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Invited Talk

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

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.

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)

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

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

The generation and verification of quantum states are fundamental tasks for quantum information processing that have recently been investigated by Irani, Natarajan, Nirkhe, Rao and Yuen [CCC 2022], Rosenthal and Yuen [ITCS 2022], Metger and Yuen [QIP 2023] under the term state synthesis. This paper studies this concept from the viewpoint of quantum distributed computing, and especially distributed quantum Merlin-Arthur (dQMA) protocols. We first introduce a novel task, on a line, called state generation with distributed inputs (SGDI). In this task, the goal is to generate the quantum state U|ψ⟩ at the rightmost node of the line, where |ψ⟩ is a quantum state given at the leftmost node and U is a unitary matrix whose description is distributed over the nodes of the line. We give a dQMA protocol for SGDI and utilize this protocol to construct a dQMA protocol for the Set Equality problem studied by Naor, Parter and Yogev [SODA 2020], and complement our protocol by showing classical lower bounds for this problem. Our second contribution is a dQMA protocol, based on a recent work by Zhu and Hayashi [Physical Review A, 2019], to create EPR-pairs between adjacent nodes of a network without quantum communication. As an application of this dQMA protocol, we prove a general result showing how to convert any dQMA protocol on an arbitrary network into another dQMA protocol where the verification stage does not require any quantum communication.

François Le Gall, Masayuki Miyamoto, and Harumichi Nishimura. Distributed Merlin-Arthur Synthesis of Quantum States and Its Applications. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{legall_et_al:LIPIcs.MFCS.2023.63, author = {Le Gall, Fran\c{c}ois and Miyamoto, Masayuki and Nishimura, Harumichi}, title = {{Distributed Merlin-Arthur Synthesis of Quantum States and Its Applications}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {63:1--63:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.63}, URN = {urn:nbn:de:0030-drops-185975}, doi = {10.4230/LIPIcs.MFCS.2023.63}, annote = {Keywords: distributed quantum Merlin-Arthur, distributed verification, quantum computation} }

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Track A: Algorithms, Complexity and Games

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

Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with 6-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to 1/2 with a ground state.
In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry.

Chris Cade, Marten Folkertsma, Sevag Gharibian, Ryu Hayakawa, François Le Gall, Tomoyuki Morimae, and Jordi Weggemans. Improved Hardness Results for the Guided Local Hamiltonian Problem. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cade_et_al:LIPIcs.ICALP.2023.32, author = {Cade, Chris and Folkertsma, Marten and Gharibian, Sevag and Hayakawa, Ryu and Le Gall, Fran\c{c}ois and Morimae, Tomoyuki and Weggemans, Jordi}, title = {{Improved Hardness Results for the Guided Local Hamiltonian Problem}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {32:1--32:19}, 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.2023.32}, URN = {urn:nbn:de:0030-drops-180840}, doi = {10.4230/LIPIcs.ICALP.2023.32}, annote = {Keywords: Quantum computing, Quantum advantage, Quantum Chemistry, Guided Local Hamiltonian Problem} }

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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

The study of distributed interactive proofs was initiated by Kol, Oshman, and Saxena [PODC 2018] as a generalization of distributed decision mechanisms (proof-labeling schemes, etc.), and has received a lot of attention in recent years. In distributed interactive proofs, the nodes of an n-node network G can exchange short messages (called certificates) with a powerful prover. The goal is to decide if the input (including G itself) belongs to some language, with as few turns of interaction and as few bits exchanged between nodes and the prover as possible. There are several results showing that the size of certificates can be reduced drastically with a constant number of interactions compared to non-interactive distributed proofs.
In this paper, we introduce the quantum counterpart of distributed interactive proofs: certificates can now be quantum bits, and the nodes of the network can perform quantum computation. The first result of this paper shows that by using distributed quantum interactive proofs, the number of interactions can be significantly reduced. More precisely, our result shows that for any constant k, the class of languages that can be decided by a k-turn classical (i.e., non-quantum) distributed interactive protocol with f(n)-bit certificate size is contained in the class of languages that can be decided by a 5-turn distributed quantum interactive protocol with O(f(n))-bit certificate size. We also show that if we allow to use shared randomness, the number of turns can be reduced to three. Since no similar turn-reduction classical technique is currently known, our result gives evidence of the power of quantum computation in the setting of distributed interactive proofs as well.
As a corollary of our results, we show that there exist 5-turn/3-turn distributed quantum interactive protocols with small certificate size for problems that have been considered in prior works on distributed interactive proofs such as [Kol, Oshman, and Saxena PODC 2018, Naor, Parter, and Yogev SODA 2020].
We then utilize the framework of the distributed quantum interactive proofs to test closeness of two quantum states each of which is distributed over the entire network.

François Le Gall, Masayuki Miyamoto, and Harumichi Nishimura. Distributed Quantum Interactive Proofs. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 42:1-42:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{legall_et_al:LIPIcs.STACS.2023.42, author = {Le Gall, Fran\c{c}ois and Miyamoto, Masayuki and Nishimura, Harumichi}, title = {{Distributed Quantum Interactive Proofs}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {42:1--42:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.42}, URN = {urn:nbn:de:0030-drops-176949}, doi = {10.4230/LIPIcs.STACS.2023.42}, annote = {Keywords: distributed interactive proofs, distributed verification, quantum computation} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Recently, Chia, Chung and Lai (STOC 2020) and Coudron and Menda (STOC 2020) have shown that there exists an oracle 𝒪 such that BQP^𝒪 ≠ (BPP^BQNC)^𝒪 ∪ (BQNC^BPP)^𝒪. In fact, Chia et al. proved a stronger statement: for any depth parameter d, there exists an oracle that separates quantum depth d and 2d+1, when polynomial-time classical computation is allowed. This implies that relative to an oracle, doubling quantum depth gives classical and quantum hybrid schemes more computational power.
In this paper, we show that for any depth parameter d, there exists an oracle that separates quantum depth d and d+1, when polynomial-time classical computation is allowed. This gives an optimal oracle separation of classical and quantum hybrid schemes. To prove our result, we consider d-Bijective Shuffling Simon’s Problem (which is a variant of d-Shuffling Simon’s Problem considered by Chia et al.) and an oracle inspired by an "in-place" permutation oracle.

Atsuya Hasegawa and François Le Gall. An Optimal Oracle Separation of Classical and Quantum Hybrid Schemes. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hasegawa_et_al:LIPIcs.ISAAC.2022.6, author = {Hasegawa, Atsuya and Le Gall, Fran\c{c}ois}, title = {{An Optimal Oracle Separation of Classical and Quantum Hybrid Schemes}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {6:1--6:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.6}, URN = {urn:nbn:de:0030-drops-172918}, doi = {10.4230/LIPIcs.ISAAC.2022.6}, annote = {Keywords: small-depth quantum circuit, hybrid quantum computer, oracle separation} }

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Brief Announcement

**Published in:** LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

The study of distributed interactive proofs was initiated by Kol, Oshman, and Saxena [PODC 2018] as a generalization of distributed decision mechanisms (proof-labeling schemes, etc.), and has received a lot of attention in recent years. In distributed interactive proofs, the nodes of an n-node network G can exchange short messages (called certificates) with a powerful prover. The goal is to decide if the input (including G itself) belongs to some language, with as few turns of interaction and as few bits exchanged between nodes and the prover as possible. There are several results showing that the size of certificates can be reduced drastically with a constant number of interactions compared to non-interactive distributed proofs.
In this brief announcement, we introduce the quantum counterpart of distributed interactive proofs: certificates can now be quantum bits, and the nodes of the network can perform quantum computation. The main result of this paper shows that by using quantum distributed interactive proofs, the number of interactions can be significantly reduced. More precisely, our main result shows that for any constant k, the class of languages that can be decided by a k-turn classical (i.e., non-quantum) distributed interactive protocol with f(n)-bit certificate size is contained in the class of languages that can be decided by a 5-turn distributed quantum interactive protocol with O(f(n))-bit certificate size. We also show that if we allow to use shared randomness, the number of turns can be reduced to 3-turn. Since no similar turn-reduction classical technique is currently known, our result gives evidence of the power of quantum computation in the setting of distributed interactive proofs as well.

François Le Gall, Masayuki Miyamoto, and Harumichi Nishimura. Brief Announcement: Distributed Quantum Interactive Proofs. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 48:1-48:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{legall_et_al:LIPIcs.DISC.2022.48, author = {Le Gall, Fran\c{c}ois and Miyamoto, Masayuki and Nishimura, Harumichi}, title = {{Brief Announcement: Distributed Quantum Interactive Proofs}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {48:1--48:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-255-6}, ISSN = {1868-8969}, year = {2022}, volume = {246}, editor = {Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.48}, URN = {urn:nbn:de:0030-drops-172396}, doi = {10.4230/LIPIcs.DISC.2022.48}, annote = {Keywords: distributed interactive proofs, distributed verification, quantum computation} }

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Complete Volume

**Published in:** LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

LIPIcs, Volume 232, TQC 2022, Complete Volume

17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 1-218, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{legall_et_al:LIPIcs.TQC.2022, title = {{LIPIcs, Volume 232, TQC 2022, Complete Volume}}, booktitle = {17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)}, pages = {1--218}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-237-2}, ISSN = {1868-8969}, year = {2022}, volume = {232}, editor = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022}, URN = {urn:nbn:de:0030-drops-165067}, doi = {10.4230/LIPIcs.TQC.2022}, annote = {Keywords: LIPIcs, Volume 232, TQC 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

Front Matter, Table of Contents, Preface, Conference Organization

17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{legall_et_al:LIPIcs.TQC.2022.0, author = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)}, pages = {0:i--0:xii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-237-2}, ISSN = {1868-8969}, year = {2022}, volume = {232}, editor = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.0}, URN = {urn:nbn:de:0030-drops-165071}, doi = {10.4230/LIPIcs.TQC.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

The possibilities offered by quantum computing have drawn attention in the distributed computing community recently, with several breakthrough results showing quantum distributed algorithms that run faster than the fastest known classical counterparts, and even separations between the two models. A prime example is the result by Izumi, Le Gall, and Magniez [STACS 2020], who showed that triangle detection by quantum distributed algorithms is easier than triangle listing, while an analogous result is not known in the classical case.
In this paper we present a framework for fast quantum distributed clique detection. This improves upon the state-of-the-art for the triangle case, and is also more general, applying to larger clique sizes.
Our main technical contribution is a new approach for detecting cliques by encapsulating this as a search task for nodes that can be added to smaller cliques. To extract the best complexities out of our approach, we develop a framework for nested distributed quantum searches, which employ checking procedures that are quantum themselves.
Moreover, we show a circuit-complexity barrier on proving a lower bound of the form Ω(n^{3/5+ε}) for K_p-detection for any p ≥ 4, even in the classical (non-quantum) distributed CONGEST setting.

