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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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**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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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} }

Document

**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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