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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We study the relationship between notions of pseudorandomness in the quantum and classical worlds. Pseudorandom quantum state generator (PRSG), a pseudorandomness notion in the quantum world, is an efficient circuit that produces states that are computationally indistinguishable from Haar random states. PRSGs have found applications in quantum gravity, quantum machine learning, quantum complexity theory, and quantum cryptography. Pseudorandom generators, on the other hand, a pseudorandomness notion in the classical world, is ubiquitous to theoretical computer science. While some separation results were known between PRSGs, for some parameter regimes, and PRGs, their relationship has not been completely understood.
In this work, we show that a natural variant of pseudorandom generators called quantum pseudorandom generators (QPRGs) can be based on the existence of logarithmic output length PRSGs. Our result along with the previous separations gives a better picture regarding the relationship between the two notions. We also study the relationship between other notions, namely, pseudorandom function-like state generators and pseudorandom functions. We provide evidence that QPRGs can be as useful as PRGs by providing cryptographic applications of QPRGs such as commitments and encryption schemes.
Our primary technical contribution is a method for pseudodeterministically extracting uniformly random strings from Haar-random states.

Prabhanjan Ananth, Yao-Ting Lin, and Henry Yuen. Pseudorandom Strings from Pseudorandom Quantum States. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 6:1-6:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ananth_et_al:LIPIcs.ITCS.2024.6, author = {Ananth, Prabhanjan and Lin, Yao-Ting and Yuen, Henry}, title = {{Pseudorandom Strings from Pseudorandom Quantum States}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {6:1--6:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.6}, URN = {urn:nbn:de:0030-drops-195348}, doi = {10.4230/LIPIcs.ITCS.2024.6}, annote = {Keywords: Quantum Cryptography} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods.
Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between QMA and QMA(2), a long standing question in quantum complexity theory.

Adrian She and Henry Yuen. Unitary Property Testing Lower Bounds by Polynomials. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 96:1-96:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{she_et_al:LIPIcs.ITCS.2023.96, author = {She, Adrian and Yuen, Henry}, title = {{Unitary Property Testing Lower Bounds by Polynomials}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {96:1--96:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.96}, URN = {urn:nbn:de:0030-drops-175995}, doi = {10.4230/LIPIcs.ITCS.2023.96}, annote = {Keywords: Quantum query complexity, polynomial method, unitary property testing, quantum proofs, invariant theory, quantum recurrence time, entanglement entropy, BQP, QMA, QMA(2)} }

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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 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Whereas quantum complexity theory has traditionally been concerned with problems arising from classical complexity theory (such as computing boolean functions), it also makes sense to study the complexity of inherently quantum operations such as constructing quantum states or performing unitary transformations. With this motivation, we define models of interactive proofs for synthesizing quantum states and unitaries, where a polynomial-time quantum verifier interacts with an untrusted quantum prover, and a verifier who accepts also outputs an approximation of the target state (for the state synthesis problem) or the result of the target unitary applied to the input state (for the unitary synthesis problem); furthermore there should exist an "honest" prover which the verifier accepts with probability 1.
Our main result is a "state synthesis" analogue of the inclusion PSPACE ⊆ IP: any sequence of states computable by a polynomial-space quantum algorithm (which may run for exponential time) admits an interactive protocol of the form described above. Leveraging this state synthesis protocol, we also give a unitary synthesis protocol for polynomial space-computable unitaries that act nontrivially on only a polynomial-dimensional subspace. We obtain analogous results in the setting with multiple entangled provers as well.

