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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 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)

The private simultaneous messages (PSM) model is a non-interactive version of the multiparty secure computation (MPC), which has been intensively studied to examine the communication cost of the secure computation. We consider its quantum counterpart, the private simultaneous quantum messages (PSQM) model, and examine the advantages of quantum communication and prior entanglement of this model.
In the PSQM model, k parties P₁,…,P_k initially share a common random string (or entangled states in a stronger setting), and they have private classical inputs x₁,…, x_k. Every P_i generates a quantum message from the private input x_i and the shared random string (entangled states), and then sends it to the referee R. Receiving the messages from the k parties, R computes F(x₁,…,x_k) from the messages. Then, R learns nothing except for F(x₁,…,x_k) as the privacy condition.
We obtain the following results for this PSQM model. (i) We demonstrate that the privacy condition inevitably increases the communication cost in the two-party PSQM model as well as in the classical case presented by Applebaum, Holenstein, Mishra, and Shayevitz [Journal of Cryptology(3), 916-953 (2020)]. In particular, we prove a lower bound (3-o(1))n of the communication complexity in PSQM protocols with a shared random string for random Boolean functions of 2n-bit input, which is larger than the trivial upper bound 2n of the communication complexity without the privacy condition. (ii) We demonstrate a factor two gap between the communication complexity of PSQM protocols with shared entangled states and with shared random strings by designing a multiparty PSQM protocol with shared entangled states for a total function that extends the two-party equality function. (iii) We demonstrate an exponential gap between the communication complexity of PSQM protocols with shared entangled states and with shared random strings for a two-party partial function.

Akinori Kawachi and Harumichi Nishimura. Communication Complexity of Private Simultaneous Quantum Messages Protocols. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kawachi_et_al:LIPIcs.ITC.2021.20, author = {Kawachi, Akinori and Nishimura, Harumichi}, title = {{Communication Complexity of Private Simultaneous Quantum Messages Protocols}}, booktitle = {2nd Conference on Information-Theoretic Cryptography (ITC 2021)}, pages = {20:1--20:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-197-9}, ISSN = {1868-8969}, year = {2021}, volume = {199}, editor = {Tessaro, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.20}, URN = {urn:nbn:de:0030-drops-143393}, doi = {10.4230/LIPIcs.ITC.2021.20}, annote = {Keywords: Communication complexity, private simultaneous messages, quantum protocols, secure multi-party computation} }

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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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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 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 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

This paper investigates the power of polynomial-time quantum computation in which only a very limited number of qubits are initially clean in the |0> state, and all the remaining qubits are initially in the totally mixed state. No initializations of qubits are allowed during the computation, nor are intermediate measurements. The main contribution of this paper is to develop unexpectedly strong error-reduction methods for such quantum computations that simultaneously reduce the number of necessary clean qubits. It is proved that any problem solvable by a polynomialtime quantum computation with one-sided bounded error that uses logarithmically many clean qubits is also solvable with exponentially small one-sided error using just two clean qubits, and with polynomially small one-sided error using just one clean qubit. It is further proved in the twosided-error case that any problem solvable by such a computation with a constant gap between completeness and soundness using logarithmically many clean qubits is also solvable with exponentially small two-sided error using just two clean qubits. If only one clean qubit is available, the problem is again still solvable with exponentially small error in one of the completeness and soundness and with polynomially small error in the other. An immediate consequence is that the Trace Estimation problem defined with fixed constant threshold parameters is complete for BQ_{[1]}P and BQ_{log}P, the classes of problems solvable by polynomial-time quantum computations with completeness 2/3 and soundness 1/3 using just one and logarithmically many clean qubits, respectively. The techniques used for proving the error-reduction results may be of independent interest in themselves, and one of the technical tools can also be used to show the hardness of weak classical simulations of one-clean-qubit computations (i.e., DQC1 computations).

Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani. Power of Quantum Computation with Few Clean Qubits. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fujii_et_al:LIPIcs.ICALP.2016.13, author = {Fujii, Keisuke and Kobayashi, Hirotada and Morimae, Tomoyuki and Nishimura, Harumichi and Tamate, Shuhei and Tani, Seiichiro}, title = {{Power of Quantum Computation with Few Clean Qubits}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.13}, URN = {urn:nbn:de:0030-drops-62960}, doi = {10.4230/LIPIcs.ICALP.2016.13}, annote = {Keywords: DQC1, quantum computing, complete problems, error reduction} }

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

This paper presents a general space-efficient method for error reduction for unitary quantum computation. Consider a polynomial-time quantum computation with completeness c and soundness s, either with or without a witness (corresponding to QMA and BQP, respectively). To convert this computation into a new computation with error at most 2^{-p}, the most space-efficient method known requires extra workspace of O(p*log(1/(c-s))) qubits. This space requirement is too large for scenarios like logarithmic-space quantum computations. This paper shows an errorreduction method for unitary quantum computations (i.e., computations without intermediate measurements) that requires extra workspace of just O(log(p/(c-s))) qubits. This in particular gives the first method of strong amplification for logarithmic-space unitary quantum computations with two-sided bounded error. This also leads to a number of consequences in complexity theory, such as the uselessness of quantum witnesses in bounded-error logarithmic-space unitary quantum computations, the PSPACE upper bound for QMA with exponentially-small completeness-soundness gap, and strong amplification for matchgate computations.

Bill Fefferman, Hirotada Kobayashi, Cedric Yen-Yu Lin, Tomoyuki Morimae, and Harumichi Nishimura. Space-Efficient Error Reduction for Unitary Quantum Computations. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fefferman_et_al:LIPIcs.ICALP.2016.14, author = {Fefferman, Bill and Kobayashi, Hirotada and Yen-Yu Lin, Cedric and Morimae, Tomoyuki and Nishimura, Harumichi}, title = {{Space-Efficient Error Reduction for Unitary Quantum Computations}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.14}, URN = {urn:nbn:de:0030-drops-62975}, doi = {10.4230/LIPIcs.ICALP.2016.14}, annote = {Keywords: space-bounded computation, quantum Merlin-Arthur proof systems, error reduction, quantum computing} }

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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:** Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

Since quantum information is continuous, its handling is sometimes
surprisingly harder than the classical counterpart. A typical
example is cloning; making a copy of digital information is
straightforward but it is not possible exactly for quantum
information. The question in this paper is whether or not {em
quantum} network coding is possible. Its classical counterpart is
another good example to show that digital information flow can be done
much more efficiently than conventional (say, liquid) flow.
Our answer to the question is similar to the case of cloning, namely,
it is shown that quantum network coding is possible if approximation
is allowed, by using a simple network model called Butterfly. In this
network, there are two flow paths, $s_1$ to $t_1$ and $s_2$ to $t_2$,
which shares a single bottleneck channel of capacity one. In the
classical case, we can send two bits simultaneously, one for each
path, in spite of the bottleneck. Our results for quantum network
coding include: (i) We can send any quantum state $|psi_1
angle$
from $s_1$ to $t_1$ and $|psi_2
angle$ from $s_2$ to $t_2$
simultaneously with a fidelity strictly greater than $1/2$. (ii) If
one of $|psi_1
angle$ and $|psi_2
angle$ is classical, then the
fidelity can be improved to $2/3$. (iii) Similar improvement is also
possible if $|psi_1
angle$ and $|psi_2
angle$ are restricted to
only a finite number of (previously known) states. (iv) Several
impossibility results including the general upper bound of the fidelity
are also given.

Masahito Hayashi, Kazuo Iwama, Harumichi Nishimura, Rudy Raymond, and Shigeru Yamashita. Quantum Network Coding. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{hayashi_et_al:DagSemProc.06111.14, author = {Hayashi, Masahito and Iwama, Kazuo and Nishimura, Harumichi and Raymond, Rudy and Yamashita, Shigeru}, title = {{Quantum Network Coding}}, booktitle = {Complexity of Boolean Functions}, pages = {1--17}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6111}, editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.14}, URN = {urn:nbn:de:0030-drops-6080}, doi = {10.4230/DagSemProc.06111.14}, annote = {Keywords: Network coding, quantum computation, quantum information} }

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