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ACM Symposium on Principles of Distributed Computing (PODC)

Can quantum communication speed up distributed computation?

Michael Elkin, Hartmut Klauck, Danupon Nanongkai, Gopal Pandurangan

Published Jul 15, 2014• 39 citations
Publisher

Abstract

The focus of this paper is on quantum distributed computation, where we investigate whether quantum communication can help in speeding up distributed network algorithms. Our main result is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, quantum communication does not help in substantially speeding up distributed algorithms for these problems compared to the classical setting.

DOI
10.1145/2611462.2611488

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The focus of this paper is on {\em quantum distributed} computation, where we investigate whether quantum communication can help in {\em speeding up} distributed network algorithms. Our main result is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, quantum communication {\em does not} help in substantially speeding up distributed algorithms for these problems compared to the classical setting. In order to obtain this result, we extend the technique of Das Sarma et al. [SICOMP 2012] to obtain a uniform approach to prove non-trivial lower bounds for quantum distributed algorithms for several graph optimization (both exact and approximate versions) as well as verification problems, some of which are new even in the classical setting, e.g. tight randomized lower bounds for Hamiltonian cycle and spanning tree verification, answering an open problem of Das Sarma et al., and a lower bound in terms of the weight aspect ratio, matching the upper bounds of Elkin [STOC 2004]. Our approach introduces the {\em Server model} and {\em Quantum Simulation Theorem} which together provide a connection between distributed algorithms and communication complexity. The Server model is the standard two-party communication complexity model augmented with additional power; yet, most of the hardness in the two-party model is carried over to this new model. The Quantum Simulation Theorem carries this hardness further to quantum distributed computing. Our techniques, except the proof of the hardness in the Server model, require very little knowledge in quantum computing, and this can help overcoming a usual impediment in proving bounds on quantum distributed algorithms.