Keren Censor-Hillel, Orr Fischer, François Le Gall, Dean Leitersdorf, and Rotem Oshman. Quantum Distributed Algorithms for Detection of Cliques. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 35:1-35:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{censorhillel_et_al:LIPIcs.ITCS.2022.35, author = {Censor-Hillel, Keren and Fischer, Orr and Le Gall, Fran\c{c}ois and Leitersdorf, Dean and Oshman, Rotem}, title = {{Quantum Distributed Algorithms for Detection of Cliques}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {35:1--35:25}, 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.2022.35}, URN = {urn:nbn:de:0030-drops-156319}, doi = {10.4230/LIPIcs.ITCS.2022.35}, annote = {Keywords: distributed graph algorithms, quantum algorithms, cycles, cliques, Congested Clique, CONGEST} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Longest common substring (LCS), longest palindrome substring (LPS), and Ulam distance (UL) are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear time quantum algorithms for these problems along with quantum lower bounds. Our results shed light on a very surprising fact: Although the classic solutions for LCS and LPS are almost identical (via suffix trees), their quantum computational complexities are different. While we give an exact Õ(√n) time algorithm for LPS, we prove that LCS needs at least time ̃ Ω(n^{2/3}) even for 0/1 strings.

François Le Gall and Saeed Seddighin. Quantum Meets Fine-Grained Complexity: Sublinear Time Quantum Algorithms for String Problems. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 97:1-97:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{legall_et_al:LIPIcs.ITCS.2022.97, author = {Le Gall, Fran\c{c}ois and Seddighin, Saeed}, title = {{Quantum Meets Fine-Grained Complexity: Sublinear Time Quantum Algorithms for String Problems}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {97:1--97:23}, 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.2022.97}, URN = {urn:nbn:de:0030-drops-156934}, doi = {10.4230/LIPIcs.ITCS.2022.97}, annote = {Keywords: Longest common substring, Longest palindrome substring, Quantum algorithms, Sublinear algorithms} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

The distributed subgraph detection asks, for a fixed graph H, whether the n-node input graph contains H as a subgraph or not. In the standard CONGEST model of distributed computing, the complexity of clique/cycle detection and listing has received a lot of attention recently.
In this paper we consider the induced variant of subgraph detection, where the goal is to decide whether the n-node input graph contains H as an induced subgraph or not. We first show a Ω̃(n) lower bound for detecting the existence of an induced k-cycle for any k ≥ 4 in the CONGEST model. This lower bound is tight for k = 4, and shows that the induced variant of k-cycle detection is much harder than the non-induced version. This lower bound is proved via a reduction from two-party communication complexity. We complement this result by showing that for 5 ≤ k ≤ 7, this Ω̃(n) lower bound cannot be improved via the two-party communication framework.
We then show how to prove stronger lower bounds for larger values of k. More precisely, we show that detecting an induced k-cycle for any k ≥ 8 requires Ω̃(n^{2-Θ{(1/k)}}) rounds in the CONGEST model, nearly matching the known upper bound Õ(n^{2-Θ{(1/k)}}) of the general k-node subgraph detection (which also applies to the induced version) by Eden, Fiat, Fischer, Kuhn, and Oshman [DISC 2019].
Finally, we investigate the case where H is the diamond (the diamond is obtained by adding an edge to a 4-cycle, or equivalently removing an edge from a 4-clique), and show non-trivial upper and lower bounds on the complexity of the induced version of diamond detecting and listing.

François Le Gall and Masayuki Miyamoto. Lower Bounds for Induced Cycle Detection in Distributed Computing. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 58:1-58:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{legall_et_al:LIPIcs.ISAAC.2021.58, author = {Le Gall, Fran\c{c}ois and Miyamoto, Masayuki}, title = {{Lower Bounds for Induced Cycle Detection in Distributed Computing}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {58:1--58:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.58}, URN = {urn:nbn:de:0030-drops-154919}, doi = {10.4230/LIPIcs.ISAAC.2021.58}, annote = {Keywords: Distributed computing, Lower bounds, Subgraph detection} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Recent works by Bravyi, Gosset and König (Science 2018), Bene Watts et al. (STOC 2019), Coudron, Stark and Vidick (QIP 2019) and Le Gall (CCC 2019) have shown unconditional separations between the computational powers of shallow (i.e., small-depth) quantum and classical circuits: quantum circuits can solve in constant depth computational problems that require logarithmic depth to solve with classical circuits. Using quantum error correction, Bravyi, Gosset, König and Tomamichel (Nature Physics 2020) further proved that a similar separation still persists even if quantum circuits are subject to local stochastic noise.
In this paper, we consider the case where any constant fraction of the qubits (for instance, huge blocks of qubits) may be arbitrarily corrupted at the end of the computation. We make a first step forward towards establishing a quantum advantage even in this extremely challenging setting: we show that there exists a computational problem that can be solved in constant depth by a quantum circuit but such that even solving any large subproblem of this problem requires logarithmic depth with bounded fan-in classical circuits. This gives another compelling evidence of the computational power of quantum shallow circuits.
In order to show our result, we consider the Graph State Sampling problem (which was also used in prior works) on expander graphs. We exploit the "robustness" of expander graphs against vertex corruption to show that a subproblem hard for small-depth classical circuits can still be extracted from the output of the corrupted quantum circuit.

Atsuya Hasegawa and François Le Gall. Quantum Advantage with Shallow Circuits Under Arbitrary Corruption. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 74:1-74:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hasegawa_et_al:LIPIcs.ISAAC.2021.74, author = {Hasegawa, Atsuya and Le Gall, Fran\c{c}ois}, title = {{Quantum Advantage with Shallow Circuits Under Arbitrary Corruption}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {74:1--74:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.74}, URN = {urn:nbn:de:0030-drops-155076}, doi = {10.4230/LIPIcs.ISAAC.2021.74}, annote = {Keywords: Quantum computing, circuit complexity, constant-depth circuits} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Recently Brakerski, Christiano, Mahadev, Vazirani and Vidick (FOCS 2018) have shown how to construct a test of quantumness based on the learning with errors (LWE) assumption: a test that can be solved efficiently by a quantum computer but cannot be solved by a classical polynomial-time computer under the LWE assumption. This test has lead to several cryptographic applications. In particular, it has been applied to producing certifiable randomness from a single untrusted quantum device, self-testing a single quantum device and device-independent quantum key distribution.
In this paper, we show that this test of quantumness, and essentially all the above applications, can actually be implemented by a very weak class of quantum circuits: constant-depth quantum circuits combined with logarithmic-depth classical computation. This reveals novel complexity-theoretic properties of this fundamental test of quantumness and gives new concrete evidence of the superiority of small-depth quantum circuits over classical computation.

Shuichi Hirahara and François Le Gall. Test of Quantumness with Small-Depth Quantum Circuits. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hirahara_et_al:LIPIcs.MFCS.2021.59, author = {Hirahara, Shuichi and Le Gall, Fran\c{c}ois}, title = {{Test of Quantumness with Small-Depth Quantum Circuits}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {59:1--59:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.59}, URN = {urn:nbn:de:0030-drops-144996}, doi = {10.4230/LIPIcs.MFCS.2021.59}, annote = {Keywords: Quantum computing, small-depth circuits, quantum cryptography} }

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**Published in:** LIPIcs, Volume 197, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)

Post-selection, the power of discarding all runs of a computation in which an undesirable event occurs, is an influential concept introduced to the field of quantum complexity theory by Aaronson (Proceedings of the Royal Society A, 2005). In the present paper, we initiate the study of post-selection for space-bounded quantum complexity classes. Our main result shows the identity PostBQL = PL, i.e., the class of problems that can be solved by a bounded-error (polynomial-time) logarithmic-space quantum algorithm with post-selection (PostBQL) is equal to the class of problems that can be solved by unbounded-error logarithmic-space classical algorithms (PL). This result gives a space-bounded version of the well-known result PostBQP = PP proved by Aaronson for polynomial-time quantum computation. As a by-product, we also show that PL coincides with the class of problems that can be solved by bounded-error logarithmic-space quantum algorithms that have no time bound.

François Le Gall, Harumichi Nishimura, and Abuzer Yakaryılmaz. Quantum Logarithmic Space and Post-Selection. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{legall_et_al:LIPIcs.TQC.2021.10, author = {Le Gall, Fran\c{c}ois and Nishimura, Harumichi and Yakary{\i}lmaz, Abuzer}, title = {{Quantum Logarithmic Space and Post-Selection}}, booktitle = {16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)}, pages = {10:1--10:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-198-6}, ISSN = {1868-8969}, year = {2021}, volume = {197}, editor = {Hsieh, Min-Hsiu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.10}, URN = {urn:nbn:de:0030-drops-140054}, doi = {10.4230/LIPIcs.TQC.2021.10}, annote = {Keywords: computational complexity, space-bounded quantum computation, post-selection} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

This paper tackles the issue of checking that all copies of a large data set replicated at several nodes of a network are identical. The fact that the replicas may be located at distant nodes prevents the system from verifying their equality locally, i.e., by having each node consult only nodes in its vicinity. On the other hand, it remains possible to assign certificates to the nodes, so that verifying the consistency of the replicas can be achieved locally. However, we show that, as the replicated data is large, classical certification mechanisms, including distributed Merlin-Arthur protocols, cannot guarantee good completeness and soundness simultaneously, unless they use very large certificates. The main result of this paper is a distributed quantum Merlin-Arthur protocol enabling the nodes to collectively check the consistency of the replicas, based on small certificates, and in a single round of message exchange between neighbors, with short messages. In particular, the certificate-size is logarithmic in the size of the data set, which gives an exponential advantage over classical certification mechanisms. We propose yet another usage of a fundamental quantum primitive, called the SWAP test, in order to show our main result.

