Distributed Quantum Interactive Proofs
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.
distributed interactive proofs
distributed verification
quantum computation
Theory of computation~Distributed algorithms
Theory of computation~Quantum computation theory
42:1-42:21
Regular Paper
FLG was supported by the JSPS KAKENHI grants JP16H01705, JP19H04066, JP20H00579, JP20H04139, JP20H05966, JP21H04879 and by the MEXT Q-LEAP grants JPMXS0118067394 and JPMXS0120319794. MM would like to take this opportunity to thank the “Nagoya University Interdisciplinary Frontier Fellowship” supported by Nagoya University and JST, the establishment of university fellowships towards the creation of science technology innovation, Grant Number JPMJFS212. HN was supported by the JSPS KAKENHI grants JP19H04066, JP20H05966, JP21H04879, JP22H00522 and by the MEXT Q-LEAP grants JPMXS0120319794.
https://arxiv.org/abs/2210.01390
François
Le Gall
François Le Gall
Graduate School of Mathematics, Nagoya University, Japan
Masayuki
Miyamoto
Masayuki Miyamoto
Graduate School of Mathematics, Nagoya University, Japan
Harumichi
Nishimura
Harumichi Nishimura
Graduate School of Informatics, Nagoya University, Japan
10.4230/LIPIcs.STACS.2023.42
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François Le Gall, Masayuki Miyamoto, and Harumichi Nishimura
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