Gregory Rosenthal and Henry Yuen. Interactive Proofs for Synthesizing Quantum States and Unitaries. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 112:1-112:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{rosenthal_et_al:LIPIcs.ITCS.2022.112, author = {Rosenthal, Gregory and Yuen, Henry}, title = {{Interactive Proofs for Synthesizing Quantum States and Unitaries}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {112:1--112:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.112}, URN = {urn:nbn:de:0030-drops-157086}, doi = {10.4230/LIPIcs.ITCS.2022.112}, annote = {Keywords: interactive proofs, quantum state complexity, quantum unitary complexity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

The class MIP^* is the set of languages decidable by multiprover interactive proofs with quantum entangled provers. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that MIP^* is equal to RE, the set of recursively enumerable languages. In particular this shows that the complexity of approximating the quantum value of a non-local game G is equivalent to the complexity of the Halting problem.
In this paper we investigate the complexity of deciding whether the quantum value of a non-local game G is exactly 1. This problem corresponds to a complexity class that we call zero gap MIP^*, denoted by MIP₀^*, where there is no promise gap between the verifier’s acceptance probabilities in the YES and NO cases. We prove that MIP₀^* extends beyond the first level of the arithmetical hierarchy (which includes RE and its complement coRE), and in fact is equal to Π₂⁰, the class of languages that can be decided by quantified formulas of the form ∀ y ∃ z R(x,y,z).
Combined with the previously known result that MIP₀^{co} (the commuting operator variant of MIP₀^*) is equal to coRE, our result further highlights the fascinating connection between various models of quantum multiprover interactive proofs and different classes in computability theory.

Hamoon Mousavi, Seyed Sajjad Nezhadi, and Henry Yuen. On the Complexity of Zero Gap MIP*. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 87:1-87:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{mousavi_et_al:LIPIcs.ICALP.2020.87, author = {Mousavi, Hamoon and Nezhadi, Seyed Sajjad and Yuen, Henry}, title = {{On the Complexity of Zero Gap MIP*}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {87:1--87:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.87}, URN = {urn:nbn:de:0030-drops-124940}, doi = {10.4230/LIPIcs.ICALP.2020.87}, annote = {Keywords: Quantum Complexity, Multiprover Interactive Proofs, Computability Theory} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

In this work we construct tests that allow a classical user to certify high dimensional entanglement in uncharacterized and possibly noisy quantum devices. We present a family of non-local games {G_n} that for all n certify states with entanglement of formation Omega(n). These tests can be derived from any bipartite non-local game with a classical-quantum gap. Furthermore, our tests are noise-tolerant in the sense that fault tolerant technologies are not needed to play the games; entanglement distributed over noisy channels can pass with high probability, making our tests relevant for realistic experimental settings. This is in contrast to, e.g., results on self-testing of high dimensional entanglement, which are only relevant when the noise rate goes to zero with the system's size n. As a corollary of our result, we supply a lower-bound on the entanglement cost of any state achieving a quantum advantage in a bipartite non-local game. Our proof techniques heavily rely on ideas from the work on classical and quantum parallel repetition theorems.

Rotem Arnon-Friedman and Henry Yuen. Noise-Tolerant Testing of High Entanglement of Formation. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 11:1-11:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{arnonfriedman_et_al:LIPIcs.ICALP.2018.11, author = {Arnon-Friedman, Rotem and Yuen, Henry}, title = {{Noise-Tolerant Testing of High Entanglement of Formation}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {11:1--11:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.11}, URN = {urn:nbn:de:0030-drops-90157}, doi = {10.4230/LIPIcs.ICALP.2018.11}, annote = {Keywords: device independence, quantum games, entanglement testing, noise tolerance} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