Pierre Fraigniaud, François Le Gall, Harumichi Nishimura, and Ami Paz. Distributed Quantum Proofs for Replicated Data. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fraigniaud_et_al:LIPIcs.ITCS.2021.28, author = {Fraigniaud, Pierre and Le Gall, Fran\c{c}ois and Nishimura, Harumichi and Paz, Ami}, title = {{Distributed Quantum Proofs for Replicated Data}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {28:1--28:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.28}, URN = {urn:nbn:de:0030-drops-135679}, doi = {10.4230/LIPIcs.ITCS.2021.28}, annote = {Keywords: Quantum Computing, Distributed Network Computing, Algorithmic Aspects of Networks} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

In this paper we give fast distributed graph algorithms for detecting and listing small subgraphs, and for computing or approximating the girth. Our algorithms improve upon the state of the art by polynomial factors, and for girth, we obtain a constant-time algorithm for additive +1 approximation in Congested Clique, and the first parametrized algorithm for exact computation in Congest.
In the Congested Clique model, we first develop a technique for learning small neighborhoods, and apply it to obtain an O(1)-round algorithm that computes the girth with only an additive +1 error. Next, we introduce a new technique (the partition tree technique) allowing for efficiently listing all copies of any subgraph, which is deterministic and improves upon the state-of the-art for non-dense graphs. We give two concrete applications of the partition tree technique: First we show that for constant k, it is possible to solve C_{2k}-detection in O(1) rounds in the Congested Clique, improving on prior work, which used fast matrix multiplication and thus had polynomial round complexity. Second, we show that in triangle-free graphs, the girth can be exactly computed in time polynomially faster than the best known bounds for general graphs. We remark that no analogous result is currently known for sequential algorithms.
In the Congest model, we describe a new approach for finding cycles, and instantiate it in two ways: first, we show a fast parametrized algorithm for girth with round complexity Õ(min{g⋅ n^{1-1/Θ(g)},n}) for any girth g; and second, we show how to find small even-length cycles C_{2k} for k = 3,4,5 in O(n^{1-1/k}) rounds. This is a polynomial improvement upon the previous running times; for example, our C₆-detection algorithm runs in O(n^{2/3}) rounds, compared to O(n^{3/4}) in prior work. Finally, using our improved C₆-freeness algorithm, and the barrier on proving lower bounds on triangle-freeness of Eden et al., we show that improving the current ̃Ω(√n) lower bound for C₆-freeness of Korhonen et al. by any polynomial factor would imply strong circuit complexity lower bounds.

Keren Censor-Hillel, Orr Fischer, Tzlil Gonen, François Le Gall, Dean Leitersdorf, and Rotem Oshman. Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{censorhillel_et_al:LIPIcs.DISC.2020.33, author = {Censor-Hillel, Keren and Fischer, Orr and Gonen, Tzlil and Le Gall, Fran\c{c}ois and Leitersdorf, Dean and Oshman, Rotem}, title = {{Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {33:1--33:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.33}, URN = {urn:nbn:de:0030-drops-131115}, doi = {10.4230/LIPIcs.DISC.2020.33}, annote = {Keywords: distributed graph algorithms, cycles, girth, Congested Clique, CONGEST} }

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Brief Announcement

**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

This paper tackles the issue of checking that all copies of a large data set replicated at several nodes of a network are identical. The fact that the replicas may be located at distant nodes prevents the system from verifying their equality locally, i.e., by having each node consult only nodes in its vicinity. On the other hand, it remains possible to assign certificates to the nodes, so that verifying the consistency of the replicas can be achieved locally. However, we show that, as the replicated data is large, classical certification mechanisms, including distributed Merlin-Arthur protocols, cannot guarantee good completeness and soundness simultaneously, unless they use very large certificates. The main result of this paper is a distributed quantum Merlin-Arthur protocol enabling the nodes to collectively check the consistency of the replicas, based on small certificates, and in a single round of message exchange between neighbors, with short messages. In particular, the certificate-size is logarithmic in the size of the data set, which gives an exponential advantage over classical certification mechanisms.

Pierre Fraigniaud, François Le Gall, Harumichi Nishimura, and Ami Paz. Brief Announcement: Distributed Quantum Proofs for Replicated Data. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 43:1-43:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fraigniaud_et_al:LIPIcs.DISC.2020.43, author = {Fraigniaud, Pierre and Le Gall, Fran\c{c}ois and Nishimura, Harumichi and Paz, Ami}, title = {{Brief Announcement: Distributed Quantum Proofs for Replicated Data}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {43:1--43:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.43}, URN = {urn:nbn:de:0030-drops-131217}, doi = {10.4230/LIPIcs.DISC.2020.43}, annote = {Keywords: Quantum Computing, Distributed Network Computing, Algorithmic Aspects of Networks} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

A recent breakthrough by Tang (STOC 2019) showed how to "dequantize" the quantum algorithm for recommendation systems by Kerenidis and Prakash (ITCS 2017). The resulting algorithm, classical but "quantum-inspired", efficiently computes a low-rank approximation of the users' preference matrix. Subsequent works have shown how to construct efficient quantum-inspired algorithms for approximating the pseudo-inverse of a low-rank matrix as well, which can be used to (approximately) solve low-rank linear systems of equations. In the present paper, we pursue this line of research and develop quantum-inspired algorithms for a large class of matrix transformations that are defined via the singular value decomposition of the matrix. In particular, we obtain classical algorithms with complexity polynomially related (in most parameters) to the complexity of the best quantum algorithms for singular value transformation recently developed by Chakraborty, Gilyén and Jeffery (ICALP 2019) and Gilyén, Su, Low and Wiebe (STOC 2019).

Dhawal Jethwani, François Le Gall, and Sanjay K. Singh. Quantum-Inspired Classical Algorithms for Singular Value Transformation. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jethwani_et_al:LIPIcs.MFCS.2020.53, author = {Jethwani, Dhawal and Le Gall, Fran\c{c}ois and Singh, Sanjay K.}, title = {{Quantum-Inspired Classical Algorithms for Singular Value Transformation}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {53:1--53:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.53}, URN = {urn:nbn:de:0030-drops-127193}, doi = {10.4230/LIPIcs.MFCS.2020.53}, annote = {Keywords: Sampling algorithms, quantum-inspired algorithms, linear algebra} }

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

This paper considers the triangle finding problem in the CONGEST model of distributed computing. Recent works by Izumi and Le Gall (PODC'17), Chang, Pettie and Zhang (SODA'19) and Chang and Saranurak (PODC'19) have successively reduced the classical round complexity of triangle finding (as well as triangle listing) from the trivial upper bound O(n) to Õ(n^{1/3}), where n denotes the number of vertices in the graph. In this paper we present a quantum distributed algorithm that solves the triangle finding problem in Õ(n^{1/4}) rounds in the CONGEST model. This gives another example of quantum algorithm beating the best known classical algorithms in distributed computing. Our result also exhibits an interesting phenomenon: while in the classical setting the best known upper bounds for the triangle finding and listing problems are identical, in the quantum setting the round complexities of these two problems are now Õ(n^{1/4}) and Θ~(n^{1/3}), respectively. Our result thus shows that triangle finding is easier than triangle listing in the quantum CONGEST model.

Taisuke Izumi, François Le Gall, and Frédéric Magniez. Quantum Distributed Algorithm for Triangle Finding in the CONGEST Model. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{izumi_et_al:LIPIcs.STACS.2020.23, author = {Izumi, Taisuke and Le Gall, Fran\c{c}ois and Magniez, Fr\'{e}d\'{e}ric}, title = {{Quantum Distributed Algorithm for Triangle Finding in the CONGEST Model}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {23:1--23:13}, 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.2020.23}, URN = {urn:nbn:de:0030-drops-118840}, doi = {10.4230/LIPIcs.STACS.2020.23}, annote = {Keywords: Quantum computing, distributed computing, CONGEST model} }

Document

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

Recently Bravyi, Gosset and König (Science 2018) proved an unconditional separation between the computational powers of small-depth quantum and classical circuits for a relation. In this paper we show a similar separation in the average-case setting that gives stronger evidence of the superiority of small-depth quantum computation: we construct a computational task that can be solved on all inputs by a quantum circuit of constant depth with bounded-fanin gates (a "shallow" quantum circuit) and show that any classical circuit with bounded-fanin gates solving this problem on a non-negligible fraction of the inputs must have logarithmic depth. Our results are obtained by introducing a technique to create quantum states exhibiting global quantum correlations from any graph, via a construction that we call the extended graph.
Similar results have been very recently (and independently) obtained by Coudron, Stark and Vidick (arXiv:1810.04233}), and Bene Watts, Kothari, Schaeffer and Tal (STOC 2019).

François Le Gall. Average-Case Quantum Advantage with Shallow Circuits. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{legall:LIPIcs.CCC.2019.21, author = {Le Gall, Fran\c{c}ois}, title = {{Average-Case Quantum Advantage with Shallow Circuits}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {21:1--21:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.21}, URN = {urn:nbn:de:0030-drops-108432}, doi = {10.4230/LIPIcs.CCC.2019.21}, annote = {Keywords: Quantum computing, circuit complexity, constant-depth circuits} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

There are two central models considered in (fault-free synchronous) distributed computing: the CONGEST model, in which communication channels have limited bandwidth, and the LOCAL model, in which communication channels have unlimited bandwidth. Very recently, Le Gall and Magniez (PODC 2018) showed the superiority of quantum distributed computing over classical distributed computing in the CONGEST model. In this work we show the superiority of quantum distributed computing in the LOCAL model: we exhibit two computational tasks that can be solved in a constant number of rounds in the quantum setting but require Omega(n) rounds in the classical (randomized) setting, where n denotes the size of the network.

