The All-Pairs Shortest Path problem (APSP) is one of the most central problems in distributed computation. In the CONGEST-CLIQUE model, in which n nodes communicate with each other over a fully connected network by exchanging messages of O(łog n) bits in synchronous rounds, the best known general algorithm for APSP uses Õ(n1/3) rounds. Breaking this barrier is a fundamental challenge in distributed graph algorithms. In this paper we investigate for the first time quantum distributed algorithms in the CONGEST-CLIQUE model, where nodes can exchange messages of O(log n) quantum bits, and show that this barrier can be broken: we construct a Õ(n1/4)-round quantum distributed algorithm for the APSP over directed graphs with polynomial weights in the CONGEST-CLIQUE model. This speedup in the quantum setting contrasts with the case of the standard CONGEST model, for which Elkin et al. (PODC 2014) showed that quantum communication does not offer significant advantages over classical communication.
The All-Pairs Shortest Path problem (APSP) is one of the most central problems in distributed computation. In the CONGEST-CLIQUE model, in which n nodes communicate with each other over a fully connected network by exchanging messages of O(łog n) bits in synchronous rounds, the best known general algorithm for APSP uses Õ(n1/3) rounds. Breaking this barrier is a fundamental challenge in distributed graph algorithms. In this paper we investigate for the first time quantum distributed algorithms in the CONGEST-CLIQUE model, where nodes can exchange messages of O(log n) quantum bits, and show that this barrier can be broken: we construct a Õ(n1/4)-round quantum distributed algorithm for the APSP over directed graphs with polynomial weights in the CONGEST-CLIQUE model. This speedup in the quantum setting contrasts with the case of the standard CONGEST model, for which Elkin et al. (PODC 2014) showed that quantum communication does not offer significant advantages over classical communication.
Computing the distance parameters of a network, including the diameter, radius, eccentricities and the all-pairs shortest paths (APSP) is a central problem in distributed computing. This paper investigates the distance parameters in the quantum CONGEST models and establishes almost linear lower bounds on eccentricities and APSP, which match the classical upper bounds. Our results imply that there is not quantum speedup for these two problems. In contrast with the diameter and radius, exchanging quantum messages is able to save the communication when the networks have low diameters [F. L. Gall and F. Magniez, Sublinear-time quantum computation of the diameter in CONGEST networks, in Proc. 2018 ACM Symp. Principles of Distributed Computing (PODC), (2018), pp. 337–346; F. Magniez and A. Nayak, arXiv:2002.11795 ]. We obtain the lower bounds via a reduction from the two-way quantum communication complexity of the set intersection [A. A. Razborov, Izv. Math. 159 (2003)].
Mind the O~: Asymptotically Better, but Still Impractical, Quantum Distributed Algorithms
The CONGEST and CONGEST-CLIQUE models have been carefully studied to represent situations where the communication bandwidth between processors in a network is severely limited. Messages of only O(log(n)) bits of information each may be sent between processors in each round. The quantum versions of these models allow the processors instead to communicate and compute with quantum bits under the same bandwidth limitations. This leads to the following natural research question: What problems can be solved more efficiently in these quantum models than in the classical ones? Building on existing work, we contribute to this question in two ways. Firstly, we present two algorithms in the Quantum CONGEST-CLIQUE model of distributed computation that succeed with high probability; one for producing an approximately optimal Steiner Tree, and one for producing an exact directed minimum spanning tree, each of which uses O~(n1/4) rounds of communication and O~(n9/4) messages, where n is the number of nodes in the network. The algorithms thus achieve a lower asymptotic round and message complexity than any known algorithms in the classical CONGEST-CLIQUE model. At a high level, we achieve these results by combining classical algorithmic frameworks with quantum subroutines. An existing framework for using distributed version of Grover's search algorithm to accelerate triangle finding lies at the core of the asymptotic speedup. Secondly, we carefully characterize the constants and logarithmic factors involved in our algorithms as well as related algorithms, otherwise commonly obscured by O~ notation. The analysis shows that some improvements are needed to render both our and existing related quantum and classical algorithms practical, as their asymptotic speedups only help for very large values of n.
We present two algorithms in the Quantum CONGEST-CLIQUE model of distributed computation that succeed with high probability; one for producing an approximately optimal Steiner Tree, and one for producing an exact directed minimum spanning tree, each of which usesO(n1/4)rounds of communication andO~(n9/4)messages, achieving a lower asymptotic round and message complexity than any known algorithms in the classical CONGEST-CLIQUE model. At a high level, we achieve these results by combining classical algorithmic frameworks with quantum subroutines. Additionally, we characterize the constants and logarithmic factors involved in our algorithms as well as related classical algorithms, otherwise obscured byOnotation, revealing that advances are needed to render both the quantum and classical algorithms practical.
We present two algorithms in the quantum CONGEST-CLIQUE model of distributed computation that succeed with high probability: one for producing an approximately optimal Steiner tree, and one for producing an exact directed minimum spanning tree, each of which uses O˜(n1/4) rounds of communication and O˜(n9/4) messages, achieving a lower asymptotic round and message complexity than any known algorithms in the classical CONGEST-CLIQUE model. At a high level, we achieve these results by combining classical algorithmic frameworks with quantum subroutines. Additionally, we characterize the constants and logarithmic factors involved in our algorithms as well as related classical algorithms, otherwise obscured by O˜ notation, revealing that advances are needed to render both the quantum and classical algorithms practical.
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) for leader election, broadcast, and MST, and O~(mn) for BFS (n and m 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) for leader election. Our algorithms demonstrate the significant communication advantage that quantum routing has over classical in distributed computing, since 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.
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.