[Paper Review] Quantum Distributed Network Computing: Lower Bounds and Techniques

Michael Elkin, Hartmut Klauck+2 more|arXiv (Cornell University)|Jul 22, 2012

Cryptography and Data Security63 references5 citations

TL;DR

This paper investigates whether quantum communication can accelerate distributed network algorithms for fundamental problems like minimum spanning tree, minimum cut, and shortest paths. It introduces the Server model and Quantum Simulation Theorem to extend classical communication complexity lower bounds to quantum distributed settings, proving that quantum communication does not provide substantial speedups for these problems, even for verification tasks like Hamiltonian cycle and spanning tree.

ABSTRACT

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.

Motivation & Objective

• To determine whether quantum communication can accelerate distributed network algorithms for fundamental graph problems.
• To develop a general framework for proving lower bounds in quantum distributed computing that extends classical communication complexity.
• To resolve open problems in classical distributed computing, such as tight randomized lower bounds for Hamiltonian cycle and spanning tree verification.
• To establish a connection between distributed algorithms and communication complexity using the Server model and quantum simulation.
• To minimize reliance on quantum computing expertise by making the core techniques accessible to non-experts.

Proposed method

• Introduce the Server model, a two-party communication complexity model augmented with additional computational power, to preserve classical hardness in a more powerful setting.
• Develop the Quantum Simulation Theorem, which transfers lower bounds from the Server model to quantum distributed algorithms.
• Use the Server model to prove classical lower bounds for problems like Hamiltonian cycle and spanning tree verification, resolving open questions.
• Apply the Quantum Simulation Theorem to extend these classical lower bounds to the quantum distributed setting.
• Ensure the method requires minimal quantum computing knowledge, enabling broader accessibility to researchers in distributed computing.
• Leverage existing classical lower bound techniques and adapt them to quantum settings through the new theoretical framework.

Experimental results

Research questions

• RQ1 Can quantum communication significantly speed up distributed algorithms for fundamental network problems such as minimum spanning tree and shortest paths?
• RQ2 What are the tight randomized lower bounds for classical verification problems like Hamiltonian cycle and spanning tree?
• RQ3 How does the weight aspect ratio affect the complexity of distributed graph problems, and can lower bounds be matched to existing upper bounds?
• RQ4 To what extent does the Server model preserve the hardness of classical communication complexity in a more powerful setting?
• RQ5 Can the Quantum Simulation Theorem be used to transfer classical lower bounds to quantum distributed algorithms without deep quantum knowledge?

Key findings

• Quantum communication does not provide a substantial speedup for distributed algorithms solving minimum spanning tree, minimum cut, or shortest paths problems.
• The paper establishes tight randomized lower bounds for Hamiltonian cycle and spanning tree verification, resolving an open problem posed by Das Sarma et al.
• A lower bound in terms of the weight aspect ratio is proven, matching the upper bounds from Elkin (STOC 2004), showing optimality in this parameter.
• The Server model successfully preserves classical hardness, enabling new lower bounds even in the classical setting.
• The Quantum Simulation Theorem enables the transfer of classical lower bounds to quantum distributed algorithms, providing a general method for quantum lower bound analysis.
• The proposed framework requires minimal quantum computing knowledge, making it accessible to researchers without specialized quantum expertise.