10 Search Results for "Yao, Penghui"


Document
Invited Talk
Quantum Distributed Computing: Potential and Limitations (Invited Talk)

Authors: François Le Gall

Published in: LIPIcs, Volume 286, 27th International Conference on Principles of Distributed Systems (OPODIS 2023)


Abstract
The subject of this talk is quantum distributed computing, i.e., distributed computing where the processors of the network can exchange quantum messages. In the first part of the talk I survey recent results [Taisuke Izumi and François Le Gall, 2019; Taisuke Izumi et al., 2020; François Le Gall and Frédéric Magniez, 2018; François Le Gall et al., 2019; Xudong Wu and Penghui Yao, 2022] and some older results [Michael Ben-Or and Avinatan Hassidim, 2005; Seiichiro Tani et al., 2012] that show the potential of quantum distributed algorithms. In the second part I present our recent work [Xavier Coiteux-Roy et al., 2023] showing the limitations of quantum distributed algorithms for approximate graph coloring. Finally, I mention interesting and important open questions in quantum distributed computing.

Cite as

François Le Gall. Quantum Distributed Computing: Potential and Limitations (Invited Talk). In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{legall:LIPIcs.OPODIS.2023.2,
  author =	{Le Gall, Fran\c{c}ois},
  title =	{{Quantum Distributed Computing: Potential and Limitations}},
  booktitle =	{27th International Conference on Principles of Distributed Systems (OPODIS 2023)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-308-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{286},
  editor =	{Bessani, Alysson and D\'{e}fago, Xavier and Nakamura, Junya and Wada, Koichi and Yamauchi, Yukiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2023.2},
  URN =		{urn:nbn:de:0030-drops-194925},
  doi =		{10.4230/LIPIcs.OPODIS.2023.2},
  annote =	{Keywords: Quantum computing, distributed algorithms, CONGEST model, LOCAL model}
}
Document
On the Fine-Grained Query Complexity of Symmetric Functions

Authors: Supartha Podder, Penghui Yao, and Zekun Ye

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)


Abstract
Watrous conjectured that the randomized and quantum query complexities of symmetric functions are polynomially equivalent, which was resolved by Ambainis and Aaronson [Scott Aaronson and Andris Ambainis, 2014], and was later improved in [André Chailloux, 2019; Shalev Ben-David et al., 2020]. This paper explores a fine-grained version of the Watrous conjecture, including the randomized and quantum algorithms with success probabilities arbitrarily close to 1/2. Our contributions include the following: 1) An analysis of the optimal success probability of quantum and randomized query algorithms of two fundamental partial symmetric Boolean functions given a fixed number of queries. We prove that for any quantum algorithm computing these two functions using T queries, there exist randomized algorithms using poly(T) queries that achieve the same success probability as the quantum algorithm, even if the success probability is arbitrarily close to 1/2. These two classes of functions are instrumental in analyzing general symmetric functions. 2) We establish that for any total symmetric Boolean function f, if a quantum algorithm uses T queries to compute f with success probability 1/2+β, then there exists a randomized algorithm using O(T²) queries to compute f with success probability 1/2 + Ω(δβ²) on a 1-δ fraction of inputs, where β,δ can be arbitrarily small positive values. As a corollary, we prove a randomized version of Aaronson-Ambainis Conjecture [Scott Aaronson and Andris Ambainis, 2014] for total symmetric Boolean functions in the regime where the success probability of algorithms can be arbitrarily close to 1/2. 3) We present polynomial equivalences for several fundamental complexity measures of partial symmetric Boolean functions. Specifically, we first prove that for certain partial symmetric Boolean functions, quantum query complexity is at most quadratic in approximate degree for any error arbitrarily close to 1/2. Next, we show exact quantum query complexity is at most quadratic in degree. Additionally, we give the tight bounds of several complexity measures, indicating their polynomial equivalence. Conversely, we exhibit an exponential separation between randomized and exact quantum query complexity for certain partial symmetric Boolean functions.

