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

In this work we consider the role of entanglement assistance in quantum communication protocols, focusing, in particular, on whether the type of shared entangled state can affect the quantum communication complexity of a function. This question is interesting because in some other settings in quantum information, such as non-local games, or tasks that involve quantum communication between players and referee, or simulating bipartite unitaries or communication channels, maximally entangled states are known to be less useful as a resource than some partially entangled states. By contrast, we prove that the bounded-error entanglement-assisted quantum communication complexity of a partial or total function cannot be improved by more than a constant factor by replacing maximally entangled states with arbitrary entangled states. In particular, we show that every quantum communication protocol using Q qubits of communication and arbitrary shared entanglement can be epsilon-approximated by a protocol using O(Q/epsilon+log(1/epsilon)/epsilon) qubits of communication and only EPR pairs as shared entanglement. This conclusion is opposite of the common wisdom in the study of non-local games, where it has been shown, for example, that the I3322 inequality has a non-local strategy using a non-maximally entangled state, which surpasses the winning probability achievable by any strategy using a maximally entangled state of any dimension [Vidick and Wehner, 2011]. We leave open the question of how much the use of a shared maximally entangled state can reduce the quantum communication complexity of a function.
Our second result concerns an old question in quantum information theory: How much quantum communication is required to approximately convert one pure bipartite entangled state into another? We give simple and efficiently computable upper and lower bounds. Given two bipartite states |chi> and |upsilon>, we define a natural quantity, d_{infty}(|chi>, |upsilon>), which we call the l_{infty} Earth Mover’s distance, and we show that the communication cost of converting between |chi> and |upsilon> is upper bounded by a constant multiple of d_{infty}(|chi>, |upsilon>). Here d_{infty}(|chi>, |upsilon>) may be informally described as the minimum over all transports between the log of the Schmidt coefficients of |chi> and those of |upsilon>, of the maximum distance that any amount of mass must be moved in that transport. A precise definition is given in the introduction. Furthermore, we prove a complementary lower bound on the cost of state conversion by the epsilon-Smoothed l_{infty}-Earth Mover’s Distance, which is a natural smoothing of the l_{infty}-Earth Mover’s Distance that we will define via a connection with optimal transport theory.

Matthew Coudron and Aram W. Harrow. Universality of EPR Pairs in Entanglement-Assisted Communication Complexity, and the Communication Cost of State Conversion. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 20:1-20:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{coudron_et_al:LIPIcs.CCC.2019.20, author = {Coudron, Matthew and Harrow, Aram W.}, title = {{Universality of EPR Pairs in Entanglement-Assisted Communication Complexity, and the Communication Cost of State Conversion}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {20:1--20:25}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.20}, URN = {urn:nbn:de:0030-drops-108421}, doi = {10.4230/LIPIcs.CCC.2019.20}, annote = {Keywords: Entanglement, quantum communication complexity} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Entangled games are a quantum analog of constraint satisfaction problems and have had important applications to quantum complexity theory, quantum cryptography, and the foundations of quantum mechanics. Given a game, the basic computational problem is to compute its entangled value: the supremum success probability attainable by a quantum strategy. We study the complexity of computing the (commuting-operator) entangled value omega^* of entangled XOR games with any number of players. Based on a duality theory for systems of operator equations, we introduce necessary and sufficient criteria for an XOR game to have omega^* = 1, and use these criteria to derive the following results:
1) An algorithm for symmetric games that decides in polynomial time whether omega^* = 1 or omega^* < 1, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known.
2) A family of games with three players and with omega^* < 1, where it takes doubly exponential time for the ncSoS algorithm to witness this. By contrast, our algorithm runs in polynomial time.
3) Existence of an unsatisfiable phase for random (non-symmetric) XOR games. We show that there exists a constant C_k^{unsat} depending only on the number k of players, such that a random k-XOR game over an alphabet of size n has omega^* < 1 with high probability when the number of clauses is above C_k^{unsat} n.
4) A lower bound of Omega(n log(n)/log log(n)) on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the (3n)^{th} level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.

Adam Bene Watts, Aram W. Harrow, Gurtej Kanwar, and Anand Natarajan. Algorithms, Bounds, and Strategies for Entangled XOR Games. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{benewatts_et_al:LIPIcs.ITCS.2019.10, author = {Bene Watts, Adam and Harrow, Aram W. and Kanwar, Gurtej and Natarajan, Anand}, title = {{Algorithms, Bounds, and Strategies for Entangled XOR Games}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {10:1--10:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.10}, URN = {urn:nbn:de:0030-drops-101032}, doi = {10.4230/LIPIcs.ITCS.2019.10}, annote = {Keywords: Nonlocal games, XOR Games, Pseudotelepathy games, Multipartite entanglement} }

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

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.

