有界度图模型中二部性与扩张测试的近最优量子查询下界
Near-optimal quantum query lower bounds on bipartiteness and expansion testing in the bounded-degree graph model
浏览论文内容
中文总结 AI 辅助
本研究在有界度图模型中，通过多项式方法和归约，证明了二部性与扩张测试的量子查询复杂度下界为$\widetilde{\Omega}(N^{1/3})$，从而在多项式对数因子内完全刻画了这两个问题的量子查询复杂度。
中文摘要 AI 辅助
在这项工作中，我们通过量子查询复杂度的视角，研究了有界度图模型中的图性质测试中的两个经典问题：二部性测试和扩张测试。在经典设置中，已知对于这两个测试问题，$\tildetilde{\Theta}(\sqrt{N})$次查询是必要且充分的（Goldreich和Ron，1999年、2000年和2002年），其中$N$表示输入图的顶点数。鉴于这些问题的重要性，（Ambainis、Childs和Liu，2011年）在量子设置中启动了这些问题的研究，并设计了用于二部性和扩张测试的量子算法，这些算法执行$\widetilde{O}(N^{1/3})$次查询，显示出多项式加速。他们还证明了扩张测试需要$\widetilde{\Omega}(N^{1/4})$次查询，但二部性测试是否存在指数级量子优势的问题仍然悬而未决。尽管付出了巨大努力，但在过去十五年中，这些结果没有任何改进。在这项工作中，我们证明了对于二部性和扩张测试，本质上紧的$\widetilde{\Omega}(N^{1/3})$量子查询下界，从而在多项式对数因子范围内完全刻画了这些问题的量子查询复杂度。虽然我们的证明与Ambainis、Childs和Liu类似地使用了多项式方法，但我们使用了中间问题，并通过归约将它们与主要问题联系起来，并对所得多项式进行了更精确的分析，从而得出了近最优的下界。
英文摘要
In this work, we study bipartiteness and expansion testing, two canonical problems in graph property testing in the bounded-degree model through the lens of quantum query complexity. In the classical setting, it is known that $\widetildeΘ(\sqrt{N})$ queries are necessary and sufficient for both these testing problems (Goldreich and Ron, 1999, 2000 & 2002), where $N$ denotes the number of vertices of the input graph. Due to their significance, (Ambainis, Childs, and Liu, 2011) initiated the study of these problems in the quantum setting and designed quantum algorithms for bipartiteness and expansion testing that perform $\widetilde{O}(N^{1/3})$ queries, showing a polynomial speedup. They also proved that $\widetildeΩ(N^{1/4})$ queries are necessary for expansion testing, but the possibility of an exponential quantum advantage for bipartiteness testing remained open. Despite significant effort, there has been no improvement in these results in the last decade and a half. In this work, we prove essentially tight $\widetildeΩ(N^{1/3})$ quantum query lower bounds for both bipartiteness and expansion testing, thereby completely characterizing the quantum query complexity of these problems up to polylogarithmic factors. While our proofs use the polynomial method similarly to Ambainis, Childs, and Liu, we use intermediate problems that we relate to the main problems via reductions, and perform a more precise analysis of the resulting polynomials, leading to the near-optimal lower bounds.
发表机构
- Université Paris Cité, CNRS, IRIF（巴黎西岱大学、法国国家科学研究中心、IRIF）
- Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
- Ludwig-Maximilians-Universität München & MCQST（慕尼黑路德维希-马克西米利安大学与MCQST）
机构由 AI 辅助整理，请以论文原文为准。
