5 Search Results for "Høyer, Peter"


Document
Symmetry and Quantum Query-To-Communication Simulation

Authors: Sourav Chakraborty, Arkadev Chattopadhyay, Peter Høyer, Nikhil S. Mande, Manaswi Paraashar, and Ronald de Wolf

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)


Abstract
Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}ⁿ → {-1,1} and G ∈ {AND₂, XOR₂}, the bounded-error quantum communication complexity of the composed function f∘G equals O(𝖰(f) log n), where 𝖰(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that 𝖱^{cc}(f∘G) ≤ 2𝖱(f), where 𝖱^{cc} and 𝖱 denote bounded-error communication and query complexity, respectively. Chakraborty et al. (CCC'20) exhibited a total function for which the log n overhead in the BCW simulation is required. This established the somewhat surprising fact that quantum reductions are in some cases inherently more expensive than classical reductions. We improve upon their result in several ways. - We show that the log n overhead is not required when f is symmetric (i.e., depends only on the Hamming weight of its input), generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). Our upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication. - In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. This also provides a different, and arguably simpler, proof of Aaronson and Ambainis’s O(√n) communication upper bound for Set-Disjointness. - In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2 (this corresponds to the unbounded-error model of communication). - We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free.

Cite as

Sourav Chakraborty, Arkadev Chattopadhyay, Peter Høyer, Nikhil S. Mande, Manaswi Paraashar, and Ronald de Wolf. Symmetry and Quantum Query-To-Communication Simulation. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{chakraborty_et_al:LIPIcs.STACS.2022.20,
  author =	{Chakraborty, Sourav and Chattopadhyay, Arkadev and H{\o}yer, Peter and Mande, Nikhil S. and Paraashar, Manaswi and de Wolf, Ronald},
  title =	{{Symmetry and Quantum Query-To-Communication Simulation}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{20:1--20:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.20},
  URN =		{urn:nbn:de:0030-drops-158309},
  doi =		{10.4230/LIPIcs.STACS.2022.20},
  annote =	{Keywords: Classical and quantum communication complexity, query-to-communication-simulation, quantum computing}
}
Document
Bounding Quantum-Classical Separations for Classes of Nonlocal Games

Authors: Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)


Abstract
We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1/m + (m-1)/m t^{1-t}. Secondly, for free XOR games, in which the input distribution is of product form, we show beta(G) >= beta^*(G)^{2^t} where beta(G) and beta^*(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1-epsilon then the classical value is at least 1-O(sqrt{epsilon log k}) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.

Cite as

Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee. Bounding Quantum-Classical Separations for Classes of Nonlocal Games. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 12:1-12:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{bannink_et_al:LIPIcs.STACS.2019.12,
  author =	{Bannink, Tom and Bri\"{e}t, Jop and Buhrman, Harry and Labib, Farrokh and Lee, Troy},
  title =	{{Bounding Quantum-Classical Separations for Classes of Nonlocal Games}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{12:1--12:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.12},
  URN =		{urn:nbn:de:0030-drops-102512},
  doi =		{10.4230/LIPIcs.STACS.2019.12},
  annote =	{Keywords: Nonlocal games, communication complexity, bounded separations, semidefinite programming, pseudorandomness, Gowers norms}
}
Document
Provably Secure Key Establishment Against Quantum Adversaries

Authors: Aleksandrs Belovs, Gilles Brassard, Peter Høyer, Marc Kaplan, Sophie Laplante, and Louis Salvail

Published in: LIPIcs, Volume 73, 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)


Abstract
At Crypto 2011, some of us had proposed a family of cryptographic protocols for key establishment capable of protecting quantum and classical legitimate parties unconditionally against a quantum eavesdropper in the query complexity model. Unfortunately, our security proofs were unsatisfactory from a cryptographically meaningful perspective because they were sound only in a worst-case scenario. Here, we extend our results and prove that for any \eps > 0, there is a classical protocol that allows the legitimate parties to establish a common key after O(N) expected queries to a random oracle, yet any quantum eavesdropper will have a vanishing probability of learning their key after O(N^(1.5-\eps)) queries to the same oracle. The vanishing probability applies to a typical run of the protocol. If we allow the legitimate parties to use a quantum computer as well, their advantage over the quantum eavesdropper becomes arbitrarily close to the quadratic advantage that classical legitimate parties enjoyed over classical eavesdroppers in the seminal 1974 work of Ralph Merkle. Along the way, we develop new tools to give lower bounds on the number of quantum queries required to distinguish two probability distributions. This method in itself could have multiple applications in cryptography. We use it here to study average-case quantum query complexity, for which we develop a new composition theorem of independent interest.

