10 Search Results for "Yao, Penghui"

Document
DOI: 10.4230/LIPIcs.DISC.2024.4
Almost Optimal Algorithms for Token Collision in Anonymous Networks

Authors: Sirui Bai, Xinyu Fu, Xudong Wu, Penghui Yao, and Chaodong Zheng

Published in: LIPIcs, Volume 319, 38th International Symposium on Distributed Computing (DISC 2024)

Abstract
In distributed systems, situations often arise where some nodes each holds a collection of tokens, and all nodes collectively need to determine whether all tokens are distinct. For example, if each token represents a logged-in user, the problem corresponds to checking whether there are duplicate logins. Similarly, if each token represents a data object or a timestamp, the problem corresponds to checking whether there are conflicting operations in distributed databases. In distributed computing theory, unique identifiers generation is also related to this problem: each node generates one token, which is its identifier, then a verification phase is needed to ensure that all identifiers are unique. In this paper, we formalize and initiate the study of token collision. In this problem, a collection of k tokens, each represented by some length-L bit string, are distributed to n nodes of an anonymous CONGEST network in an arbitrary manner. The nodes need to determine whether there are tokens with an identical value. We present near optimal deterministic algorithms for the token collision problem with Õ(D+k⋅L/log n) round complexity, where D denotes the network diameter. Besides high efficiency, the prior knowledge required by our algorithms is also limited. For completeness, we further present a near optimal randomized algorithm for token collision.
Cite as

Sirui Bai, Xinyu Fu, Xudong Wu, Penghui Yao, and Chaodong Zheng. Almost Optimal Algorithms for Token Collision in Anonymous Networks. In 38th International Symposium on Distributed Computing (DISC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 319, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{bai_et_al:LIPIcs.DISC.2024.4,
  author =	{Bai, Sirui and Fu, Xinyu and Wu, Xudong and Yao, Penghui and Zheng, Chaodong},
  title =	{{Almost Optimal Algorithms for Token Collision in Anonymous Networks}},
  booktitle =	{38th International Symposium on Distributed Computing (DISC 2024)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-352-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{319},
  editor =	{Alistarh, Dan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2024.4},
  URN =		{urn:nbn:de:0030-drops-212319},
  doi =		{10.4230/LIPIcs.DISC.2024.4},
  annote =	{Keywords: Token collision, anonymous networks, deterministic algorithms}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.30
The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise

Authors: Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
The class MIP^* of quantum multiprover interactive proof systems with entanglement is much more powerful than its classical counterpart MIP [Babai et al., 1991; Zhengfeng Ji et al., 2020; Zhengfeng Ji et al., 2020]: while MIP = NEXP, the quantum class MIP^* is equal to RE, a class including the halting problem. This is because the provers in MIP^* can share unbounded quantum entanglement. However, recent works [Qin and Yao, 2021; Qin and Yao, 2023] have shown that this advantage is significantly reduced if the provers' shared state contains noise. This paper attempts to exactly characterize the effect of noise on the computational power of quantum multiprover interactive proof systems. We investigate the quantum two-prover one-round interactive system MIP^*[poly,O(1)], where the verifier sends polynomially many bits to the provers and the provers send back constantly many bits. We show noise completely destroys the computational advantage given by shared entanglement in this model. Specifically, we show that if the provers are allowed to share arbitrarily many EPR states, where each EPR state is affected by an arbitrarily small constant amount of noise, the resulting complexity class is equivalent to NEXP = MIP. This improves significantly on the previous best-known bound of NEEEXP (nondeterministic triply exponential time) [Qin and Yao, 2021]. We also show that this collapse in power is due to the noise, rather than the O(1) answer size, by showing that allowing for noiseless EPR states gives the class the full power of RE = MIP^*[poly, poly]. Along the way, we develop two technical tools of independent interest. First, we give a new, deterministic tester for the positivity of an exponentially large matrix, provided it has a low-degree Fourier decomposition in terms of Pauli matrices. Secondly, we develop a new invariance principle for smooth matrix functions having bounded third-order Fréchet derivatives or which are Lipschitz continuous.
Cite as

Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao. The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 30:1-30:71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{dong_et_al:LIPIcs.CCC.2024.30,
  author =	{Dong, Yangjing and Fu, Honghao and Natarajan, Anand and Qin, Minglong and Xu, Haochen and Yao, Penghui},
  title =	{{The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{30:1--30:71},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.30},
  URN =		{urn:nbn:de:0030-drops-204263},
  doi =		{10.4230/LIPIcs.CCC.2024.30},
  annote =	{Keywords: Interactive proofs, Quantum complexity theory, Quantum entanglement, Fourier analysis, Matrix analysis, Invariance principle, Derandomization, PCP, Locally testable code, Positivity testing}
}
Document
DOI: 10.4230/LIPIcs.STACS.2024.39
Quantum and Classical Communication Complexity of Permutation-Invariant Functions

Authors: Ziyi Guan, Yunqi Huang, Penghui Yao, and Zekun Ye

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract
This paper gives a nearly tight characterization of the quantum communication complexity of the permutation-invariant Boolean functions. With such a characterization, we show that the quantum and randomized communication complexity of the permutation-invariant Boolean functions are quadratically equivalent (up to a logarithmic factor). Our results extend a recent line of research regarding query complexity [Scott Aaronson and Andris Ambainis, 2014; André Chailloux, 2019; Shalev Ben-David et al., 2020] to communication complexity, showing symmetry prevents exponential quantum speedups. Furthermore, we show the Log-rank Conjecture holds for any non-trivial total permutation-invariant Boolean function. Moreover, we establish a relationship between the quantum/classical communication complexity and the approximate rank of permutation-invariant Boolean functions. This implies the correctness of the Log-approximate-rank Conjecture for permutation-invariant Boolean functions in both randomized and quantum settings (up to a logarithmic factor).
Cite as

Ziyi Guan, Yunqi Huang, Penghui Yao, and Zekun Ye. Quantum and Classical Communication Complexity of Permutation-Invariant Functions. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{guan_et_al:LIPIcs.STACS.2024.39,
  author =	{Guan, Ziyi and Huang, Yunqi and Yao, Penghui and Ye, Zekun},
  title =	{{Quantum and Classical Communication Complexity of Permutation-Invariant Functions}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{39:1--39:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.39},
  URN =		{urn:nbn:de:0030-drops-197498},
  doi =		{10.4230/LIPIcs.STACS.2024.39},
  annote =	{Keywords: Communication complexity, Permutation-invariant functions, Log-rank Conjecture, Quantum advantages}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.OPODIS.2023.2
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 (2024)

Copy BibTex To Clipboard

@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
DOI: 10.4230/LIPIcs.ISAAC.2023.55
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)

Copy BibTex To Clipboard

@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
DOI: 10.4230/LIPIcs.ICALP.2023.97
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)

Copy BibTex To Clipboard

@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
DOI: 10.4230/LIPIcs.ICALP.2022.108
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)

Copy BibTex To Clipboard

@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
DOI: 10.4230/LIPIcs.ITC.2020.8
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)

Copy BibTex To Clipboard

@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
DOI: 10.4230/LIPIcs.STACS.2020.45
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)

Copy BibTex To Clipboard

@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
DOI: 10.4230/LIPIcs.TQC.2016.3
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)

Copy BibTex To Clipboard

@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}
}
  • Refine by Author
  • 7 Yao, Penghui
  • 2 Qin, Minglong
  • 2 Ye, Zekun
  • 1 Anshu, Anurag
  • 1 Arunachalam, Srinivasan
  • Show More...

  • Refine by Classification

  • Refine by Keyword
  • 2 Fourier analysis
  • 2 Quantum advantages
  • 1 CONGEST model
  • 1 Communication complexity
  • 1 Cryptanalysis
  • Show More...

  • Refine by Type
  • 10 document

  • Refine by Publication Year
  • 4 2024
  • 2 2020
  • 2 2023
  • 1 2016
  • 1 2022

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact