发表机构
University of Washington; Sandia National Laboratories; Purdue University（华盛顿大学; 桑迪亚国家实验室; 普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于噪声算子的新方法，首次证明排序问题的完全通用量子时间-空间权衡下界，并推广到哈希函数计算，突破了以往仅输出无关下界的限制。
AI 中文摘要
时间和空间（内存）是计算中两个最重要的代价度量，对于量子计算而言更是如此。在量子计算中，我们用于证明时间与空间之间无条件权衡的工具出奇地有限。第一个量子时间-空间权衡下界是由Klauck、Špalek和de Wolf为排序问题证明的。不幸的是，他们的方法仅限于证明输出无关的下界（即下界仅适用于具有非自适应输出调度的算法），而其他方法在输出无关下界之外一无所获。我们证明了排序问题的第一个完全通用的量子时间-空间权衡下界。为此，我们引入了一种基于噪声算子的新方法，以扩充用于证明量子查询和时间-空间权衡下界的分析工具集。通过将我们得到的量子噪声稳定性界与量子记录查询方法相结合，我们证明了一个$\Omega(n^{4/3} (\log \log n)/(S^{1/3} \log n))$的下界，该下界针对的是内存至多$S$个量子比特的完全通用量子算法对来自$[n^2]$的$n$个数字进行排序所需的查询次数。应用我们的噪声算子论证涉及纯经典论证，这使得它特别易于使用。我们还用它来证明，对于任何从$n$比特到$m$比特的强通用（两两独立）哈希函数族$H$，$H$中几乎所有的哈希函数都要求内存至多$S$个量子比特的量子算法对输入$x$进行$\Omega(nm/S)$次查询才能计算$h(x)$，即使成功概率非常小。此前，Mansour、Nisan和Tiwari使用他们的哈希混合引理证明了类似的经典下界。我们的噪声算子方法允许我们使用哈希函数的一个相关但更简单的性质来证明我们的下界。
英文摘要
Time and space (memory) are two of the most important measures of cost in computation, even more so for quantum computation. Yet, our tools for proving unconditional quantum tradeoffs between time and space are surprisingly limited. The first quantum time-space tradeoff lower bounds were proven for sorting by Klauck, Špalek and de Wolf. Unfortunately, their method is limited to proving output-oblivious lower bounds (i.e. the lower bounds only apply to algorithms with a non-adaptive output schedule) and other methods have yielded nothing beyond output-oblivious lower bounds for sorting. Here, we prove the first fully general quantum time-space tradeoff lower bound for sorting. We do so by introducing a novel method based on the noise operator to add to the analysis toolkit for proving quantum query and time-space tradeoff lower bounds. By combining our resulting quantum noise stability bound with quantum recording query methods, we prove an $Ω(n^{4/3} (\log \log n)/(S^{1/3} \log n))$ lower bound on the number of queries that a fully general quantum algorithm with at most $S$ qubits of memory requires to sort $n$ numbers from $[n^2]$. Applying our noise operator argument involves purely classical reasoning, which makes it particularly simple to use. We also use it to prove that, for any strongly universal (pairwise independent) hash function family $H$ from $n$ bits to $m$ bits, almost all hash functions in $H$ require any algorithm with at most $S$ qubits of memory to make $Ω(nm/S)$ quantum queries to input $x$ in order to compute $h(x)$, even with very small success probability. Previously, Mansour, Nisan, and Tiwari had shown a matching classical lower bound for computing $h(x)$ with both $h$ and $x$ as inputs using their hash mixing lemma. Our noise operator method allows us to use a related property of hash functions to prove our quantum lower bounds.
Comments46 pages, submitted to QIP 2027. New version fixes the displayed abstract and makes some minor corrections to fix the parameters used in results