Cite as

Supartha Podder, Penghui Yao, and Zekun Ye. On the Fine-Grained Query Complexity of Symmetric Functions. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 55:1-55:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{podder_et_al:LIPIcs.ISAAC.2023.55,
  author =	{Podder, Supartha and Yao, Penghui and Ye, Zekun},
  title =	{{On the Fine-Grained Query Complexity of Symmetric Functions}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{55:1--55:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.55},
  URN =		{urn:nbn:de:0030-drops-193570},
  doi =		{10.4230/LIPIcs.ISAAC.2023.55},
  annote =	{Keywords: Query complexity, Symmetric functions, Quantum advantages}
}
Document
Track A: Algorithms, Complexity and Games
Decidability of Fully Quantum Nonlocal Games with Noisy Maximally Entangled States

Authors: Minglong Qin and Penghui Yao

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


Abstract
This paper considers the decidability of fully quantum nonlocal games with noisy maximally entangled states. Fully quantum nonlocal games are a generalization of nonlocal games, where both questions and answers are quantum and the referee performs a binary POVM measurement to decide whether they win the game after receiving the quantum answers from the players. The quantum value of a fully quantum nonlocal game is the supremum of the probability that they win the game, where the supremum is taken over all the possible entangled states shared between the players and all the valid quantum operations performed by the players. The seminal work MIP^* = RE [Zhengfeng Ji et al., 2020; Zhengfeng Ji et al., 2020] implies that it is undecidable to approximate the quantum value of a fully nonlocal game. This still holds even if the players are only allowed to share (arbitrarily many copies of) maximally entangled states. This paper investigates the case that the shared maximally entangled states are noisy. We prove that there is a computable upper bound on the copies of noisy maximally entangled states for the players to win a fully quantum nonlocal game with a probability arbitrarily close to the quantum value. This implies that it is decidable to approximate the quantum values of these games. Hence, the hardness of approximating the quantum value of a fully quantum nonlocal game is not robust against the noise in the shared states. This paper is built on the framework for the decidability of non-interactive simulations of joint distributions [Badih Ghazi et al., 2016; De et al., 2018; Ghazi et al., 2018] and generalizes the analogous result for nonlocal games in [Qin and Yao, 2021]. We extend the theory of Fourier analysis to the space of super-operators and prove several key results including an invariance principle and a dimension reduction for super-operators. These results are interesting in their own right and are believed to have further applications.

Cite as

Minglong Qin and Penghui Yao. Decidability of Fully Quantum Nonlocal Games with Noisy Maximally Entangled States. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 97:1-97:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{qin_et_al:LIPIcs.ICALP.2023.97,
  author =	{Qin, Minglong and Yao, Penghui},
  title =	{{Decidability of Fully Quantum Nonlocal Games with Noisy Maximally Entangled States}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{97:1--97:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.97},
  URN =		{urn:nbn:de:0030-drops-181499},
  doi =		{10.4230/LIPIcs.ICALP.2023.97},
  annote =	{Keywords: Fully quantum nonlocal games, Fourier analysis, Dimension reduction}
}
Document
Track A: Algorithms, Complexity and Games
Polynomial-Time Approximation of Zero-Free Partition Functions

Authors: Penghui Yao, Yitong Yin, and Xinyuan Zhang

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)


Abstract
Zero-free based algorithms are a major technique for deterministic approximate counting. In Barvinok’s original framework [Barvinok, 2017], by calculating truncated Taylor expansions, a quasi-polynomial time algorithm was given for estimating zero-free partition functions. Patel and Regts [Patel and Regts, 2017] later gave a refinement of Barvinok’s framework, which gave a polynomial-time algorithm for a class of zero-free graph polynomials that can be expressed as counting induced subgraphs in bounded-degree graphs. In this paper, we give a polynomial-time algorithm for estimating classical and quantum partition functions specified by local Hamiltonians with bounded maximum degree, assuming a zero-free property for the temperature. Consequently, when the inverse temperature is close enough to zero by a constant gap, we have a polynomial-time approximation algorithm for all such partition functions. Our result is based on a new abstract framework that extends and generalizes the approach of Patel and Regts.