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The focus of this paper is on quantum distributed computation, where we investigate whether quantum communication can help in speeding up distributed network algorithms. Our main result is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, quantum communication does not help in substantially speeding up distributed algorithms for these problems compared to the classical setting.
The focus of this paper is on {\em quantum distributed} computation, where we investigate whether quantum communication can help in {\em speeding up} distributed network algorithms. Our main result is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, quantum communication {\em does not} help in substantially speeding up distributed algorithms for these problems compared to the classical setting. In order to obtain this result, we extend the technique of Das Sarma et al. [SICOMP 2012] to obtain a uniform approach to prove non-trivial lower bounds for quantum distributed algorithms for several graph optimization (both exact and approximate versions) as well as verification problems, some of which are new even in the classical setting, e.g. tight randomized lower bounds for Hamiltonian cycle and spanning tree verification, answering an open problem of Das Sarma et al., and a lower bound in terms of the weight aspect ratio, matching the upper bounds of Elkin [STOC 2004]. Our approach introduces the {\em Server model} and {\em Quantum Simulation Theorem} which together provide a connection between distributed algorithms and communication complexity. The Server model is the standard two-party communication complexity model augmented with additional power; yet, most of the hardness in the two-party model is carried over to this new model. The Quantum Simulation Theorem carries this hardness further to quantum distributed computing. Our techniques, except the proof of the hardness in the Server model, require very little knowledge in quantum computing, and this can help overcoming a usual impediment in proving bounds on quantum distributed algorithms.
We present new distributed quantum algorithms for fundamental distributed computing problems, namely, leader election, broadcast, Minimum Spanning Tree (MST), and Breadth-First Search (BFS) tree, in arbitrary networks. These algorithms are (essentially) optimal with respect to their communication (message) complexity in the {\em quantum routing model} introduced in [PODC 2025]. The message complexity of our algorithms is O~(n)\tilde{O}(n)O~(n) for leader election, broadcast, and MST, and O~(mn)\tilde{O}(\sqrt{mn})O~(mn​) for BFS (nnn and mmm are the number of nodes and edges of the network, respectively). These message bounds are nearly tight in the quantum routing model since we show almost matching corresponding quantum message lower bounds. Our results significantly improve on the prior work of [PODC 2025], who presented distributed quantum algorithms under the same model that had a message complexity of O~(mn)\tilde{O}(\sqrt{mn})O~(mn​) for leader election. Our algorithms demonstrate the significant communication advantage that quantum routing has over classical in distributed computing, since Ω(m)Ω(m)Ω(m) is a well-established classical message lower bound for leader election, broadcast, MST, and BFS that applies even to randomized Monte-Carlo algorithms [JACM 2015]. Thus, our quantum algorithms can, in general, give a quadratic advantage in the communication cost for these fundamental problems. A main technical tool we use to design our distributed algorithms is quantum walks based on electric networks. We posit a framework for using quantum walks in the distributed setting to design communication-efficient distributed quantum algorithms. Our framework can be used as a black box to significantly reduce communication costs and may be of independent interest. Additionally, our lower-bound technique for establishing distributed quantum message lower bounds can also be applied to other problems.
We study lower bounds for quantum distributed computing, where a set of nodes (representing quantum computers) interconnected by an underlying network consisting of (bandwidth-restricted) links, communicate using quantum communication. Nodes have unlimited computational power and may share an unlimited number of entangled qubits. Our main contribution is a simple uniform technique for proving lower bounds for quantum distributed algorithms. In particular, we identify a new quantum communication model called the Server model which provides a connection between distributed algorithms and communication complexity: it is strong enough to capture the hardness of several distributed computing problems, while weak enough that several hard problems in two-party communication complexity remain hard; to this end, we identify a set of communication complexity techniques that can be carried over to the Server model, namely techniques based on nonlocal games. We show that these techniques serve as a fundamental tool in proving lower bounds in quantum distributed computing. The new techniques help us to prove several non-trivial quantum distributed lower bounds (which are the first-known quantum bounds for fundamental global problems such as minimum spanning tree, shortest paths etc.), some of which are new even in the classical setting. First, it allows us to show that all previous classical lower bounds of more than twenty verification and optimization graph problems in [Das Sarma et al., STOC 11] also hold in the quantum setting. Many of these bounds are tight, implying a large class of problems that do not gain an advantage from quantum effects. Our results also imply the following new results in the classical communication models: (1) the first randomized lower bounds for Hamiltonian cycle and spanning tree verification problems in both the distributed computing and the communication complexity model, answering the open problem of Das Sarma et al. and subsuming many bounds in [Babai, Frankl, and Simon, FOCS 96], and (2) the first lower bound that is tight for all weight aspect ratios, matching previous upper bounds of [Elkin, STOC 04]. Submitted for a Regular Presentation. The full version of this paper can be found as [15] at http://arxiv.org/abs/1207.5211. ∗Department of Computer Science, Ben-Gurion University, Beer-Sheva, 84105, Israel. E-mail: elkinm@cs.bgu.ac.il. †Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Centre for Quantum Technologies, National University of Singapore, Singapore 117543. E-mail: hklauck@gmail.com. Research at the Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation. ‡Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371. Work partially done while at University of Vienna, Austria. E-mail: danupon@gmail.com. §Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Department of Computer Science, Brown University, Providence, RI 02912, USA. E-mail: gopalpandurangan@gmail.com. Supported in part by the following research grants: Nanyang Technological University grant M58110000, Singapore MOE Academic Research Fund (AcRF) Tier 2 grant MOE2010-T2-2-082, and a grant from the US-Israeli Binational Science Foundation (BSF).
Quantum advantage is well-established in centralized computing, where quantum algorithms can solve certain problems exponentially faster than classical ones. In the distributed setting, significant progress has been made in bandwidth-limited networks, where quantum distributed networks have shown computational advantages over classical counterparts. However, the potential of quantum computing in networks that are constrained only by large distances is not yet understood. We focus on the LOCAL model of computation (Linial, FOCS 1987), a distributed computational model where computational power and communication bandwidth are unconstrained, and its quantum generalization. In this brief survey, we summarize recent progress on the quantum-LOCAL model outlining its limitations with respect to its classical counterpart: we discuss emerging techniques, and highlight open research questions that could guide future efforts in the field.