm that verifies*a**product*of two n*n matrices over any field*with**bounded*error in worst-case*time*n^5/3*and*expected*time*n^5/3 / min(w,sqrt(n))^1/3, where w is the number ... This improves the previous best algorithm that runs in*time*n^7/4. We also present*a**quantum*matrix multiplication algorithm that is efficient when the result has few nonzero entries. ... Acknowledgments We thank Ronald de Wolf*and*Troy Lee for useful discussions. We thank anonymous referees for their valuable comments. ...##
###
Element Distinctness, Frequency Moments, and Sliding Windows
[article]

2013
*
arXiv
*
pre-print

We derive

arXiv:1309.3690v1
fatcat:3ypzlfgfqzhjvgbw7yfdlruyrq
*new**time*-*space**tradeoff**lower**bounds**and*algorithms for exactly computing statistics of input data, including frequency moments, element distinctness,*and*order statistics, that are simple*to*... In contrast, we show*a**time*-*space**tradeoff**lower**bound*of T in Omega(n^2/S) for randomized branching programs*to*compute the number of distinct elements over sliding windows. ... Acknowledgements The authors would like*to*thank Aram Harrow, Ely Porat*and*Shachar Lovett for*a*number of insightful discussions*and*helpful comments during the preparation of this paper. ...##
###
Fundamental rate-loss tradeoff for optical quantum key distribution

2014
*
Nature Communications
*

*A*natural

*and*fundamental question is then whether there are yet-

*to*-be discovered optical QKD protocols (without

*quantum*repeaters) that could circumvent this rate-distance

*tradeoff*. ... Since 1984, various optical

*quantum*key distribution (QKD) protocols have been proposed

*and*examined. In all of them, the rate of secret key generation decays exponentially

*with*distance. ... We also acknowledge Mark Byrd, Eric Chitambar

*and*the other participants of the Boris Musulin Workshop on Open

*Quantum*Systems

*and*Information Processing for helpful feedback. ...

##
###
Tight Quantum Time-Space Tradeoffs for Function Inversion
[article]

2020
*
arXiv
*
pre-print

*To*prove this result, we develop

*a*general framework for establishing

*quantum*

*time*-

*space*

*lower*

*bounds*. ... We further demonstrate the power of our framework by proving

*quantum*

*time*-

*space*

*lower*

*bounds*for Yao's box problem

*and*salted cryptography. ... Open Problems

*Quantum*

*time*-

*space*

*tradeoff*

*lower*

*bounds*for permutation inversion. ...

##
###
Quantum Proofs for Classical Theorems
[article]

2011
*
arXiv
*
pre-print

In this paper we survey these results

arXiv:0910.3376v2
fatcat:poabjgl3rjfaxnpqfipt3byhue
*and*the*quantum*toolbox they use. ... Alongside the development of*quantum*algorithms*and**quantum*complexity theory in recent years,*quantum*techniques have also proved instrumental in obtaining results in classical (non-*quantum*) areas. ... We thank Scott Aaronson, Joshua Brody, Iordanis Kerenidis, Greg Kuperberg, Troy Lee, Frédéric Magniez, Ashwin Nayak, Alexander Sherstov,*and*Shengyu Zhang for helpful comments*and*pointers*to*the literature ...
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