Cite as

Penghui Yao, Yitong Yin, and Xinyuan Zhang. Polynomial-Time Approximation of Zero-Free Partition Functions. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 108:1-108:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{yao_et_al:LIPIcs.ICALP.2022.108,
  author =	{Yao, Penghui and Yin, Yitong and Zhang, Xinyuan},
  title =	{{Polynomial-Time Approximation of Zero-Free Partition Functions}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{108:1--108:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.108},
  URN =		{urn:nbn:de:0030-drops-164494},
  doi =		{10.4230/LIPIcs.ICALP.2022.108},
  annote =	{Keywords: partition function, zero-freeness, local Hamiltonian}
}
Document
Lower Bounds for Function Inversion with Quantum Advice

Authors: Kai-Min Chung, Tai-Ning Liao, and Luowen Qian

Published in: LIPIcs, Volume 163, 1st Conference on Information-Theoretic Cryptography (ITC 2020)


Abstract
Function inversion is the problem that given a random function f: [M] → [N], we want to find pre-image of any image f^{-1}(y) in time T. In this work, we revisit this problem under the preprocessing model where we can compute some auxiliary information or advice of size S that only depends on f but not on y. It is a well-studied problem in the classical settings, however, it is not clear how quantum algorithms can solve this task any better besides invoking Grover’s algorithm [Grover, 1996], which does not leverage the power of preprocessing. Nayebi et al. [Nayebi et al., 2015] proved a lower bound ST² ≥ ̃Ω(N) for quantum algorithms inverting permutations, however, they only consider algorithms with classical advice. Hhan et al. [Minki Hhan et al., 2019] subsequently extended this lower bound to fully quantum algorithms for inverting permutations. In this work, we give the same asymptotic lower bound to fully quantum algorithms for inverting functions for fully quantum algorithms under the regime where M = O(N). In order to prove these bounds, we generalize the notion of quantum random access code, originally introduced by Ambainis et al. [Ambainis et al., 1999], to the setting where we are given a list of (not necessarily independent) random variables, and we wish to compress them into a variable-length encoding such that we can retrieve a random element just using the encoding with high probability. As our main technical contribution, we give a nearly tight lower bound (for a wide parameter range) for this generalized notion of quantum random access codes, which may be of independent interest.

Cite as

Kai-Min Chung, Tai-Ning Liao, and Luowen Qian. Lower Bounds for Function Inversion with Quantum Advice. In 1st Conference on Information-Theoretic Cryptography (ITC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 163, pp. 8:1-8:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{chung_et_al:LIPIcs.ITC.2020.8,
  author =	{Chung, Kai-Min and Liao, Tai-Ning and Qian, Luowen},
  title =	{{Lower Bounds for Function Inversion with Quantum Advice}},
  booktitle =	{1st Conference on Information-Theoretic Cryptography (ITC 2020)},
  pages =	{8:1--8:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-151-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{163},
  editor =	{Tauman Kalai, Yael and Smith, Adam D. and Wichs, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2020.8},
  URN =		{urn:nbn:de:0030-drops-121134},
  doi =		{10.4230/LIPIcs.ITC.2020.8},
  annote =	{Keywords: Cryptanalysis, Data Structures, Quantum Query Complexity}
}
Document
Improved Bounds on Fourier Entropy and Min-Entropy

Authors: Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucký, Nitin Saurabh, and Ronald de Wolf

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)