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} }

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Complete Volume

**Published in:** LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

LIPIcs, Volume 27, TQC'14, Complete Volume

9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@Proceedings{flammia_et_al:LIPIcs.TQC.2014, title = {{LIPIcs, Volume 27, TQC'14, Complete Volume}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014}, URN = {urn:nbn:de:0030-drops-48241}, doi = {10.4230/LIPIcs.TQC.2014}, annote = {Keywords: Data Encryption, Coding and Information Theory, Theory of Computation} }

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Front Matter

**Published in:** LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

Front Matter, Table of Contents, Preface, Conference Organization

9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. i-xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{flammia_et_al:LIPIcs.TQC.2014.i, author = {Flammia, Steven T. and Harrow, Aram W.}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {i--xiv}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.i}, URN = {urn:nbn:de:0030-drops-48197}, doi = {10.4230/LIPIcs.TQC.2014.i}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Suppose one has access to oracles generating samples from two unknown probability distributions $p$ and $q$ on some $N$-element set. How many samples does one need to test whether the two distributions are close or far from each other in the $L_1$-norm? This and related questions have been extensively studied during the last years in the field of property testing.
In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the $L_1$-distance $\|p-q\|_1$ can be estimated with a constant precision using only $O(N^{1/2})$ queries in the quantum settings, whereas classical computers need $\Omega(N^{1-o(1)})$ queries. We also describe quantum algorithms for testing Uniformity and Orthogonality with query complexity $O(N^{1/3})$. The classical query complexity of these
problems is known to be $\Omega(N^{1/2})$. A quantum algorithm for testing Uniformity has been recently independently discovered
by Chakraborty et al. \cite{CFMW09}.

Sergey Bravyi, Aram W. Harrow, and Avinatan Hassidim. Quantum Algorithms for Testing Properties of Distributions. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 131-142, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{bravyi_et_al:LIPIcs.STACS.2010.2450, author = {Bravyi, Sergey and Harrow, Aram W. and Hassidim, Avinatan}, title = {{Quantum Algorithms for Testing Properties of Distributions}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {131--142}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2450}, URN = {urn:nbn:de:0030-drops-24502}, doi = {10.4230/LIPIcs.STACS.2010.2450}, annote = {Keywords: Quantum computing, property testing, sampling} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

Nash equilibria always exist, but are widely conjectured to require time to find that is exponential in the number of strategies, even for two-player games. By contrast, a simple quasi-polynomial time algorithm, due to Lipton, Markakis and Mehta (LMM), can find approximate Nash equilibria, in which no player can improve their utility by more than epsilon by changing their strategy. The LMM algorithm can also be used to find an approximate Nash equilibrium with near-maximal total welfare. Matching hardness results for this optimization problem re found assuming the hardness of the planted-clique problem (by Hazan and Krauthgamer) and assuming the Exponential Time Hypothesis (by Braverman, Ko and Weinstein).
In this paper we consider the application of the sum-squares (SoS) algorithm from convex optimization to the problem of optimizing over Nash equilibria. We show the first unconditional lower bounds on the number of levels of SoS needed to achieve a constant factor approximation to this problem. While it may seem that Nash equilibria do not naturally lend themselves to convex optimization, we also describe a simple LP (linear programming) hierarchy that can find an approximate Nash equilibrium in time comparable to that of the LMM algorithm, although neither algorithm is obviously a generalization of the other. This LP can be viewed as arising from the SoS algorithm at log(n) levels - matching our lower bounds. The lower bounds involve a modification of the Braverman-Ko-Weinstein embedding of CSPs into strategic games and techniques from sum-of-squares proof systems. The upper bound (i.e. analysis of the LP) uses information-theory techniques that have been recently applied to other linear- and semidefinite-programming hierarchies.

Aram Harrow, Anand V. Natarajan, and Xiaodi Wu. Tight SoS-Degree Bounds for Approximate Nash Equilibria. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 22:1-22:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{harrow_et_al:LIPIcs.CCC.2016.22, author = {Harrow, Aram and Natarajan, Anand V. and Wu, Xiaodi}, title = {{Tight SoS-Degree Bounds for Approximate Nash Equilibria}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {22:1--22:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.22}, URN = {urn:nbn:de:0030-drops-58565}, doi = {10.4230/LIPIcs.CCC.2016.22}, annote = {Keywords: Approximate Nash Equilibrium, Sum of Squares, LP, SDP} }

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