发表机构
Universiteit Leiden; Moth（莱顿大学; Moth）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种随机双人博弈的量子估计算法，利用去随机化多层蒙特卡洛与相干二分搜索，实现分支因子和精度上的双重二次加速，优于经典算法。
AI 中文摘要
我们提出了一种用于随机双人博弈的量子算法。其博弈树是经典博弈搜索中的期望最小最大树：m个对抗层与m个机会层交替出现，每个节点有deg个子节点，叶值位于长度为N的区间内。该算法以均方根误差ε估计博弈价值，对叶值的查询复杂度为O(K^D·deg^(D/4)·(N/ε)·D^(4D)·log(deg·N/ε)^(4D))，其中D=2m，K为绝对常数。若按文献中常见做法将深度D视为常数，则该界为Õ(deg^(m/2)·ε^(-1))，而经典算法为Õ(deg^m·ε^(-2))。因此，对抗树中已知的分支因子二次加速和单一期望中已知的精度二次加速，在将二者嵌套时均得以保留。我们采用去随机化的多层蒙特卡洛估计器处理机会层，并采用相干二分搜索处理对抗层。组合过程通过引入均方根误差与均匀误差保证之间的转换，以及与量子均值估计的嵌套组合来实现。该算法虽针对随机博弈提出，但适用于以下条件：父节点的价值是其子节点价值的Lipschitz函数，且机会顶点的价值对其子节点呈线性关系；然而，仅当确定性步骤具有亚线性查询复杂度的量子子程序时，加速才能得以保留。
英文摘要
We give a quantum algorithm for stochastic $2$-player games. Their game trees are the expectiminimax trees of classical game search: $m$ adversarial layers alternate with $m$ chance layers, every node has $deg$ children, and the leaf values lie in an interval of length $N$. The algorithm estimates the value of the game to root mean square error $ε$ with \[ O\!\left( K^{D}\, deg^{\frac{D}{4}} \, \frac{N}ε \, D^{4D} \log\left( \frac{deg N}ε \right)^{4D} \right), \qquad D = 2m \] queries to the leaf values, where $K$ is an absolute constant. If we treat the depth $D$ as a constant as is sometimes done in the literature, the bound is $\widetilde O(deg^{m/2}ε^{-1})$ against the classical $\widetilde O(deg^{m}ε^{-2})$. So the quadratic speedup in the branching factor known for adversarial trees and the quadratic speedup in the accuracy known for a single expectation both survive when we nest one inside the other. We use a derandomised multilevel Monte Carlo estimator for chance layers and a coherent binary search for adversarial layers. The composition is done by introducing a conversion between root-mean-square and uniform error guarantees, and a composition to nest it with quantum mean estimation. We state the algorithm for stochastic games, but it is valid for the following conditions: the value of a parent is a Lipschitz function of the values of its children and the value at a chance vertex is linear in its children, however the speedup is preserved only if the deterministic step has a quantum subroutine with sublinear query complexity.
Comments32 pages, 3 algorithms