François Le Gall, Harumichi Nishimura, and Ansis Rosmanis. Quantum Advantage for the LOCAL Model in Distributed Computing. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{legall_et_al:LIPIcs.STACS.2019.49, author = {Le Gall, Fran\c{c}ois and Nishimura, Harumichi and Rosmanis, Ansis}, title = {{Quantum Advantage for the LOCAL Model in Distributed Computing}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {49:1--49:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.49}, URN = {urn:nbn:de:0030-drops-102887}, doi = {10.4230/LIPIcs.STACS.2019.49}, annote = {Keywords: Quantum computing, distributed computing, LOCAL model} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

In this paper we consider what can be computed by a user interacting with a potentially malicious server, when the server performs polynomial-time quantum computation but the user can only perform polynomial-time classical (i.e., non-quantum) computation. Understanding the computational power of this model, which corresponds to polynomial-time quantum computation that can be efficiently verified classically, is a well-known open problem in quantum computing. Our result shows that computing the order of a solvable group, which is one of the most general problems for which quantum computing exhibits an exponential speed-up with respect to classical computing, can be realized in this model.

François Le Gall, Tomoyuki Morimae, Harumichi Nishimura, and Yuki Takeuchi. Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 26:1-26:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{legall_et_al:LIPIcs.MFCS.2018.26, author = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki and Nishimura, Harumichi and Takeuchi, Yuki}, title = {{Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {26:1--26:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.26}, URN = {urn:nbn:de:0030-drops-96087}, doi = {10.4230/LIPIcs.MFCS.2018.26}, annote = {Keywords: Quantum computing, interactive proofs, group-theoretic problems} }

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**Published in:** LIPIcs, Volume 73, 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)

Triangle finding (deciding if a graph contains a triangle or not) is a central problem in quantum query complexity. The quantum communication complexity of this problem, where the edges of the graph are distributed among the players, was considered recently by Ivanyos et al. in the two- party setting. In this paper we consider its k-party quantum communication complexity with k >= 3. Our main result is a ~O(m^(7/12))-qubit protocol, for any constant number of players k, deciding with high probability if a graph with m edges contains a triangle or not. Our approach makes connections between the multiparty quantum communication complexity of triangle finding and the quantum query complexity of graph collision, a well-studied problem in quantum query complexity.

François Le Gall and Shogo Nakajima. Multiparty Quantum Communication Complexity of Triangle Finding. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 73, pp. 6:1-6:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{legall_et_al:LIPIcs.TQC.2017.6, author = {Le Gall, Fran\c{c}ois and Nakajima, Shogo}, title = {{Multiparty Quantum Communication Complexity of Triangle Finding}}, booktitle = {12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)}, pages = {6:1--6:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-034-7}, ISSN = {1868-8969}, year = {2018}, volume = {73}, editor = {Wilde, Mark M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2017.6}, URN = {urn:nbn:de:0030-drops-85793}, doi = {10.4230/LIPIcs.TQC.2017.6}, annote = {Keywords: Quantum communication complexity, triangle finding, graph collision} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

A recent series of breakthroughs initiated by Spielman and Teng culminated in the construction of nearly linear time Laplacian solvers, approximating the solution of a linear system Lx=b, where L is the normalized Laplacian of an undirected graph. In this paper we study the space complexity of the problem. Surprisingly we are able to show a probabilistic, logspace algorithm solving the problem. We further extend the algorithm to other families of graphs like Eulerian graphs (and directed regular graphs) and graphs that mix in polynomial time.
Our approach is to pseudo-invert the Laplacian, by first "peeling-off" the problematic kernel of the operator, and then to approximate the inverse of the remaining part by using a Taylor series. We approximate the Taylor series using a previous work and the special structure of the problem. For directed graphs we exploit in the analysis the Jordan normal form and results from matrix functions.

Dean Doron, François Le Gall, and Amnon Ta-Shma. Probabilistic Logarithmic-Space Algorithms for Laplacian Solvers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 41:1-41:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{doron_et_al:LIPIcs.APPROX-RANDOM.2017.41, author = {Doron, Dean and Le Gall, Fran\c{c}ois and Ta-Shma, Amnon}, title = {{Probabilistic Logarithmic-Space Algorithms for Laplacian Solvers}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {41:1--41:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.41}, URN = {urn:nbn:de:0030-drops-75908}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.41}, annote = {Keywords: Laplacian solvers, Randomized logspace, Bounded-space complexity classes, Random walks, Matrix computation} }

Document

**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Computing set joins of two inputs is a common task in database theory. Recently, Van Gucht, Williams, Woodruff and Zhang [PODS 2015] considered the complexity of such problems in the natural model of (classical) two-party communication complexity and obtained tight bounds for the complexity of several important distributed set joins.
In this paper we initiate the study of the quantum communication complexity of distributed set joins. We design a quantum protocol for distributed Boolean matrix multiplication, which corresponds to computing the composition join of two databases, showing that the product of two n times n Boolean matrices, each owned by one of two respective parties, can be computed with widetilde-O(sqrt{n} ell^{3/4}) qubits of communication, where ell denotes the number of non-zero entries of the product. Since Van Gucht et al. showed that the classical communication complexity of this problem is widetilde-Theta(n sqrt{ell}), our quantum algorithm outperforms classical protocols whenever the output matrix is sparse. We also show a quantum lower bound and a matching classical upper bound on the communication complexity of distributed matrix multiplication over F_2.
Besides their applications to database theory, the communication complexity of set joins is interesting due to its connections to direct product theorems in communication complexity. In this work we also introduce a notion of all-pairs product theorem, and relate this notion to standard direct product theorems in communication complexity.

Stacey Jeffery and François Le Gall. Quantum Communication Complexity of Distributed Set Joins. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{jeffery_et_al:LIPIcs.MFCS.2016.54, author = {Jeffery, Stacey and Le Gall, Fran\c{c}ois}, title = {{Quantum Communication Complexity of Distributed Set Joins}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {54:1--54:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.54}, URN = {urn:nbn:de:0030-drops-64663}, doi = {10.4230/LIPIcs.MFCS.2016.54}, annote = {Keywords: quantum communication complexity, distributed quantum computing, database joins} }

Document

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

This paper investigates the role of interaction and coins in quantum Arthur-Merlin games (also called public-coin quantum interactive proof systems). While the existing model restricts the messages from the verifier to be classical even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the messages from the verifier can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. This generalization turns out to provide several novel characterizations of quantum interactive proof systems with a constant number of turns. First, it is proved that the complexity class corresponding to two-turn quantum Arthur-Merlin games where both of the two messages are quantum, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interaction prior to the communications of qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper presents a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking properties related to quantum circuits, and thus, qq-QAM may provide a good measure in computational complexity theory. It is further proved that the class qq-QAM_1, the perfect-completeness variant of qq-QAM, gives new bounds for standard well-studied classes of two-turn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum Arthur-Merlin games: it is proved that, for any constant m >= 2, the class of problems having $m$-turn quantum Arthur-Merlin proof systems is either equal to PSPACE or equal to the class of problems having two-turn quantum Arthur-Merlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.

Hirotada Kobayashi, Francois Le Gall, and Harumichi Nishimura. Generalized Quantum Arthur-Merlin Games. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 488-511, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kobayashi_et_al:LIPIcs.CCC.2015.488, author = {Kobayashi, Hirotada and Le Gall, Francois and Nishimura, Harumichi}, title = {{Generalized Quantum Arthur-Merlin Games}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {488--511}, 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.2015.488}, URN = {urn:nbn:de:0030-drops-50697}, doi = {10.4230/LIPIcs.CCC.2015.488}, annote = {Keywords: interactive proof systems, Arthur-Merlin games, quantum computing, complete problems, entanglement} }

Document

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

In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian
groups. Le Gall has constructed an efficient classical algorithm for a class of groups corresponding to one of the most natural ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $\Int_m$ with the order of $A$ coprime with $m$.
More precisely, the running time of that algorithm is almost linear in the order of the input groups.
In this paper we present a \emph{quantum} algorithm solving the same problem in time polynomial in the \emph{logarithm} of the order of the input groups. This algorithm works in the black-box setting and is the first quantum algorithm solving instances of the nonabelian group isomorphism problem exponentially faster than the best known classical algorithms.

François Le Gall. An Efficient Quantum Algorithm for Some Instances of the Group Isomorphism Problem. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 549-560, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{legall:LIPIcs.STACS.2010.2484, author = {Le Gall, Fran\c{c}ois}, title = {{An Efficient Quantum Algorithm for Some Instances of the Group Isomorphism Problem}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {549--560}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2484}, URN = {urn:nbn:de:0030-drops-24843}, doi = {10.4230/LIPIcs.STACS.2010.2484}, annote = {Keywords: Quantum algorithms, group isomorphism problem, black-box groups} }

Document

**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

The group isomorphism problem asks whether two given groups are isomorphic or not. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. In this paper we study this problem for a class of groups corresponding to one of the simplest ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $\mathbb{Z}_m$. We present an efficient algorithm solving the group isomorphism problem for all the groups of this class such that the order of $A$ is coprime with $m$. More precisely, our algorithm runs in time almost linear in the orders of the input groups and works in the general setting where the groups are given as black-boxes.

Francois Le Gall. Efficient Isomorphism Testing for a Class of Group Extensions. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 625-636, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{legall:LIPIcs.STACS.2009.1830, author = {Le Gall, Francois}, title = {{Efficient Isomorphism Testing for a Class of Group Extensions}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {625--636}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1830}, URN = {urn:nbn:de:0030-drops-18303}, doi = {10.4230/LIPIcs.STACS.2009.1830}, annote = {Keywords: Polynomial-time algorithms, Group isomorphism, Black-box groups} }

Document

Invited Talk

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

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.

