发表机构
Cambridge University; IRIF, CNRS, Université Paris Cité; Tilburg University（剑桥大学; IRIF，法国国家科学研究中心，巴黎西岱大学; 蒂尔堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究将Cousins和Vempala的随机体积估计算法量子化，利用换能器与摊销量子游走框架，将查询复杂度从O~(d^3.5+d^2.25/ε)改进至O~(d^3.5+d^1.75/ε)，克服了量子实现的关键障碍。
AI 中文摘要
体积估计问题是计算几何中的一个经典任务。针对该问题的随机算法的发展推动了与马尔可夫链蒙特卡洛和模拟退火相关的许多有影响力的算法技术的进步，并且该问题与若干重要的几何结果相关联，例如最近解决的KLS猜想。在本工作中，我们将Cousins和Vempala开发的最先进的具有$\widetilde{O}(d^{3.5}+d^3/\varepsilon^2)$查询复杂度的随机算法量子化，并获得了一个具有$\widetilde{O}(d^{3.5} + d^{1.75}/\varepsilon)$查询复杂度的量子算法，改进了最先进的$\widetilde{O}(d^{3.5} + d^{2.25}/\varepsilon)$界限。我们的关键技术贡献是一个用于摊销量子游走成本的框架。该框架基于Belovs、Jeffery和Yolcu最近引入的换能器工具包。正是这个摊销量子游走框架使我们能够利用Cousins和Vempala对球游走的摊销分析，从而克服了先前阻碍其量子实现的关键障碍。
英文摘要
The volume estimation problem is a classic task in computational geometry. The development of randomized algorithms for this problem spurred the development of many influential algorithmic techniques related to Markov Chain Monte Carlo and simulated annealing, and the problem connects to several important geometrical results, like the KLS conjecture. In this work, we quantize the state-of-the-art $\widetilde{O}(d^{3.5}+d^3/\varepsilon^2)$-query randomized algorithm developed by Cousins and Vempala, and obtain a $\widetilde{O}(d^{3.5} + d^{1.75}/\varepsilon)$-query quantum algorithm, improving over the $\widetilde{O}(d^{3.5} + d^{2.25}/\varepsilon)$ state-of-the-art bound. Our key technical contribution is a framework for amortizing the cost of a quantum walk. The framework is based on the recent transducer toolkit introduced by Belovs, Jeffery and Yolcu. It is this amortized quantum walk framework that allows us to exploit the amortized analysis of the ball walk by Cousins and Vempala, thus overcoming the key barrier that previously barred its quantum implementation.