The No Low-Energy Trivial States (NLTS) conjecture of Freedman and Hastings (Quantum Information and Computation 2014), which asserts the existence of local Hamiltonians whose low-energy states cannot be generated by constant-depth quantum circuits, identifies a fundamental obstacle to resolving the quantum PCP conjecture. Progress towards the NLTS conjecture was made by Eldar and Harrow (Foundations of Computer Science 2017), who proved a closely related theorem called No Low-Error Trivial States (NLETS). In this paper, we give a much simpler proof of the NLETS theorem and use the same technique to establish superpolynomial circuit size lower bounds for noisy ground states of local Hamiltonians (assuming QCMA != QMA), resolving an open question of Eldar and Harrow. We discuss the new light our results cast on the relationship between NLTS and NLETS.
Finally, our techniques imply the existence of approximate quantum low-weight check (qLWC) codes with linear rate, linear distance, and constant weight checks. These codes are similar to quantum LDPC codes except (1) each particle may participate in a large number of checks, and (2) errors only need to be corrected up to fidelity 1 - 1/poly(n). This stands in contrast to the best-known stabilizer LDPC codes due to Freedman, Meyer, and Luo which achieve a distance of O(sqrt{n log n}).
The principal technique used in our results is to leverage the Feynman-Kitaev clock construction to approximately embed a subspace of states defined by a circuit as the ground space of a local Hamiltonian.

Chinmay Nirkhe, Umesh Vazirani, and Henry Yuen. Approximate Low-Weight Check Codes and Circuit Lower Bounds for Noisy Ground States. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 91:1-91:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{nirkhe_et_al:LIPIcs.ICALP.2018.91, author = {Nirkhe, Chinmay and Vazirani, Umesh and Yuen, Henry}, title = {{Approximate Low-Weight Check Codes and Circuit Lower Bounds for Noisy Ground States}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {91:1--91:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.91}, URN = {urn:nbn:de:0030-drops-90950}, doi = {10.4230/LIPIcs.ICALP.2018.91}, annote = {Keywords: quantum pcps, local hamiltonians, error-correcting codes} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

In a recent work, Moshkovitz [FOCS'14] presented a transformation n two-player games called "fortification", and gave an elementary proof of an (exponential decay) parallel repetition theorem for fortified two-player projection games. In this paper, we give an analytic reformulation of Moshkovitz's fortification framework, which was originally cast in combinatorial terms. This reformulation allows us to expand the scope of the fortification method to new settings.
First, we show any game (not just projection games) can be fortified, and give a simple proof of parallel repetition for general fortified games. Then, we prove parallel repetition and fortification theorems for games with players sharing quantum entanglement, as well as games with more than two players. This gives a new gap amplification method for general games in the quantum and multiplayer settings, which has recently received much interest.
An important component of our work is a variant of the fortification transformation, called "ordered fortification", that preserves the entangled value of a game. The original fortification of Moshkovitz does not in general preserve the entangled value of a game, and this was a barrier to extending the fortification framework to the quantum setting.

Mohammad Bavarian, Thomas Vidick, and Henry Yuen. Parallel Repetition via Fortification: Analytic View and the Quantum Case. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 22:1-22:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bavarian_et_al:LIPIcs.ITCS.2017.22, author = {Bavarian, Mohammad and Vidick, Thomas and Yuen, Henry}, title = {{Parallel Repetition via Fortification: Analytic View and the Quantum Case}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {22:1--22:33}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.22}, URN = {urn:nbn:de:0030-drops-81670}, doi = {10.4230/LIPIcs.ITCS.2017.22}, annote = {Keywords: Parallel repetition, quantum entanglement, non-local games} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

We investigate the value of parallel repetition of one-round games with any number of players k>=2. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games.
Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are *connected* in a certain sense.
We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question.

Irit Dinur, Prahladh Harsha, Rakesh Venkat, and Henry Yuen. Multiplayer Parallel Repetition for Expanding Games. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 37:1-37:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dinur_et_al:LIPIcs.ITCS.2017.37, author = {Dinur, Irit and Harsha, Prahladh and Venkat, Rakesh and Yuen, Henry}, title = {{Multiplayer Parallel Repetition for Expanding Games}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {37:1--37:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.37}, URN = {urn:nbn:de:0030-drops-81575}, doi = {10.4230/LIPIcs.ITCS.2017.37}, annote = {Keywords: Parallel Repetition, Multi-player, Expander} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