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)

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

Document

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

The generation and verification of quantum states are fundamental tasks for quantum information processing that have recently been investigated by Irani, Natarajan, Nirkhe, Rao and Yuen [CCC 2022], Rosenthal and Yuen [ITCS 2022], Metger and Yuen [QIP 2023] under the term state synthesis. This paper studies this concept from the viewpoint of quantum distributed computing, and especially distributed quantum Merlin-Arthur (dQMA) protocols. We first introduce a novel task, on a line, called state generation with distributed inputs (SGDI). In this task, the goal is to generate the quantum state U|ψ⟩ at the rightmost node of the line, where |ψ⟩ is a quantum state given at the leftmost node and U is a unitary matrix whose description is distributed over the nodes of the line. We give a dQMA protocol for SGDI and utilize this protocol to construct a dQMA protocol for the Set Equality problem studied by Naor, Parter and Yogev [SODA 2020], and complement our protocol by showing classical lower bounds for this problem. Our second contribution is a dQMA protocol, based on a recent work by Zhu and Hayashi [Physical Review A, 2019], to create EPR-pairs between adjacent nodes of a network without quantum communication. As an application of this dQMA protocol, we prove a general result showing how to convert any dQMA protocol on an arbitrary network into another dQMA protocol where the verification stage does not require any quantum communication.

François Le Gall, Masayuki Miyamoto, and Harumichi Nishimura. Distributed Merlin-Arthur Synthesis of Quantum States and Its Applications. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{legall_et_al:LIPIcs.MFCS.2023.63, author = {Le Gall, Fran\c{c}ois and Miyamoto, Masayuki and Nishimura, Harumichi}, title = {{Distributed Merlin-Arthur Synthesis of Quantum States and Its Applications}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {63:1--63:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.63}, URN = {urn:nbn:de:0030-drops-185975}, doi = {10.4230/LIPIcs.MFCS.2023.63}, annote = {Keywords: distributed quantum Merlin-Arthur, distributed verification, quantum computation} }

Document

Track A: Algorithms, Complexity and Games

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

Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with 6-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to 1/2 with a ground state.
In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry.

Chris Cade, Marten Folkertsma, Sevag Gharibian, Ryu Hayakawa, François Le Gall, Tomoyuki Morimae, and Jordi Weggemans. Improved Hardness Results for the Guided Local Hamiltonian Problem. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cade_et_al:LIPIcs.ICALP.2023.32, author = {Cade, Chris and Folkertsma, Marten and Gharibian, Sevag and Hayakawa, Ryu and Le Gall, Fran\c{c}ois and Morimae, Tomoyuki and Weggemans, Jordi}, title = {{Improved Hardness Results for the Guided Local Hamiltonian Problem}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {32:1--32:19}, 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.2023.32}, URN = {urn:nbn:de:0030-drops-180840}, doi = {10.4230/LIPIcs.ICALP.2023.32}, annote = {Keywords: Quantum computing, Quantum advantage, Quantum Chemistry, Guided Local Hamiltonian Problem} }

Document

**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

The study of distributed interactive proofs was initiated by Kol, Oshman, and Saxena [PODC 2018] as a generalization of distributed decision mechanisms (proof-labeling schemes, etc.), and has received a lot of attention in recent years. In distributed interactive proofs, the nodes of an n-node network G can exchange short messages (called certificates) with a powerful prover. The goal is to decide if the input (including G itself) belongs to some language, with as few turns of interaction and as few bits exchanged between nodes and the prover as possible. There are several results showing that the size of certificates can be reduced drastically with a constant number of interactions compared to non-interactive distributed proofs.
In this paper, we introduce the quantum counterpart of distributed interactive proofs: certificates can now be quantum bits, and the nodes of the network can perform quantum computation. The first result of this paper shows that by using distributed quantum interactive proofs, the number of interactions can be significantly reduced. More precisely, our result shows that for any constant k, the class of languages that can be decided by a k-turn classical (i.e., non-quantum) distributed interactive protocol with f(n)-bit certificate size is contained in the class of languages that can be decided by a 5-turn distributed quantum interactive protocol with O(f(n))-bit certificate size. We also show that if we allow to use shared randomness, the number of turns can be reduced to three. Since no similar turn-reduction classical technique is currently known, our result gives evidence of the power of quantum computation in the setting of distributed interactive proofs as well.
As a corollary of our results, we show that there exist 5-turn/3-turn distributed quantum interactive protocols with small certificate size for problems that have been considered in prior works on distributed interactive proofs such as [Kol, Oshman, and Saxena PODC 2018, Naor, Parter, and Yogev SODA 2020].
We then utilize the framework of the distributed quantum interactive proofs to test closeness of two quantum states each of which is distributed over the entire network.

François Le Gall, Masayuki Miyamoto, and Harumichi Nishimura. Distributed Quantum Interactive Proofs. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 42:1-42:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{legall_et_al:LIPIcs.STACS.2023.42, author = {Le Gall, Fran\c{c}ois and Miyamoto, Masayuki and Nishimura, Harumichi}, title = {{Distributed Quantum Interactive Proofs}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {42:1--42:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.42}, URN = {urn:nbn:de:0030-drops-176949}, doi = {10.4230/LIPIcs.STACS.2023.42}, annote = {Keywords: distributed interactive proofs, distributed verification, quantum computation} }

Document

**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Recently, Chia, Chung and Lai (STOC 2020) and Coudron and Menda (STOC 2020) have shown that there exists an oracle 𝒪 such that BQP^𝒪 ≠ (BPP^BQNC)^𝒪 ∪ (BQNC^BPP)^𝒪. In fact, Chia et al. proved a stronger statement: for any depth parameter d, there exists an oracle that separates quantum depth d and 2d+1, when polynomial-time classical computation is allowed. This implies that relative to an oracle, doubling quantum depth gives classical and quantum hybrid schemes more computational power.
In this paper, we show that for any depth parameter d, there exists an oracle that separates quantum depth d and d+1, when polynomial-time classical computation is allowed. This gives an optimal oracle separation of classical and quantum hybrid schemes. To prove our result, we consider d-Bijective Shuffling Simon’s Problem (which is a variant of d-Shuffling Simon’s Problem considered by Chia et al.) and an oracle inspired by an "in-place" permutation oracle.

Atsuya Hasegawa and François Le Gall. An Optimal Oracle Separation of Classical and Quantum Hybrid Schemes. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hasegawa_et_al:LIPIcs.ISAAC.2022.6, author = {Hasegawa, Atsuya and Le Gall, Fran\c{c}ois}, title = {{An Optimal Oracle Separation of Classical and Quantum Hybrid Schemes}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {6:1--6:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.6}, URN = {urn:nbn:de:0030-drops-172918}, doi = {10.4230/LIPIcs.ISAAC.2022.6}, annote = {Keywords: small-depth quantum circuit, hybrid quantum computer, oracle separation} }

Document

Brief Announcement

**Published in:** LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

The study of distributed interactive proofs was initiated by Kol, Oshman, and Saxena [PODC 2018] as a generalization of distributed decision mechanisms (proof-labeling schemes, etc.), and has received a lot of attention in recent years. In distributed interactive proofs, the nodes of an n-node network G can exchange short messages (called certificates) with a powerful prover. The goal is to decide if the input (including G itself) belongs to some language, with as few turns of interaction and as few bits exchanged between nodes and the prover as possible. There are several results showing that the size of certificates can be reduced drastically with a constant number of interactions compared to non-interactive distributed proofs.
In this brief announcement, we introduce the quantum counterpart of distributed interactive proofs: certificates can now be quantum bits, and the nodes of the network can perform quantum computation. The main result of this paper shows that by using quantum distributed interactive proofs, the number of interactions can be significantly reduced. More precisely, our main result shows that for any constant k, the class of languages that can be decided by a k-turn classical (i.e., non-quantum) distributed interactive protocol with f(n)-bit certificate size is contained in the class of languages that can be decided by a 5-turn distributed quantum interactive protocol with O(f(n))-bit certificate size. We also show that if we allow to use shared randomness, the number of turns can be reduced to 3-turn. Since no similar turn-reduction classical technique is currently known, our result gives evidence of the power of quantum computation in the setting of distributed interactive proofs as well.

François Le Gall, Masayuki Miyamoto, and Harumichi Nishimura. Brief Announcement: Distributed Quantum Interactive Proofs. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 48:1-48:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{legall_et_al:LIPIcs.DISC.2022.48, author = {Le Gall, Fran\c{c}ois and Miyamoto, Masayuki and Nishimura, Harumichi}, title = {{Brief Announcement: Distributed Quantum Interactive Proofs}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {48:1--48:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-255-6}, ISSN = {1868-8969}, year = {2022}, volume = {246}, editor = {Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.48}, URN = {urn:nbn:de:0030-drops-172396}, doi = {10.4230/LIPIcs.DISC.2022.48}, annote = {Keywords: distributed interactive proofs, distributed verification, quantum computation} }

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Complete Volume

**Published in:** LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

LIPIcs, Volume 232, TQC 2022, Complete Volume

17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 1-218, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{legall_et_al:LIPIcs.TQC.2022, title = {{LIPIcs, Volume 232, TQC 2022, Complete Volume}}, booktitle = {17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)}, pages = {1--218}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-237-2}, ISSN = {1868-8969}, year = {2022}, volume = {232}, editor = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022}, URN = {urn:nbn:de:0030-drops-165067}, doi = {10.4230/LIPIcs.TQC.2022}, annote = {Keywords: LIPIcs, Volume 232, TQC 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

Front Matter, Table of Contents, Preface, Conference Organization

17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{legall_et_al:LIPIcs.TQC.2022.0, author = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)}, pages = {0:i--0:xii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-237-2}, ISSN = {1868-8969}, year = {2022}, volume = {232}, editor = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.0}, URN = {urn:nbn:de:0030-drops-165071}, doi = {10.4230/LIPIcs.TQC.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

The possibilities offered by quantum computing have drawn attention in the distributed computing community recently, with several breakthrough results showing quantum distributed algorithms that run faster than the fastest known classical counterparts, and even separations between the two models. A prime example is the result by Izumi, Le Gall, and Magniez [STACS 2020], who showed that triangle detection by quantum distributed algorithms is easier than triangle listing, while an analogous result is not known in the classical case.
In this paper we present a framework for fast quantum distributed clique detection. This improves upon the state-of-the-art for the triangle case, and is also more general, applying to larger clique sizes.
Our main technical contribution is a new approach for detecting cliques by encapsulating this as a search task for nodes that can be added to smaller cliques. To extract the best complexities out of our approach, we develop a framework for nested distributed quantum searches, which employ checking procedures that are quantum themselves.
Moreover, we show a circuit-complexity barrier on proving a lower bound of the form Ω(n^{3/5+ε}) for K_p-detection for any p ≥ 4, even in the classical (non-quantum) distributed CONGEST setting.