Abstract
Given a Boolean function f:{-1,1}ⁿ→ {-1,1}, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f̂(S)². The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [E. Friedgut and G. Kalai, 1996] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C>0 such that ℍ(f̂²)≤ C⋅ Inf(f), where ℍ(f̂²) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f? In this paper we present three new contributions towards the FEI conjecture: ii) Our first contribution shows that ℍ(f̂²) ≤ 2⋅ aUC^⊕(f), where aUC^⊕(f) is the average unambiguous parity-certificate complexity of f. This improves upon several bounds shown by Chakraborty et al. [S. Chakraborty et al., 2016]. We further improve this bound for unambiguous DNFs. iii) We next consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture posed by O'Donnell and others [R. O'Donnell et al., 2011; R. O'Donnell, 2014] which asks if ℍ_{∞}(f̂²) ≤ C⋅ Inf(f), where ℍ_{∞}(f̂²) is the min-entropy of the Fourier distribution. We show ℍ_{∞}(f̂²) ≤ 2⋅?_{min}^⊕(f), where ?_{min}^⊕(f) is the minimum parity certificate complexity of f. We also show that for all ε ≥ 0, we have ℍ_{∞}(f̂²) ≤ 2log (‖f̂‖_{1,ε}/(1-ε)), where ‖f̂‖_{1,ε} is the approximate spectral norm of f. As a corollary, we verify the FMEI conjecture for the class of read-k DNFs (for constant k). iv) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2^ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.

Cite as

Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucký, Nitin Saurabh, and Ronald de Wolf. Improved Bounds on Fourier Entropy and Min-Entropy. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 45:1-45:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{arunachalam_et_al:LIPIcs.STACS.2020.45,
  author =	{Arunachalam, Srinivasan and Chakraborty, Sourav and Kouck\'{y}, Michal and Saurabh, Nitin and de Wolf, Ronald},
  title =	{{Improved Bounds on Fourier Entropy and Min-Entropy}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{45:1--45:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.45},
  URN =		{urn:nbn:de:0030-drops-119062},
  doi =		{10.4230/LIPIcs.STACS.2020.45},
  annote =	{Keywords: Fourier analysis of Boolean functions, FEI conjecture, query complexity, polynomial approximation, approximate degree, certificate complexity}
}
Document
Optimality of Linear Sketching Under Modular Updates

Authors: Kaave Hosseini, Shachar Lovett, and Grigory Yaroslavtsev

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
We study the relation between streaming algorithms and linear sketching algorithms, in the context of binary updates. We show that for inputs in n dimensions, the existence of efficient streaming algorithms which can process Omega(n^2) updates implies efficient linear sketching algorithms with comparable cost. This improves upon the previous work of Li, Nguyen and Woodruff [Yi Li et al., 2014] and Ai, Hu, Li and Woodruff [Yuqing Ai et al., 2016] which required a triple-exponential number of updates to achieve a similar result for updates over integers. We extend our results to updates modulo p for integers p >= 2, and to approximation instead of exact computation.

Cite as

Kaave Hosseini, Shachar Lovett, and Grigory Yaroslavtsev. Optimality of Linear Sketching Under Modular Updates. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{hosseini_et_al:LIPIcs.CCC.2019.13,
  author =	{Hosseini, Kaave and Lovett, Shachar and Yaroslavtsev, Grigory},
  title =	{{Optimality of Linear Sketching Under Modular Updates}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{13:1--13:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.13},
  URN =		{urn:nbn:de:0030-drops-108355},
  doi =		{10.4230/LIPIcs.CCC.2019.13},
  annote =	{Keywords: communication complexity, linear sketching, streaming algorithm}
}
Document
Lower Bound on Expected Communication Cost of Quantum Huffman Coding

Authors: Anurag Anshu, Ankit Garg, Aram W. Harrow, and Penghui Yao

Published in: LIPIcs, Volume 61, 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)


Abstract
Data compression is a fundamental problem in quantum and classical information theory. A typical version of the problem is that the sender Alice receives a (classical or quantum) state from some known ensemble and needs to transmit them to the receiver Bob with average error below some specified bound. We consider the case in which the message can have a variable length and the goal is to minimize its expected length. For classical messages this problem has a well-known solution given by Huffman coding. In this scheme, the expected length of the message is equal to the Shannon entropy of the source (with a constant additive factor) and the scheme succeeds with zero error. This is a single-shot result which implies the asymptotic result, viz. Shannon's source coding theorem, by encoding each state sequentially. For the quantum case, the asymptotic compression rate is given by the von-Neumann entropy. However, we show that there is no one-shot scheme which is able to match this rate, even if interactive communication is allowed. This is a relatively rare case in quantum information theory when the cost of a quantum task is significantly different than the classical analogue. Our result has implications for direct sum theorems in quantum communication complexity and one-shot formulations of Quantum Reverse Shannon theorem.

