列表搜索的量子查询复杂度
Quantum Query Complexity for List Search
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- Mie University（三重大学）
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中文总结 AI 辅助
本文研究量子查询模型中的链表搜索问题，证明判定和搜索版本的量子查询复杂度为$\Theta(\min\{\ell,(N\ell)^{1/4}\})$，揭示了地址空间可带来量子优势，并推广到双向链表。
中文摘要 AI 辅助
在链表中搜索是经典算法中最基本的问题之一。尽管链表的节点带有内存地址，但在经典情况下，这些地址在搜索成本中不起作用：只需从头节点开始，沿着后继指针寻找元素即可。在本文中，我们证明量子设置是不同的。在这里，列表顶点所取自的周围地址空间本身可以影响查询复杂度。我们研究了查询复杂度模型中与链表搜索类似的问题：输入包括一个地址宇宙$[N]$、一个公共起始符号$s$、一个后继预言机$f$，其非$\perp$值追踪一条隐藏的简单路径$s \to a_1 \to a_2 \to \cdots \to a_\ell \to \perp$，以及一个标记预言机$g$，它最多标记一个列表顶点。任务是判断列表中是否包含被标记的顶点。我们证明了对于所有$N \ge \ell \ge 1$，该问题的判定版本和搜索版本都具有量子查询复杂度$\Theta\\!\bigl(\min\{\ell,(N\ell)^{1/4}\}\bigr)$。因此，相当令人惊讶的是，当$N < \ell^3$时，最优量子复杂度为$(N\ell)^{1/4}$，这严格小于普通链表遍历的$\Theta(\ell)$成本。这给出了周围地址空间何时为链表搜索带来真正量子优势的精确刻画。我们还扩展了结果，为自然的双向链表版本给出了相同的紧渐近界。
英文摘要
Searching in a linked list is one of the most basic problems in classical algorithms. Although the nodes of the list come with memory addresses, classically those addresses play no role in the cost of search: one simply starts at the head and follows successor pointers to search for an element. In this paper, we show that the quantum setting is different. Here, the ambient address space from which the list vertices are drawn can itself affect the query complexity. We study the following problem analogous to search in a linked list in the query complexity model: the input consists of an address universe $[N]$, a public start symbol $s$, a successor oracle $f$ whose non-$\perp$ values trace a hidden simple path $s \to a_1 \to a_2 \to \cdots \to a_\ell \to \perp$, and a marking oracle $g$ that marks at most one list vertex. The task is to decide whether the list contains a marked vertex. We prove that both the decision and search versions of this problem for all $N \ge \ell \ge 1$ have quantum query complexity $Θ\!\bigl(\min\{\ell,(N\ell)^{1/4}\}\bigr)$. Thus, quite surprisingly, when $N < \ell^3$ the optimal quantum complexity is $(N\ell)^{1/4}$, which is strictly smaller than the $Θ(\ell)$ cost of ordinary linked-list traversal. This gives a precise characterization of when the ambient address space yields a genuine quantum advantage for linked-list search. We extend our results and give the same tight asymptotic bounds for the natural double linked-list version as well.
