前几何中的量子查询复杂度与跨度程序
Quantum Query Complexity and Span Programs from Pre-Geometry
- George Mason University（乔治梅森大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过拟阵化组织跨度程序优化，分离查询依赖与程序结构，推导精确约简和数值界，并构造量子算法族，实现次线性查询复杂度。
AI中文摘要：
跨度程序为设计量子查询算法提供了一种通用方法，对于布尔坐标查询，其最优代价与一般对抗界一致。我们发展了一种拟阵化的组织方式，将查询问题所决定的依赖数据与跨度程序所选择的依赖结构分离开来。由此得到的源拟阵记录查询依赖、目标确定和分离数据，而程序拟阵记录候选程序中目标与响应标记输入元素之间的依赖关系。当查询问题允许源拟阵时，我们利用其结构来组织对抗上界和下界。在此设置中分离二元查询集时，全局拟阵-跨度程序最优值等于$\mathrm{Adv}^{\pm}_{\mathcal Q}(f)$。对于固定的正则程序拟阵，投影唯一性将实数表示上的优化简化为正元素权重。我们将正负代价表示为加权基比率，并推导出精确的1-、2-和3-和约简，从而沿Seymour分解给出固定权重目标的组合评估。对于尖点$R_{10}$闭包，精确证书给出$3.9087553<\mathrm{Adv}^{\pm}_{\mathcal Q_R}(f_R)<3.9301$。自然$R_{10}$程序具有优化代价5，而另一个正则程序代价至多为$\sqrt{87/5}$。递归自组合产生一个在$N$个输入比特上的族，其随机查询复杂度为$\Omega(N^{0.7324867\ldots})$，一个显式量子算法使用$O(N^{0.6500178\ldots})$次查询，且$Q(F)=\Theta(N^\alpha)$，其中$0.6204277\ldots<\alpha<0.6229063\ldots$。
英文摘要：
Span programs provide a general method for designing quantum query algorithms, and for Boolean coordinate queries their optimum cost agrees with the general adversary bound. We develop a matroidal organization of this optimization that separates dependence data determined by the query problem from the dependence structure chosen by a span program. The resulting source matroid records query dependence, target determination, and separation data, while a program matroid records dependencies among the target and response-labeled input elements of a candidate program. When a query problem admits a source matroid, we use its structure to organize adversary upper and lower bounds. For separating binary query sets in this setting, the global matroid-span-program optimum equals $\mathrm{Adv}^{\pm}_{\mathcal Q}(f)$. For a fixed regular program matroid, projective uniqueness reduces optimization over real representations to positive element weights. We express positive and negative costs as weighted basis ratios and derive exact $1$-, $2$-, and $3$-sum reductions, giving a compositional evaluation of the fixed-weight objective along a Seymour decomposition. For pointed $R_{10}$ closure, exact certificates give $3.9087553<\mathrm{Adv}^{\pm}_{\mathcal Q_R}(f_R)<3.9301$. The natural $R_{10}$ program has optimized cost $5$, while another regular program has cost at most $\sqrt{87/5}$. Recursive self-composition yields a family on $N$ input bits with randomized query complexity $Ω(N^{0.7324867\ldots})$, an explicit quantum algorithm using $O(N^{0.6500178\ldots})$ queries, and $Q(F)=Θ(N^α)$ for $0.6204277\ldots<α<0.6229063\ldots$.