Cite as

Anurag Anshu, Ankit Garg, Aram W. Harrow, and Penghui Yao. Lower Bound on Expected Communication Cost of Quantum Huffman Coding. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 61, pp. 3:1-3:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{anshu_et_al:LIPIcs.TQC.2016.3,
  author =	{Anshu, Anurag and Garg, Ankit and Harrow, Aram W. and Yao, Penghui},
  title =	{{Lower Bound on Expected Communication Cost of Quantum Huffman Coding}},
  booktitle =	{11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)},
  pages =	{3:1--3:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-019-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{61},
  editor =	{Broadbent, Anne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2016.3},
  URN =		{urn:nbn:de:0030-drops-66843},
  doi =		{10.4230/LIPIcs.TQC.2016.3},
  annote =	{Keywords: Quantum information, quantum communication, expected communica- tion cost, huffman coding}
}
Document
Parallel Repetition for Entangled k-player Games via Fast Quantum Search

Authors: Kai-Min Chung, Xiaodi Wu, and Henry Yuen

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)


Abstract
We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a k-player free game G with entangled value val^*(G) = 1 - epsilon, the n-fold repetition of G has entangled value val^*(G^(\otimes n)) at most (1 - epsilon^(3/2))^(Omega(n/sk^4)), where s is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is val(G^(\otimes n)) <= (1 - epsilon^2)^(Omega(n/s)), due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition. Our second theorem handles the broader class of free games G where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by (1 - epsilon^2)^(Omega(n/sk^2)). We also show that the dependence of the exponent on k is necessary: we exhibit a k-player free game G and n >= 1 such that val^*(G^(\otimes n)) >= val^*(G)^(n/k). Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: in particular, our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic Grover speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.

Cite as

Kai-Min Chung, Xiaodi Wu, and Henry Yuen. Parallel Repetition for Entangled k-player Games via Fast Quantum Search. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 512-536, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{chung_et_al:LIPIcs.CCC.2015.512,
  author =	{Chung, Kai-Min and Wu, Xiaodi and Yuen, Henry},
  title =	{{Parallel Repetition for Entangled k-player Games via Fast Quantum Search}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{512--536},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.512},
  URN =		{urn:nbn:de:0030-drops-50727},
  doi =		{10.4230/LIPIcs.CCC.2015.512},
  annote =	{Keywords: Parallel repetition, quantum entanglement, communication complexity}
}
Document
Certifying Equality With Limited Interaction

Authors: Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)


Abstract
The EQUALITY problem is usually one’s first encounter with communication complexity and is one of the most fundamental problems in the field. Although its deterministic and randomized communication complexity were settled decades ago, we find several new things to say about the problem by focusing on three subtle aspects. The first is to consider the expected communication cost (at a worst-case input) for a protocol that uses limited interaction—i.e., a bounded number of rounds of communication—and whose error probability is zero or close to it. The second is to treat the false negative error rate separately from the false positive error rate. The third is to consider the information cost of such protocols. We obtain asymptotically optimal rounds-versus-cost tradeoffs for EQUALITY: both expected communication cost and information cost scale as Theta(log log ... log n), with r-1 logs, where r is the number of rounds. These bounds hold even when the false negative rate approaches 1. For the case of zero-error communication cost, we obtain essentially matching bounds, up to a tiny additive constant. We also provide some applications.

Cite as

Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev. Certifying Equality With Limited Interaction. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 545-581, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


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@InProceedings{brody_et_al:LIPIcs.APPROX-RANDOM.2014.545,
  author =	{Brody, Joshua and Chakrabarti, Amit and Kondapally, Ranganath and Woodruff, David P. and Yaroslavtsev, Grigory},
  title =	{{Certifying Equality With Limited Interaction}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{545--581},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.545},
  URN =		{urn:nbn:de:0030-drops-47229},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.545},
  annote =	{Keywords: equality, communication complexity, information complexity}
}
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