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

Andris Ambainis, Ronald de Wolf
2006 arXiv   pre-print
Finally, we present 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.  ... 
arXiv:quant-ph/0511200v2 fatcat:udxouud47nd53ji3v2jmjpg27q

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

Andris Ambainis, Robert Špalek, Ronald de Wolf
2007 Algorithmica  
Finally, we present 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.  ... 
doi:10.1007/s00453-007-9022-9 fatcat:5zy2kbfg6nhgfjoexvi3uxqwzi

Quantum and Classical Strong Direct Product Theorems and Optimal Time‐Space Tradeoffs

Hartmut Klauck, Robert Špalek, Ronald de Wolf
2007 SIAM journal on computing (Print)  
Our 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.  ... 
doi:10.1137/05063235x fatcat:yxa4y3g4snhrlhnxa3jjjwztd4

Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs [article]

Hartmut Klauck
2004 arXiv   pre-print
Our 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.  ... 
arXiv:quant-ph/0402123v2 fatcat:zjptysi4afchxotkemro3wgake

Quantum and Classical Communication-Space Tradeoffs from Rectangle Bounds [chapter]

Hartmut Klauck
2004 Lecture Notes in Computer Science  
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 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  ... 
doi:10.1007/978-3-540-30538-5_32 fatcat:y3gj4isd7ra2xazlrfo5x4t5ge

Quantum and Classical Communication-Space Tradeoffs from Rectangle Bounds [article]

Hartmut Klauck
2004 arXiv   pre-print
We then turn 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.  ... 
arXiv:quant-ph/0412088v1 fatcat:l5kyxfgztrel3e5wk3maiey3nq

A new quantum lower bound method,

Andris Ambainis, Robert Špalek, Ronald de Wolf
2006 Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06  
Finally, we present 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.  ... 
doi:10.1145/1132516.1132604 dblp:conf/stoc/AmbainisSW06 fatcat:4sc24jzya5gy7kywpckfqkdaka

Time-space tradeoffs for two-way finite automata [article]

Shenggen Zheng, Daowen Qiu, Jozef Gruska
2016 arXiv   pre-print
We prove: (1) 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.  ... 
arXiv:1507.01346v2 fatcat:rgel6tnjpnc45opahfeb4ghe2y

A Strong Direct Product Theorem for Disjointness [article]

Hartmut Klauck
2010 arXiv   pre-print
This also implies 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.  ... 
arXiv:0908.2940v3 fatcat:peg2x6wjjffgfcuswgihxyvlsu

Remote Preparation of Quantum States

C.H. Bennett, P. Hayden, D.W. Leung, P.W. Shor, A. Winter
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, 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.  ... 
doi:10.1109/tit.2004.839476 fatcat:nishrbydifdb7pztykahq4ooka

Quantum Verification of Matrix Products [article]

Harry Buhrman, Robert Spalek
2005 arXiv   pre-print
We present 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.  ... 
arXiv:quant-ph/0409035v2 fatcat:wg56esv63vho3m6qayj37jztgu

Element Distinctness, Frequency Moments, and Sliding Windows [article]

Paul Beame, Raphael Clifford, Widad Machmouchi
2013 arXiv   pre-print
We derive 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.  ... 
arXiv:1309.3690v1 fatcat:3ypzlfgfqzhjvgbw7yfdlruyrq

Fundamental rate-loss tradeoff for optical quantum key distribution

Masahiro Takeoka, Saikat Guha, Mark M. Wilde
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.  ... 
doi:10.1038/ncomms6235 pmid:25341406 fatcat:6m57yhlvgjdqpdb5catk3cf5he

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

Kai-Min Chung, Siyao Guo, Qipeng Liu, Luowen Qian
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.  ... 
arXiv:2006.05650v2 fatcat:tib4rclksbf7lm7dnubrwobjtm

Quantum Proofs for Classical Theorems [article]

Andrew Drucker
2011 arXiv   pre-print
In this paper we survey these results 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  ... 
arXiv:0910.3376v2 fatcat:poabjgl3rjfaxnpqfipt3byhue
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