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

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.

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)

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@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} }

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**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

It is a long-standing open question in quantum complexity theory whether the definition of non-deterministic quantum computation requires quantum witnesses (QMA) or if classical witnesses suffice (QCMA). We make progress on this question by constructing a randomized classical oracle separating the respective computational complexity classes. Previous separations [Aaronson and Kuperberg, 2007; Bill Fefferman and Shelby Kimmel, 2018] required a quantum unitary oracle. The separating problem is deciding whether a distribution supported on regular un-directed graphs either consists of multiple connected components (yes instances) or consists of one expanding connected component (no instances) where the graph is given in an adjacency-list format by the oracle. Therefore, the oracle is a distribution over n-bit boolean functions.

Anand Natarajan and Chinmay Nirkhe. A Distribution Testing Oracle Separating QMA and QCMA. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 22:1-22:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{natarajan_et_al:LIPIcs.CCC.2023.22, author = {Natarajan, Anand and Nirkhe, Chinmay}, title = {{A Distribution Testing Oracle Separating QMA and QCMA}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {22:1--22:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.22}, URN = {urn:nbn:de:0030-drops-182928}, doi = {10.4230/LIPIcs.CCC.2023.22}, annote = {Keywords: quantum non-determinism, complexity theory} }

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**Published in:** LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

It is a useful fact in classical computer science that many search problems are reducible to decision problems; this has led to decision problems being regarded as the de facto computational task to study in complexity theory. In this work, we explore search-to-decision reductions for quantum search problems, wherein a quantum algorithm makes queries to a classical decision oracle to output a desired quantum state. In particular, we focus on search-to-decision reductions for QMA, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a QMA problem up to inverse polynomial precision by making one query to a PP decision oracle. We complement this result by showing that QMA-search does not reduce to QMA-decision in polynomial-time, relative to a quantum oracle.
We also explore the more general state synthesis problem, in which the goal is to efficiently synthesize a target state by making queries to a classical oracle encoding the state. We prove that there exists a classical oracle with which any quantum state can be synthesized to inverse polynomial precision using only one oracle query and to inverse exponential precision using two oracle queries. This answers an open question of Aaronson [Aaronson, 2016], who presented a state synthesis algorithm that makes O(n) queries to a classical oracle to prepare an n-qubit state, and asked if the query complexity could be made sublinear.

Sandy Irani, Anand Natarajan, Chinmay Nirkhe, Sujit Rao, and Henry Yuen. Quantum Search-To-Decision Reductions and the State Synthesis Problem. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{irani_et_al:LIPIcs.CCC.2022.5, author = {Irani, Sandy and Natarajan, Anand and Nirkhe, Chinmay and Rao, Sujit and Yuen, Henry}, title = {{Quantum Search-To-Decision Reductions and the State Synthesis Problem}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {5:1--5:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.5}, URN = {urn:nbn:de:0030-drops-165674}, doi = {10.4230/LIPIcs.CCC.2022.5}, annote = {Keywords: Search-to-decision, state synthesis, quantum computing} }

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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.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 102, 33rd Computational Complexity Conference (CCC 2018)

The article, published on June 4th, 2018 in the CCC 2018 proceedings, has been retracted by agreement between the authors, the editor(s), and the publisher Schloss Dagstuhl / LIPIcs. The retraction has been agreed due to an error in the proof of the main result. This error is carried over from an error in the referenced paper “Three-player entangled XOR games are NP-hard to approximate” by Thomas Vidick (SICOMP ’16). That paper was used in an essential way to obtain the present result, and the error cannot be addressed through an erratum. See Retraction Notice on the last page of the PDF.
We show that it is NP-hard to approximate, to within an additive constant, the maximum success probability of players sharing quantum entanglement in a two-player game with classical questions of logarithmic length and classical answers of constant length. As a corollary, the inclusion NEXP subseteq MIP^*, first shown by Ito and Vidick (FOCS'12) with three provers, holds with two provers only. The proof is based on a simpler, improved analysis of the low-degree test of Raz and Safra (STOC'97) against two entangled provers.

Anand Natarajan and Thomas Vidick. Retracted: Two-Player Entangled Games are NP-Hard. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{natarajan_et_al:LIPIcs.CCC.2018.20, author = {Natarajan, Anand and Vidick, Thomas}, title = {{Retracted: Two-Player Entangled Games are NP-Hard}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {20:1--20:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.20}, URN = {urn:nbn:de:0030-drops-88696}, doi = {10.4230/LIPIcs.CCC.2018.20}, annote = {Keywords: low-degree testing, entangled nonlocal games, multi-prover interactive proof systems} }