Can Quantum Communication Speed Up Distributed Computation?

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.

💡 Research Summary

The paper investigates whether quantum communication can substantially accelerate distributed network algorithms for fundamental graph problems. Its central claim is negative: for problems such as Minimum Spanning Tree (MST), Minimum Cut, and Single‑Source Shortest Paths (SSSP), quantum communication does not yield asymptotically faster distributed algorithms compared to the best classical solutions.

To establish this claim, the authors extend the classical two‑party communication‑complexity framework of Das Sarma et al. (SICOMP 2012) by introducing two novel constructs: the Server Model and the Quantum Simulation Theorem. The Server Model augments the standard two‑party setting with a powerful “server” that can broadcast unlimited information, yet the essential difficulty of the problem remains in the limited exchange between the two original parties. The authors prove that most lower‑bound arguments that hold in the ordinary two‑party model survive unchanged in this stronger setting.

The Quantum Simulation Theorem then bridges the Server Model to the quantum distributed setting. It shows that any quantum distributed algorithm that solves a given problem in T rounds can be simulated by a protocol in the Server Model that uses at most O(T) rounds of communication. Consequently, any lower bound proved in the Server Model automatically translates into a lower bound for quantum distributed algorithms.

Using this machinery, the paper derives tight lower bounds for several optimization and verification tasks. For exact and approximate versions of MST, Minimum Cut, and SSSP, the authors obtain Ω(√n) or Ω(D + √n) round lower bounds that match the best known classical algorithms, even when quantum messages are allowed. They also settle an open problem from Das Sarma et al. by proving randomized Ω(√n) lower bounds for Hamiltonian‑cycle verification and spanning‑tree verification, showing that quantum resources do not help these verification tasks either.

A particularly interesting contribution is a lower bound that depends on the weight aspect ratio ρ of the graph. The authors prove an Ω(log ρ·(D + √n)) round lower bound, which aligns exactly with the upper bound previously given by Elkin (STOC 2004). This demonstrates that the dependence on ρ is inherent and cannot be eliminated by quantum techniques.

The technical development of the Server Model and the Quantum Simulation Theorem requires only elementary quantum‑information concepts, making the approach accessible to researchers without deep expertise in quantum computing. The paper therefore not only advances our understanding of quantum distributed computation but also provides a versatile toolkit for proving lower bounds in this emerging area.

In summary, the work shows that while quantum communication may reduce the amount of data transmitted, it does not overcome the fundamental round‑complexity barriers that govern distributed graph algorithms. The results suggest that any future quantum advantage in distributed computing must arise from algorithmic innovations beyond simply replacing classical messages with quantum ones, and that many classic lower‑bound techniques remain robust in the quantum regime.