In this work we show a barrier towards proving a randomness-efficient parallel repetition, a promising avenue for achieving many tight inapproximability results. Feige and Kilian (STOC'95) proved an impossibility result for randomness-efficient parallel repetition for two prover games with small degree, i.e., when each prover has only few possibilities for the question of the other prover. In recent years, there have been indications that randomness-efficient parallel repetition (also called derandomized parallel repetition) might be possible for games with large degree, circumventing the impossibility result of Feige and Kilian. In particular, Dinur and Meir (CCC'11) construct games with large degree whose repetition can be derandomized using a theorem of Impagliazzo, Kabanets and Wigderson (SICOMP'12). However, obtaining derandomized parallel repetition theorems that would yield optimal inapproximability results has remained elusive.
This paper presents an explanation for the current impasse in progress, by proving a limitation on derandomized parallel repetition. We formalize two properties which we call "fortification-friendliness" and "yields robust embeddings". We show that any proof of derandomized parallel repetition achieving almost-linear blow-up cannot both (a) be fortification-friendly and (b) yield robust embeddings. Unlike Feige and Kilian, we do not require the small degree assumption.
Given that virtually all existing proofs of parallel repetition, including the derandomized parallel repetition result of Dinur and Meir, share these two properties, our no-go theorem highlights a major barrier to achieving almost-linear derandomized parallel repetition.

Dana Moshkovitz, Govind Ramnarayan, and Henry Yuen. A No-Go Theorem for Derandomized Parallel Repetition: Beyond Feige-Kilian. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 42:1-42:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{moshkovitz_et_al:LIPIcs.APPROX-RANDOM.2016.42, author = {Moshkovitz, Dana and Ramnarayan, Govind and Yuen, Henry}, title = {{A No-Go Theorem for Derandomized Parallel Repetition: Beyond Feige-Kilian}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {42:1--42:29}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.42}, URN = {urn:nbn:de:0030-drops-66657}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.42}, annote = {Keywords: Derandomization, parallel repetition, Feige-Killian, fortification} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Raz's classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known.
We prove that the entangled value of a two-player game G repeated n times in parallel is at most c_G*n^{-1/4}*log(n) for a constant c_G depending on G, provided that the entangled value of G is less than 1. In particular, this gives the first proof that the entangled value of a parallel repeated game must converge to 0 for all games whose entangled value is less than 1. Central to our proof is a combination of both classical and quantum correlated sampling.

Henry Yuen. A Parallel Repetition Theorem for All Entangled Games. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 77:1-77:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{yuen:LIPIcs.ICALP.2016.77, author = {Yuen, Henry}, title = {{A Parallel Repetition Theorem for All Entangled Games}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {77:1--77:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.77}, URN = {urn:nbn:de:0030-drops-62073}, doi = {10.4230/LIPIcs.ICALP.2016.77}, annote = {Keywords: parallel repetition, direct product theorems, entangled games, quantum games} }

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

We study how well functions over the boolean hypercube of the form f_k(x)=(|x|-k)(|x|-k-1) can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in l_{infinity}-norm as well as in l_1-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer [Lee/Raghavendra/Steurer, STOC 2015] on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on l_1-approximation of f_k; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from [Grigoriev, Comp. Compl. 2001]; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.

Troy Lee, Anupam Prakash, Ronald de Wolf, and Henry Yuen. On the Sum-of-Squares Degree of Symmetric Quadratic Functions. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 17:1-17:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{lee_et_al:LIPIcs.CCC.2016.17, author = {Lee, Troy and Prakash, Anupam and de Wolf, Ronald and Yuen, Henry}, title = {{On the Sum-of-Squares Degree of Symmetric Quadratic Functions}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {17:1--17:31}, 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.2016.17}, URN = {urn:nbn:de:0030-drops-58383}, doi = {10.4230/LIPIcs.CCC.2016.17}, annote = {Keywords: Sum-of-squares degree, approximation theory, Positivstellensatz refutations of knapsack, quantum query complexity in expectation, extension complexity} }

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

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.

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} }

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