Keren Censor-Hillel, Orr Fischer, François Le Gall, Dean Leitersdorf, and Rotem Oshman. Quantum Distributed Algorithms for Detection of Cliques. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 35:1-35:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{censorhillel_et_al:LIPIcs.ITCS.2022.35, author = {Censor-Hillel, Keren and Fischer, Orr and Le Gall, Fran\c{c}ois and Leitersdorf, Dean and Oshman, Rotem}, title = {{Quantum Distributed Algorithms for Detection of Cliques}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {35:1--35:25}, 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.2022.35}, URN = {urn:nbn:de:0030-drops-156319}, doi = {10.4230/LIPIcs.ITCS.2022.35}, annote = {Keywords: distributed graph algorithms, quantum algorithms, cycles, cliques, Congested Clique, CONGEST} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Longest common substring (LCS), longest palindrome substring (LPS), and Ulam distance (UL) are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear time quantum algorithms for these problems along with quantum lower bounds. Our results shed light on a very surprising fact: Although the classic solutions for LCS and LPS are almost identical (via suffix trees), their quantum computational complexities are different. While we give an exact Õ(√n) time algorithm for LPS, we prove that LCS needs at least time ̃ Ω(n^{2/3}) even for 0/1 strings.

François Le Gall and Saeed Seddighin. Quantum Meets Fine-Grained Complexity: Sublinear Time Quantum Algorithms for String Problems. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 97:1-97:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{legall_et_al:LIPIcs.ITCS.2022.97, author = {Le Gall, Fran\c{c}ois and Seddighin, Saeed}, title = {{Quantum Meets Fine-Grained Complexity: Sublinear Time Quantum Algorithms for String Problems}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {97:1--97:23}, 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.2022.97}, URN = {urn:nbn:de:0030-drops-156934}, doi = {10.4230/LIPIcs.ITCS.2022.97}, annote = {Keywords: Longest common substring, Longest palindrome substring, Quantum algorithms, Sublinear algorithms} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

The distributed subgraph detection asks, for a fixed graph H, whether the n-node input graph contains H as a subgraph or not. In the standard CONGEST model of distributed computing, the complexity of clique/cycle detection and listing has received a lot of attention recently.
In this paper we consider the induced variant of subgraph detection, where the goal is to decide whether the n-node input graph contains H as an induced subgraph or not. We first show a Ω̃(n) lower bound for detecting the existence of an induced k-cycle for any k ≥ 4 in the CONGEST model. This lower bound is tight for k = 4, and shows that the induced variant of k-cycle detection is much harder than the non-induced version. This lower bound is proved via a reduction from two-party communication complexity. We complement this result by showing that for 5 ≤ k ≤ 7, this Ω̃(n) lower bound cannot be improved via the two-party communication framework.
We then show how to prove stronger lower bounds for larger values of k. More precisely, we show that detecting an induced k-cycle for any k ≥ 8 requires Ω̃(n^{2-Θ{(1/k)}}) rounds in the CONGEST model, nearly matching the known upper bound Õ(n^{2-Θ{(1/k)}}) of the general k-node subgraph detection (which also applies to the induced version) by Eden, Fiat, Fischer, Kuhn, and Oshman [DISC 2019].
Finally, we investigate the case where H is the diamond (the diamond is obtained by adding an edge to a 4-cycle, or equivalently removing an edge from a 4-clique), and show non-trivial upper and lower bounds on the complexity of the induced version of diamond detecting and listing.

François Le Gall and Masayuki Miyamoto. Lower Bounds for Induced Cycle Detection in Distributed Computing. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 58:1-58:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{legall_et_al:LIPIcs.ISAAC.2021.58, author = {Le Gall, Fran\c{c}ois and Miyamoto, Masayuki}, title = {{Lower Bounds for Induced Cycle Detection in Distributed Computing}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {58:1--58:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.58}, URN = {urn:nbn:de:0030-drops-154919}, doi = {10.4230/LIPIcs.ISAAC.2021.58}, annote = {Keywords: Distributed computing, Lower bounds, Subgraph detection} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Recent works by Bravyi, Gosset and König (Science 2018), Bene Watts et al. (STOC 2019), Coudron, Stark and Vidick (QIP 2019) and Le Gall (CCC 2019) have shown unconditional separations between the computational powers of shallow (i.e., small-depth) quantum and classical circuits: quantum circuits can solve in constant depth computational problems that require logarithmic depth to solve with classical circuits. Using quantum error correction, Bravyi, Gosset, König and Tomamichel (Nature Physics 2020) further proved that a similar separation still persists even if quantum circuits are subject to local stochastic noise.
In this paper, we consider the case where any constant fraction of the qubits (for instance, huge blocks of qubits) may be arbitrarily corrupted at the end of the computation. We make a first step forward towards establishing a quantum advantage even in this extremely challenging setting: we show that there exists a computational problem that can be solved in constant depth by a quantum circuit but such that even solving any large subproblem of this problem requires logarithmic depth with bounded fan-in classical circuits. This gives another compelling evidence of the computational power of quantum shallow circuits.
In order to show our result, we consider the Graph State Sampling problem (which was also used in prior works) on expander graphs. We exploit the "robustness" of expander graphs against vertex corruption to show that a subproblem hard for small-depth classical circuits can still be extracted from the output of the corrupted quantum circuit.

Atsuya Hasegawa and François Le Gall. Quantum Advantage with Shallow Circuits Under Arbitrary Corruption. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 74:1-74:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hasegawa_et_al:LIPIcs.ISAAC.2021.74, author = {Hasegawa, Atsuya and Le Gall, Fran\c{c}ois}, title = {{Quantum Advantage with Shallow Circuits Under Arbitrary Corruption}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {74:1--74:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.74}, URN = {urn:nbn:de:0030-drops-155076}, doi = {10.4230/LIPIcs.ISAAC.2021.74}, annote = {Keywords: Quantum computing, circuit complexity, constant-depth circuits} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Recently Brakerski, Christiano, Mahadev, Vazirani and Vidick (FOCS 2018) have shown how to construct a test of quantumness based on the learning with errors (LWE) assumption: a test that can be solved efficiently by a quantum computer but cannot be solved by a classical polynomial-time computer under the LWE assumption. This test has lead to several cryptographic applications. In particular, it has been applied to producing certifiable randomness from a single untrusted quantum device, self-testing a single quantum device and device-independent quantum key distribution.
In this paper, we show that this test of quantumness, and essentially all the above applications, can actually be implemented by a very weak class of quantum circuits: constant-depth quantum circuits combined with logarithmic-depth classical computation. This reveals novel complexity-theoretic properties of this fundamental test of quantumness and gives new concrete evidence of the superiority of small-depth quantum circuits over classical computation.

Shuichi Hirahara and François Le Gall. Test of Quantumness with Small-Depth Quantum Circuits. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hirahara_et_al:LIPIcs.MFCS.2021.59, author = {Hirahara, Shuichi and Le Gall, Fran\c{c}ois}, title = {{Test of Quantumness with Small-Depth Quantum Circuits}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {59:1--59:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.59}, URN = {urn:nbn:de:0030-drops-144996}, doi = {10.4230/LIPIcs.MFCS.2021.59}, annote = {Keywords: Quantum computing, small-depth circuits, quantum cryptography} }

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**Published in:** LIPIcs, Volume 197, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)

Post-selection, the power of discarding all runs of a computation in which an undesirable event occurs, is an influential concept introduced to the field of quantum complexity theory by Aaronson (Proceedings of the Royal Society A, 2005). In the present paper, we initiate the study of post-selection for space-bounded quantum complexity classes. Our main result shows the identity PostBQL = PL, i.e., the class of problems that can be solved by a bounded-error (polynomial-time) logarithmic-space quantum algorithm with post-selection (PostBQL) is equal to the class of problems that can be solved by unbounded-error logarithmic-space classical algorithms (PL). This result gives a space-bounded version of the well-known result PostBQP = PP proved by Aaronson for polynomial-time quantum computation. As a by-product, we also show that PL coincides with the class of problems that can be solved by bounded-error logarithmic-space quantum algorithms that have no time bound.

François Le Gall, Harumichi Nishimura, and Abuzer Yakaryılmaz. Quantum Logarithmic Space and Post-Selection. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{legall_et_al:LIPIcs.TQC.2021.10, author = {Le Gall, Fran\c{c}ois and Nishimura, Harumichi and Yakary{\i}lmaz, Abuzer}, title = {{Quantum Logarithmic Space and Post-Selection}}, booktitle = {16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)}, pages = {10:1--10:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-198-6}, ISSN = {1868-8969}, year = {2021}, volume = {197}, editor = {Hsieh, Min-Hsiu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.10}, URN = {urn:nbn:de:0030-drops-140054}, doi = {10.4230/LIPIcs.TQC.2021.10}, annote = {Keywords: computational complexity, space-bounded quantum computation, post-selection} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

This paper tackles the issue of checking that all copies of a large data set replicated at several nodes of a network are identical. The fact that the replicas may be located at distant nodes prevents the system from verifying their equality locally, i.e., by having each node consult only nodes in its vicinity. On the other hand, it remains possible to assign certificates to the nodes, so that verifying the consistency of the replicas can be achieved locally. However, we show that, as the replicated data is large, classical certification mechanisms, including distributed Merlin-Arthur protocols, cannot guarantee good completeness and soundness simultaneously, unless they use very large certificates. The main result of this paper is a distributed quantum Merlin-Arthur protocol enabling the nodes to collectively check the consistency of the replicas, based on small certificates, and in a single round of message exchange between neighbors, with short messages. In particular, the certificate-size is logarithmic in the size of the data set, which gives an exponential advantage over classical certification mechanisms. We propose yet another usage of a fundamental quantum primitive, called the SWAP test, in order to show our main result.

