[论文解读] Quantum Lower Bounds for Distributed Network Computing

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

Complexity and Algorithms in Graphs参考文献 37被引用 1

一句话总结

本文提出了 Server 模型，这是一种新颖的量子通信框架，连接了分布式计算与通信复杂性，首次为最小生成树和最短路径等基础全局问题建立了量子下界。通过将通信复杂性中的非局域游戏技术应用于该框架，作者建立了紧致的量子下界，表明许多经典下界在量子环境下依然成立，证明了对于一大类问题，量子优势是有限的。

ABSTRACT

We study lower bounds for quantum distributed computing, where a set of nodes (representing quantum computers) interconnected by an underlying network consisting of (bandwidth-restricted) links, communicate using quantum communication. Nodes have unlimited computational power and may share an unlimited number of entangled qubits. Our main contribution is a simple uniform technique for proving lower bounds for quantum distributed algorithms. In particular, we identify a new quantum communication model called the Server model which provides a connection between distributed algorithms and communication complexity: it is strong enough to capture the hardness of several distributed computing problems, while weak enough that several hard problems in two-party communication complexity remain hard; to this end, we identify a set of communication complexity techniques that can be carried over to the Server model, namely techniques based on nonlocal games. We show that these techniques serve as a fundamental tool in proving lower bounds in quantum distributed computing. The new techniques help us to prove several non-trivial quantum distributed lower bounds (which are the first-known quantum bounds for fundamental global problems such as minimum spanning tree, shortest paths etc.), some of which are new even in the classical setting. First, it allows us to show that all previous classical lower bounds of more than twenty verification and optimization graph problems in [Das Sarma et al., STOC 11] also hold in the quantum setting. Many of these bounds are tight, implying a large class of problems that do not gain an advantage from quantum effects. Our results also imply the following new results in the classical communication models: (1) the first randomized lower bounds for Hamiltonian cycle and spanning tree verification problems in both the distributed computing and the communication complexity model, answering the open problem of Das Sarma et al. and subsuming many bounds in [Babai, Frankl, and Simon, FOCS 96], and (2) the first lower bound that is tight for all weight aspect ratios, matching previous upper bounds of [Elkin, STOC 04]. Submitted for a Regular Presentation. The full version of this paper can be found as [15] at http://arxiv.org/abs/1207.5211. ∗Department of Computer Science, Ben-Gurion University, Beer-Sheva, 84105, Israel. E-mail: [e-mail]. †Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Centre for Quantum Technologies, National University of Singapore, Singapore 117543. E-mail: [e-mail]. Research at the Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation. ‡Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371. Work partially done while at University of Vienna, Austria. E-mail: [e-mail]. §Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Department of Computer Science, Brown University, Providence, RI 02912, USA. E-mail: [e-mail]. Supported in part by the following research grants: Nanyang Technological University grant M58110000, Singapore MOE Academic Research Fund (AcRF) Tier 2 grant MOE2010-T2-2-082, and a grant from the US-Israeli Binational Science Foundation (BSF).

研究动机与目标

• 开发一个统一的框架，用于在带宽受限的分布式网络中证明量子下界。
• 识别哪些基本分布式问题即使在存在纠缠和无限本地计算能力的情况下也无法从量子加速中受益。
• 将经典通信复杂性技术——特别是非局域游戏——扩展到量子分布式设置中。
• 通过为哈密顿回路和生成树验证问题推导新的随机化下界，解决经典分布式计算中的开放问题。
• 在所有权重比范围内，建立与现有上界匹配的紧致下界，涵盖网络优化中的各类问题。

提出的方法

• 提出 Server 模型，这是一种量子通信模型，能够捕捉分布式问题的困难性，同时保留两方通信复杂性问题的难度。
• 将通信复杂性中非局域游戏技术的适应性应用于分析 Server 模型中的量子协议。
• 利用 Server 模型模拟和分析量子分布式算法，重点关注通信成本和纠缠资源的使用。
• 建立从两方通信复杂性中已知难题到分布式网络问题的归约。
• 利用非局域游戏的结构，推导出分布式设置下量子通信复杂性的下界。
• 通过证明经典下界在量子环境下依然成立，证明量子协议无法在广泛类别的全局图问题上超越经典方案。

实验结果

研究问题

• RQ1 能否将通信复杂性中的非局域游戏技术适应于证明分布式网络中的量子下界？
• RQ2 对于最小生成树和最短路径等基本全局问题，量子分布式算法是否相较于经典算法具有优势？
• RQ3 在分布式网络中，验证与优化问题的经典下界是否同样适用于量子设置？
• RQ4 在随机化模型中，哈密顿回路和生成树验证等分布式问题的最紧致下界是什么？
• RQ5 能否使用量子通信复杂性技术证明与已知上界在所有权重比范围内都匹配的紧致下界？

主要发现

• 首次为最小生成树和最短路径等基础全局问题建立了量子下界，表明这些问题不存在量子优势。
• Das Sarma 等人（STOC 2011）针对二十余个验证与优化问题的经典下界，在量子设置中依然成立，意味着这一大类问题不存在量子加速。
• 本文首次在分布式计算与通信复杂性模型中，为哈密顿回路和生成树验证问题提供了随机化下界。
• 证明了一个紧致下界，该下界与 Elkin（STOC 2004）的上界在所有权重比范围内完全匹配，解决了网络优化中的一个开放问题。
• Server 模型成功地将基于非局域游戏的技术从两方通信复杂性迁移至分布式设置，实现了新的下界证明。
• 结果表明，量子纠缠和无限本地计算能力无法克服许多分布式图问题中固有的通信复杂性障碍。