图之字形中持久性统计量的量子查询复杂度
Quantum Query Complexity of Persistence Statistics in Graph Zigzags
- California State University, Fresno（弗雷斯诺加州州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出量子算法，利用精确恒等式将图之字形条形寿命统计量转化为期望广义秩，实现最优查询复杂度，并证明经典下界。
AI中文摘要：
我们研究了从快照-邻接比特估计之字形条形寿命的标量汇总的查询复杂度。对于$n$个标记顶点上的图$G_1,\ldots,G_m$，令$\ell_b$为交集之字形中度为一的条形$b$的快照寿命。对于概率生成函数$\phi(x)=\mathbb{E}[x^R]$，统计量$F_\phi=\sum_b\phi(\ell_b/m)$包括归一化的$r$阶总持久性和均匀时间窗口上的平均广义秩。我们的算法基于一个精确恒等式：采样$R$个均匀时间；其最小值和最大值之间的期望广义秩等于$F_\phi$。对于这些图，该秩是交集图的电路秩，因此非线性条形码泛函变成边数和分量数的平均值，无需计算条形码。在没有谱隙、同调态或QRAM假设的情况下，当提供界限$K\ge F_\phi$时，这给出了一个具有加性误差$\varepsilon n$和$\widetilde O(\sqrt{m(K+n)}/\varepsilon)$次查询的量子估计器，而经典方法需要$\widetilde O(m\min\{n^2,(K+n)/\varepsilon^2\})$次查询，并且还有一个具有相同实例依赖性的自适应量子变体。这些估计器在两个区域中是最优的。对于每个固定幂权重$x^r$（$r\ge2$）和均匀窗口均值，最坏情况复杂度为量子$\widetilde\Theta(n\sqrt m/\varepsilon)$和经典$\Theta(n^2m)$。在稀疏实例上，在显式分裂泄漏承诺下（该承诺由对数度的幂权重和二项式权重满足）以及承诺$F_\phi\le K$，它们分别为$\widetilde\Theta(\sqrt{mK}/\varepsilon)$和$\widetilde\Theta(m\min\{n^2,K/\varepsilon^2\})$。经典下界适用于完全自适应算法，并且少于$m$个这样的统计量无法确定正寿命直方图。所有界限都涉及快照访问；对于显式更新流，已知有近线性全条形码算法。
英文摘要：
We study the query complexity of estimating scalar summaries of zigzag bar lifetimes from snapshot-adjacency bits. For graphs $G_1,\ldots,G_m$ on $n$ labeled vertices, let $\ell_b$ be the snapshot lifetime of a degree-one bar $b$ of the intersection zigzag. For a probability generating function $ϕ(x)=\mathbb{E}[x^R]$, the statistic $F_ϕ=\sum_bϕ(\ell_b/m)$ includes normalized degree-$r$ total persistence and the mean generalized rank over a uniform time window. An exact identity underlies our algorithm: sample $R$ uniform times; the expected generalized rank between their minimum and maximum equals $F_ϕ$. For graphs that rank is the circuit rank of an intersection graph, so a nonlinear barcode functional becomes an average of edge and component counts, and no barcode is computed. Without spectral-gap, homology-state, or QRAM assumptions, this gives a quantum estimator with additive error $\varepsilon n$ and $\widetilde O(\sqrt{m(K+n)}/\varepsilon)$ queries when a bound $K\ge F_ϕ$ is supplied, against $\widetilde O(m\min\{n^2,(K+n)/\varepsilon^2\})$ classically, and an adaptive quantum variant with the same instance dependence. These estimators are optimal in two regimes. For every fixed power weight $x^r$, $r\ge2$, and for the uniform-window mean, the worst-case complexities are $\widetildeΘ(n\sqrt m/\varepsilon)$ quantum and $Θ(n^2m)$ classical. On sparse instances, under an explicit split-leakage promise met by power and binomial weights of logarithmic degree and the promise $F_ϕ\le K$, they are $\widetildeΘ(\sqrt{mK}/\varepsilon)$ and $\widetildeΘ(m\min\{n^2,K/\varepsilon^2\})$. The classical lower bounds hold against fully adaptive algorithms, and fewer than $m$ such statistics cannot determine the positive-lifetime histogram. All bounds concern snapshot access; with an explicit update stream, near-linear full-barcode algorithms are known.
