A Framework for Distributed Quantum Queries in the CONGEST Model

Abstract

The Quantum CONGEST model is a variant of the CONGEST model, where messages consist of O(log(n)) qubits. In this paper, we give a general framework for implementing quantum query algorithms efficiently in a Quantum CONGEST network, using the concept of parallel-query quantum algorithms. We apply our framework for distributed quantum queries in two settings: problems where data is distributed over the network, and graph theoretical problems where the network defines the input. The first setting is slightly unusual in CONGEST but here our results follow almost directly from the quantum query setting. The second setting is more traditional for the CONGEST model but here our framework requires also some classical CONGEST steps to apply. In the setting with distributed data, we show how a network can pick one of k dates for a meeting such that a maximum number of nodes is available, using Õ(g (kD)+D) rounds, with D the network diameter. The classical complexity is linear in k. We also give an efficient algorithm for element distinctness: if all nodes together holds a list of k numbers, we show that the nodes can determine whether there are any duplicates in Õ(k2/3D1/3+D) rounds. Classically this problem requires ∼ω(k+D) rounds. We also generalize the protocol for the distributed Deutsch-Jozsa problem from the two-party setting considered by Buhrman, Cleve, and Wigderson [4] to general networks. This gives a novel separation between exact classical and exact quantum protocols in the CONGEST model. In the setting where the input is the network structure itself, our framework almost directly allows us to recover the Õ(g nD) round diameter computation algorithm of Le Gall and Magniez [21]. We extend this approach to also compute the radius in the same number of rounds, and to give an -additive approximation of the average eccentricity in Õ(D3/2/) rounds. Finally, we give the first quantum speedups over classical CONGEST for the problems of cycle detection and girth computation. We detect whether a graph has a cycle of length at most k in O(k+(kn)1/2-1/Θ(k)) rounds. For girth computation, we give an Õ(g+(gn)1/2-1/Θ(g)) round algorithm for graphs with girth g, beating the classical ω(g n) round lower bound by Frischknecht, Holzer, and Wattenhofer [12].