Pierre Fraigniaud, François Le Gall, Harumichi Nishimura, and Ami Paz. Distributed Quantum Proofs for Replicated Data. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fraigniaud_et_al:LIPIcs.ITCS.2021.28, author = {Fraigniaud, Pierre and Le Gall, Fran\c{c}ois and Nishimura, Harumichi and Paz, Ami}, title = {{Distributed Quantum Proofs for Replicated Data}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {28:1--28:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.28}, URN = {urn:nbn:de:0030-drops-135679}, doi = {10.4230/LIPIcs.ITCS.2021.28}, annote = {Keywords: Quantum Computing, Distributed Network Computing, Algorithmic Aspects of Networks} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

In this paper we give fast distributed graph algorithms for detecting and listing small subgraphs, and for computing or approximating the girth. Our algorithms improve upon the state of the art by polynomial factors, and for girth, we obtain a constant-time algorithm for additive +1 approximation in Congested Clique, and the first parametrized algorithm for exact computation in Congest.
In the Congested Clique model, we first develop a technique for learning small neighborhoods, and apply it to obtain an O(1)-round algorithm that computes the girth with only an additive +1 error. Next, we introduce a new technique (the partition tree technique) allowing for efficiently listing all copies of any subgraph, which is deterministic and improves upon the state-of the-art for non-dense graphs. We give two concrete applications of the partition tree technique: First we show that for constant k, it is possible to solve C_{2k}-detection in O(1) rounds in the Congested Clique, improving on prior work, which used fast matrix multiplication and thus had polynomial round complexity. Second, we show that in triangle-free graphs, the girth can be exactly computed in time polynomially faster than the best known bounds for general graphs. We remark that no analogous result is currently known for sequential algorithms.
In the Congest model, we describe a new approach for finding cycles, and instantiate it in two ways: first, we show a fast parametrized algorithm for girth with round complexity Õ(min{g⋅ n^{1-1/Θ(g)},n}) for any girth g; and second, we show how to find small even-length cycles C_{2k} for k = 3,4,5 in O(n^{1-1/k}) rounds. This is a polynomial improvement upon the previous running times; for example, our C₆-detection algorithm runs in O(n^{2/3}) rounds, compared to O(n^{3/4}) in prior work. Finally, using our improved C₆-freeness algorithm, and the barrier on proving lower bounds on triangle-freeness of Eden et al., we show that improving the current ̃Ω(√n) lower bound for C₆-freeness of Korhonen et al. by any polynomial factor would imply strong circuit complexity lower bounds.

Keren Censor-Hillel, Orr Fischer, Tzlil Gonen, François Le Gall, Dean Leitersdorf, and Rotem Oshman. Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{censorhillel_et_al:LIPIcs.DISC.2020.33, author = {Censor-Hillel, Keren and Fischer, Orr and Gonen, Tzlil and Le Gall, Fran\c{c}ois and Leitersdorf, Dean and Oshman, Rotem}, title = {{Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {33:1--33:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.33}, URN = {urn:nbn:de:0030-drops-131115}, doi = {10.4230/LIPIcs.DISC.2020.33}, annote = {Keywords: distributed graph algorithms, cycles, girth, Congested Clique, CONGEST} }

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Brief Announcement

**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

This paper tackles the issue of checking that all copies of a large data set replicated at several nodes of a network are identical. The fact that the replicas may be located at distant nodes prevents the system from verifying their equality locally, i.e., by having each node consult only nodes in its vicinity. On the other hand, it remains possible to assign certificates to the nodes, so that verifying the consistency of the replicas can be achieved locally. However, we show that, as the replicated data is large, classical certification mechanisms, including distributed Merlin-Arthur protocols, cannot guarantee good completeness and soundness simultaneously, unless they use very large certificates. The main result of this paper is a distributed quantum Merlin-Arthur protocol enabling the nodes to collectively check the consistency of the replicas, based on small certificates, and in a single round of message exchange between neighbors, with short messages. In particular, the certificate-size is logarithmic in the size of the data set, which gives an exponential advantage over classical certification mechanisms.

Pierre Fraigniaud, François Le Gall, Harumichi Nishimura, and Ami Paz. Brief Announcement: Distributed Quantum Proofs for Replicated Data. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 43:1-43:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fraigniaud_et_al:LIPIcs.DISC.2020.43, author = {Fraigniaud, Pierre and Le Gall, Fran\c{c}ois and Nishimura, Harumichi and Paz, Ami}, title = {{Brief Announcement: Distributed Quantum Proofs for Replicated Data}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {43:1--43:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.43}, URN = {urn:nbn:de:0030-drops-131217}, doi = {10.4230/LIPIcs.DISC.2020.43}, annote = {Keywords: Quantum Computing, Distributed Network Computing, Algorithmic Aspects of Networks} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

A recent breakthrough by Tang (STOC 2019) showed how to "dequantize" the quantum algorithm for recommendation systems by Kerenidis and Prakash (ITCS 2017). The resulting algorithm, classical but "quantum-inspired", efficiently computes a low-rank approximation of the users' preference matrix. Subsequent works have shown how to construct efficient quantum-inspired algorithms for approximating the pseudo-inverse of a low-rank matrix as well, which can be used to (approximately) solve low-rank linear systems of equations. In the present paper, we pursue this line of research and develop quantum-inspired algorithms for a large class of matrix transformations that are defined via the singular value decomposition of the matrix. In particular, we obtain classical algorithms with complexity polynomially related (in most parameters) to the complexity of the best quantum algorithms for singular value transformation recently developed by Chakraborty, Gilyén and Jeffery (ICALP 2019) and Gilyén, Su, Low and Wiebe (STOC 2019).

Dhawal Jethwani, François Le Gall, and Sanjay K. Singh. Quantum-Inspired Classical Algorithms for Singular Value Transformation. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jethwani_et_al:LIPIcs.MFCS.2020.53, author = {Jethwani, Dhawal and Le Gall, Fran\c{c}ois and Singh, Sanjay K.}, title = {{Quantum-Inspired Classical Algorithms for Singular Value Transformation}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {53:1--53:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.53}, URN = {urn:nbn:de:0030-drops-127193}, doi = {10.4230/LIPIcs.MFCS.2020.53}, annote = {Keywords: Sampling algorithms, quantum-inspired algorithms, linear algebra} }

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

This paper considers the triangle finding problem in the CONGEST model of distributed computing. Recent works by Izumi and Le Gall (PODC'17), Chang, Pettie and Zhang (SODA'19) and Chang and Saranurak (PODC'19) have successively reduced the classical round complexity of triangle finding (as well as triangle listing) from the trivial upper bound O(n) to Õ(n^{1/3}), where n denotes the number of vertices in the graph. In this paper we present a quantum distributed algorithm that solves the triangle finding problem in Õ(n^{1/4}) rounds in the CONGEST model. This gives another example of quantum algorithm beating the best known classical algorithms in distributed computing. Our result also exhibits an interesting phenomenon: while in the classical setting the best known upper bounds for the triangle finding and listing problems are identical, in the quantum setting the round complexities of these two problems are now Õ(n^{1/4}) and Θ~(n^{1/3}), respectively. Our result thus shows that triangle finding is easier than triangle listing in the quantum CONGEST model.

Taisuke Izumi, François Le Gall, and Frédéric Magniez. Quantum Distributed Algorithm for Triangle Finding in the CONGEST Model. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{izumi_et_al:LIPIcs.STACS.2020.23, author = {Izumi, Taisuke and Le Gall, Fran\c{c}ois and Magniez, Fr\'{e}d\'{e}ric}, title = {{Quantum Distributed Algorithm for Triangle Finding in the CONGEST Model}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {23:1--23:13}, 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.2020.23}, URN = {urn:nbn:de:0030-drops-118840}, doi = {10.4230/LIPIcs.STACS.2020.23}, annote = {Keywords: Quantum computing, distributed computing, CONGEST model} }

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

Recently Bravyi, Gosset and König (Science 2018) proved an unconditional separation between the computational powers of small-depth quantum and classical circuits for a relation. In this paper we show a similar separation in the average-case setting that gives stronger evidence of the superiority of small-depth quantum computation: we construct a computational task that can be solved on all inputs by a quantum circuit of constant depth with bounded-fanin gates (a "shallow" quantum circuit) and show that any classical circuit with bounded-fanin gates solving this problem on a non-negligible fraction of the inputs must have logarithmic depth. Our results are obtained by introducing a technique to create quantum states exhibiting global quantum correlations from any graph, via a construction that we call the extended graph.
Similar results have been very recently (and independently) obtained by Coudron, Stark and Vidick (arXiv:1810.04233}), and Bene Watts, Kothari, Schaeffer and Tal (STOC 2019).

François Le Gall. Average-Case Quantum Advantage with Shallow Circuits. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{legall:LIPIcs.CCC.2019.21, author = {Le Gall, Fran\c{c}ois}, title = {{Average-Case Quantum Advantage with Shallow Circuits}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {21:1--21:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.21}, URN = {urn:nbn:de:0030-drops-108432}, doi = {10.4230/LIPIcs.CCC.2019.21}, annote = {Keywords: Quantum computing, circuit complexity, constant-depth circuits} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

There are two central models considered in (fault-free synchronous) distributed computing: the CONGEST model, in which communication channels have limited bandwidth, and the LOCAL model, in which communication channels have unlimited bandwidth. Very recently, Le Gall and Magniez (PODC 2018) showed the superiority of quantum distributed computing over classical distributed computing in the CONGEST model. In this work we show the superiority of quantum distributed computing in the LOCAL model: we exhibit two computational tasks that can be solved in a constant number of rounds in the quantum setting but require Omega(n) rounds in the classical (randomized) setting, where n denotes the size of the network.