Cite as

Aleksandrs Belovs, Gilles Brassard, Peter Høyer, Marc Kaplan, Sophie Laplante, and Louis Salvail. Provably Secure Key Establishment Against Quantum Adversaries. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 73, pp. 3:1-3:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{belovs_et_al:LIPIcs.TQC.2017.3,
  author =	{Belovs, Aleksandrs and Brassard, Gilles and H{\o}yer, Peter and Kaplan, Marc and Laplante, Sophie and Salvail, Louis},
  title =	{{Provably Secure Key Establishment Against Quantum Adversaries}},
  booktitle =	{12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)},
  pages =	{3:1--3:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-034-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{73},
  editor =	{Wilde, Mark M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2017.3},
  URN =		{urn:nbn:de:0030-drops-85816},
  doi =		{10.4230/LIPIcs.TQC.2017.3},
  annote =	{Keywords: Merkle puzzles, Key establishment schemes, Quantum cryptography, Adversary method, Average-case analysis}
}
Document
Controlled Quantum Amplification

Authors: Catalin Dohotaru and Peter Høyer

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)


Abstract
We propose a new framework for turning quantum search algorithms that decide into quantum algorithms for finding a solution. Consider we are given an abstract quantum search algorithm A that can determine whether a target g exists or not. We give a general construction of another operator U that both determines and finds the target, whenever one exists. Our amplification method at most doubles the cost over using A, has little overhead, and works by controlling the evolution of A. This is the first known general framework to the open question of turning abstract quantum search algorithms into quantum algorithms for finding a solution. We next apply the framework to random walks. We develop a new classical algorithm and a new quantum algorithm for finding a unique marked element. Our new random walk finds a unique marked element using H update operations and 1/eps checking operations. Here H is the hitting time, and eps is the probability that the stationary distribution of the walk is in the marked state. Our classical walk is derived via quantum arguments. Our new quantum algorithm finds a unique marked element using H^(1/2) update operations and 1/eps^(1/2) checking operations, up to logarithmic factors. This is the first known quantum algorithm being simultaneously quadratically faster in both parameters. We also show that the framework can simulate Grover's quantum search algorithm, amplitude amplification, Szegedy's quantum walks, and quantum interpolated walks.

Cite as

Catalin Dohotaru and Peter Høyer. Controlled Quantum Amplification. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{dohotaru_et_al:LIPIcs.ICALP.2017.18,
  author =	{Dohotaru, Catalin and H{\o}yer, Peter},
  title =	{{Controlled Quantum Amplification}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{18:1--18:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.18},
  URN =		{urn:nbn:de:0030-drops-74893},
  doi =		{10.4230/LIPIcs.ICALP.2017.18},
  annote =	{Keywords: Quantum algorithms, quantum walks, random walks, quantum search}
}
Document
Efficient Quantum Walk on the Grid with Multiple Marked Elements

Authors: Peter Hoyer and Mojtaba Komeili

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)


Abstract
We give a quantum algorithm for finding a marked element on the grid when there are multiple marked elements. Our algorithm uses quadratically fewer steps than a random walk on the grid, ignoring logarithmic factors. This is the first known quantum walk that finds a marked element in a number of steps less than the square-root of the extended hitting time. We also give a new tighter upper bound on the extended hitting time of a marked subset, expressed in terms of the hitting times of its members.

Cite as

Peter Hoyer and Mojtaba Komeili. Efficient Quantum Walk on the Grid with Multiple Marked Elements. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{hoyer_et_al:LIPIcs.STACS.2017.42,
  author =	{Hoyer, Peter and Komeili, Mojtaba},
  title =	{{Efficient Quantum Walk on the Grid with Multiple Marked Elements}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{42:1--42:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.42},
  URN =		{urn:nbn:de:0030-drops-69902},
  doi =		{10.4230/LIPIcs.STACS.2017.42},
  annote =	{Keywords: Quantum walks, random walks, query complexity, spatial search}
}
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