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. 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 * The full version of this paper is available at http://arxiv.org/ abs/1207.5211 † 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 Server model and Quantum Simulation Theorem which together provide a connection between distributed algorithms and communication complexity. The Server model is the standard twoparty 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. In particular, if one can prove a lower bound for distributed algorithms for a certain problem using the technique of Das Sarma et al., it is likely that such lower bound can be extended to the quantum setting using tools provided in this paper and without the need of knowledge in quantum computing. Problems Our results Ham, ST, MST verification Ω( n/B log n) deterministic, classical communication [13, 36 ] Ω( n/B log n) two-sided error, quantum communication with entanglement Conn and other verification problems from [13] Ω( n/B log n) two-sided error, classical communication [13] α-approx MST and other optimization problems from [13] Ω( n/B log n) Monte Carlo, classical communication for W = Ω(αn) [13]
Recently, Elkin et al. REF proved that quantum communication cannot substantially speed up algorithms for some fundamental problems, such as the minimum spanning tree, compared to the classical setting.
Can quantum communication speed up distributed computation?
PODC '14
Computer Science Physics