量子算法估计凸体体积
Quantum algorithm for estimating volumes of convex bodies
AI总结：
提出首个量子体积估计算法，利用改进的模拟退火框架和连续空间量子游走，在查询和算术复杂度上优于经典算法，并证明在$1/ε$上最优。
摘要：
估计凸体的体积是凸几何中的一个核心问题，可视为计数的连续版本。我们提出一种量子算法，利用$\tilde{O}(n^{3}+n^{2.5}/ε)$次成员查询和$\tilde{O}(n^{5}+n^{4.5}/ε)$次额外算术运算，在乘法误差$ε$内估计$n$维凸体的体积。相比之下，已知最优经典算法需要$\tilde{O}(n^{4}+n^{3}/ε^{2})$次查询和$\tilde{O}(n^{6}+n^{5}/ε^{2})$次额外算术运算。据我们所知，这是首个用于体积估计的量子加速算法。我们的算法基于一个加速模拟退火算法的改进框架，该框架可能具有独立的研究价值。此框架适用于“切比雪夫冷却”设置，其中解表示为具有有界方差的比值望远镜乘积。在实现该框架时，我们开发了多种新技术，包括具有严格离散化误差界的连续空间量子游走理论。为补充我们的量子算法，我们还证明了体积估计需要$Ω(\n+1/ε)$次量子成员查询，这排除了在$n$上实现指数级量子加速的可能性，并表明我们的算法在$1/ε$上达到最优（至多相差多对数因子）。
英文摘要：
Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an $n$-dimensional convex body within multiplicative error $ε$ using $\tilde{O}(n^{3}+n^{2.5}/ε)$ queries to a membership oracle and $\tilde{O}(n^{5}+n^{4.5}/ε)$ additional arithmetic operations. For comparison, the best known classical algorithm uses $\tilde{O}(n^{4}+n^{3}/ε^{2})$ queries and $\tilde{O}(n^{6}+n^{5}/ε^{2})$ additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of "Chebyshev cooling", where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requires $Ω(\sqrt n+1/ε)$ quantum membership queries, which rules out the possibility of exponential quantum speedup in $n$ and shows optimality of our algorithm in $1/ε$ up to poly-logarithmic factors.
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