François Le Gall, Harumichi Nishimura, and Ansis Rosmanis. Quantum Advantage for the LOCAL Model in Distributed Computing. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{legall_et_al:LIPIcs.STACS.2019.49, author = {Le Gall, Fran\c{c}ois and Nishimura, Harumichi and Rosmanis, Ansis}, title = {{Quantum Advantage for the LOCAL Model in Distributed Computing}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {49:1--49:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.49}, URN = {urn:nbn:de:0030-drops-102887}, doi = {10.4230/LIPIcs.STACS.2019.49}, annote = {Keywords: Quantum computing, distributed computing, LOCAL model} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

In this paper we consider what can be computed by a user interacting with a potentially malicious server, when the server performs polynomial-time quantum computation but the user can only perform polynomial-time classical (i.e., non-quantum) computation. Understanding the computational power of this model, which corresponds to polynomial-time quantum computation that can be efficiently verified classically, is a well-known open problem in quantum computing. Our result shows that computing the order of a solvable group, which is one of the most general problems for which quantum computing exhibits an exponential speed-up with respect to classical computing, can be realized in this model.

François Le Gall, Tomoyuki Morimae, Harumichi Nishimura, and Yuki Takeuchi. Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 26:1-26:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{legall_et_al:LIPIcs.MFCS.2018.26, author = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki and Nishimura, Harumichi and Takeuchi, Yuki}, title = {{Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {26:1--26:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.26}, URN = {urn:nbn:de:0030-drops-96087}, doi = {10.4230/LIPIcs.MFCS.2018.26}, annote = {Keywords: Quantum computing, interactive proofs, group-theoretic problems} }

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**Published in:** LIPIcs, Volume 73, 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)

Triangle finding (deciding if a graph contains a triangle or not) is a central problem in quantum query complexity. The quantum communication complexity of this problem, where the edges of the graph are distributed among the players, was considered recently by Ivanyos et al. in the two- party setting. In this paper we consider its k-party quantum communication complexity with k >= 3. Our main result is a ~O(m^(7/12))-qubit protocol, for any constant number of players k, deciding with high probability if a graph with m edges contains a triangle or not. Our approach makes connections between the multiparty quantum communication complexity of triangle finding and the quantum query complexity of graph collision, a well-studied problem in quantum query complexity.

François Le Gall and Shogo Nakajima. Multiparty Quantum Communication Complexity of Triangle Finding. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 73, pp. 6:1-6:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{legall_et_al:LIPIcs.TQC.2017.6, author = {Le Gall, Fran\c{c}ois and Nakajima, Shogo}, title = {{Multiparty Quantum Communication Complexity of Triangle Finding}}, booktitle = {12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)}, pages = {6:1--6:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-034-7}, ISSN = {1868-8969}, year = {2018}, volume = {73}, editor = {Wilde, Mark M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2017.6}, URN = {urn:nbn:de:0030-drops-85793}, doi = {10.4230/LIPIcs.TQC.2017.6}, annote = {Keywords: Quantum communication complexity, triangle finding, graph collision} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

A recent series of breakthroughs initiated by Spielman and Teng culminated in the construction of nearly linear time Laplacian solvers, approximating the solution of a linear system Lx=b, where L is the normalized Laplacian of an undirected graph. In this paper we study the space complexity of the problem. Surprisingly we are able to show a probabilistic, logspace algorithm solving the problem. We further extend the algorithm to other families of graphs like Eulerian graphs (and directed regular graphs) and graphs that mix in polynomial time.
Our approach is to pseudo-invert the Laplacian, by first "peeling-off" the problematic kernel of the operator, and then to approximate the inverse of the remaining part by using a Taylor series. We approximate the Taylor series using a previous work and the special structure of the problem. For directed graphs we exploit in the analysis the Jordan normal form and results from matrix functions.

Dean Doron, François Le Gall, and Amnon Ta-Shma. Probabilistic Logarithmic-Space Algorithms for Laplacian Solvers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 41:1-41:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{doron_et_al:LIPIcs.APPROX-RANDOM.2017.41, author = {Doron, Dean and Le Gall, Fran\c{c}ois and Ta-Shma, Amnon}, title = {{Probabilistic Logarithmic-Space Algorithms for Laplacian Solvers}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {41:1--41:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.41}, URN = {urn:nbn:de:0030-drops-75908}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.41}, annote = {Keywords: Laplacian solvers, Randomized logspace, Bounded-space complexity classes, Random walks, Matrix computation} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Computing set joins of two inputs is a common task in database theory. Recently, Van Gucht, Williams, Woodruff and Zhang [PODS 2015] considered the complexity of such problems in the natural model of (classical) two-party communication complexity and obtained tight bounds for the complexity of several important distributed set joins.
In this paper we initiate the study of the quantum communication complexity of distributed set joins. We design a quantum protocol for distributed Boolean matrix multiplication, which corresponds to computing the composition join of two databases, showing that the product of two n times n Boolean matrices, each owned by one of two respective parties, can be computed with widetilde-O(sqrt{n} ell^{3/4}) qubits of communication, where ell denotes the number of non-zero entries of the product. Since Van Gucht et al. showed that the classical communication complexity of this problem is widetilde-Theta(n sqrt{ell}), our quantum algorithm outperforms classical protocols whenever the output matrix is sparse. We also show a quantum lower bound and a matching classical upper bound on the communication complexity of distributed matrix multiplication over F_2.
Besides their applications to database theory, the communication complexity of set joins is interesting due to its connections to direct product theorems in communication complexity. In this work we also introduce a notion of all-pairs product theorem, and relate this notion to standard direct product theorems in communication complexity.

Stacey Jeffery and François Le Gall. Quantum Communication Complexity of Distributed Set Joins. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{jeffery_et_al:LIPIcs.MFCS.2016.54, author = {Jeffery, Stacey and Le Gall, Fran\c{c}ois}, title = {{Quantum Communication Complexity of Distributed Set Joins}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {54:1--54:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.54}, URN = {urn:nbn:de:0030-drops-64663}, doi = {10.4230/LIPIcs.MFCS.2016.54}, annote = {Keywords: quantum communication complexity, distributed quantum computing, database joins} }

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

This paper investigates the role of interaction and coins in quantum Arthur-Merlin games (also called public-coin quantum interactive proof systems). While the existing model restricts the messages from the verifier to be classical even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the messages from the verifier can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. This generalization turns out to provide several novel characterizations of quantum interactive proof systems with a constant number of turns. First, it is proved that the complexity class corresponding to two-turn quantum Arthur-Merlin games where both of the two messages are quantum, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interaction prior to the communications of qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper presents a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking properties related to quantum circuits, and thus, qq-QAM may provide a good measure in computational complexity theory. It is further proved that the class qq-QAM_1, the perfect-completeness variant of qq-QAM, gives new bounds for standard well-studied classes of two-turn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum Arthur-Merlin games: it is proved that, for any constant m >= 2, the class of problems having $m$-turn quantum Arthur-Merlin proof systems is either equal to PSPACE or equal to the class of problems having two-turn quantum Arthur-Merlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.

Hirotada Kobayashi, Francois Le Gall, and Harumichi Nishimura. Generalized Quantum Arthur-Merlin Games. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 488-511, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kobayashi_et_al:LIPIcs.CCC.2015.488, author = {Kobayashi, Hirotada and Le Gall, Francois and Nishimura, Harumichi}, title = {{Generalized Quantum Arthur-Merlin Games}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {488--511}, 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.2015.488}, URN = {urn:nbn:de:0030-drops-50697}, doi = {10.4230/LIPIcs.CCC.2015.488}, annote = {Keywords: interactive proof systems, Arthur-Merlin games, quantum computing, complete problems, entanglement} }

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

In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian
groups. Le Gall has constructed an efficient classical algorithm for a class of groups corresponding to one of the most natural ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $\Int_m$ with the order of $A$ coprime with $m$.
More precisely, the running time of that algorithm is almost linear in the order of the input groups.
In this paper we present a \emph{quantum} algorithm solving the same problem in time polynomial in the \emph{logarithm} of the order of the input groups. This algorithm works in the black-box setting and is the first quantum algorithm solving instances of the nonabelian group isomorphism problem exponentially faster than the best known classical algorithms.

François Le Gall. An Efficient Quantum Algorithm for Some Instances of the Group Isomorphism Problem. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 549-560, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{legall:LIPIcs.STACS.2010.2484, author = {Le Gall, Fran\c{c}ois}, title = {{An Efficient Quantum Algorithm for Some Instances of the Group Isomorphism Problem}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {549--560}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2484}, URN = {urn:nbn:de:0030-drops-24843}, doi = {10.4230/LIPIcs.STACS.2010.2484}, annote = {Keywords: Quantum algorithms, group isomorphism problem, black-box groups} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

The group isomorphism problem asks whether two given groups are isomorphic or not. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. In this paper we study this problem for a class of groups corresponding to one of the simplest ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $\mathbb{Z}_m$. We present an efficient algorithm solving the group isomorphism problem for all the groups of this class such that the order of $A$ is coprime with $m$. More precisely, our algorithm runs in time almost linear in the orders of the input groups and works in the general setting where the groups are given as black-boxes.

Francois Le Gall. Efficient Isomorphism Testing for a Class of Group Extensions. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 625-636, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{legall:LIPIcs.STACS.2009.1830, author = {Le Gall, Francois}, title = {{Efficient Isomorphism Testing for a Class of Group Extensions}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {625--636}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1830}, URN = {urn:nbn:de:0030-drops-18303}, doi = {10.4230/LIPIcs.STACS.2009.1830}, annote = {Keywords: Polynomial-time algorithms, Group isomorphism, Black-box groups} }

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