随机垃圾将XOR与仅前向查询分离
Random Garbage Separates XOR from Forward-Only Queries
- Khalifa University of Science and Technology（哈利法科学技术大学）
- Center for Quantum and Topological Systems, NYUAD Research Institute（NYUAD研究院量子与拓扑系统中心）
- Division of Science, New York University Abu Dhabi（纽约大学阿布扎比分校科学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文证明标准XOR接口与仅前向接口存在指数级量子查询分离，解决了Aaronson的开放问题11，还将实例嵌入置换得到对应置换问题的查询复杂度结果。
AI中文摘要：
我们在标准XOR接口与既不提供伴随也不提供逆预言机的两种仅前向接口之间给出指数级量子查询分离。设$X=\text{F}_2^n$，$N=|X|$，且$f_{h,r}(x)=(h(x),x,r_x)$，其中$h:X\to X$被承诺为要么是置换要么是Simon二对一函数，而$r$是一个固定的n位标签表，不受该承诺约束且在每次查询中重复使用。由此产生的问题可通过至多$n+2$次标准XOR查询求解，但具有仅前向擦除查询复杂度$\text{\th}(\text{\radic}N)$。这肯定地回答了[Scott Aaronson. 与量子查询复杂度相关的开放问题. ACM Transactions on Quantum Computing, 2(4):14:1-14:9, 2021]中的开放问题11。我们还将这些实例嵌入置换中。详细构造在每个规定输出中保留$x$的副本，但对于这些承诺，该副本可被一个能区分每个Simon对中两个输入的位替换。这给出了大小为$L=4N^2$的置换域和一个置换问题，其具有相同的标准查询上界以及仅前向原位查询复杂度$\text{\th}(\text{\radic}N)=\text{\th}(L^{1/4})$。两个下界在干净相干旁路下仍然有效。共同下界使用仅分析的记录替换。在替换计算中，在$T$次调用后追踪固定随机标签表得到正半定算子贡献的和，每个贡献取决于$h$在不超过$T$个地址处的值。在这样的集合上，随机置换和随机Simon函数诱导的限制仅当该集合包含一个隐藏Simon对时不同，该事件的概率为$O(T^2/N)$。
英文摘要：
We give exponential quantum query separations between the standard XOR interface and two forward-only interfaces that supply neither an adjoint nor an inverse oracle. Let $X=\F_2^n$, $N=|X|$, and $f_{h,r}(x)=(h(x),x,r_x)$, where $h:X\to X$ is promised to be either a permutation or a Simon two-to-one function, and $r$ is a fixed table of $n$-bit tags, unrestricted by the promise and reused on every query. The resulting problem is solvable with at most $n+2$ standard XOR queries, but has forward-erasing query complexity $Θ(\sqrt N)$. This answers affirmatively open question 11 in [Scott Aaronson. Open problems related to quantum query complexity. ACM Transactions on Quantum Computing, 2(4):14:1-14:9, 2021] . We also embed these instances into permutations. The detailed construction retains the copy of $x$ in each prescribed output, but for these promises that copy can be replaced by one bit that distinguishes the two inputs in every Simon pair. This gives a permutation domain of size $L=4N^2$ and a permutation problem with the same standard-query upper bound and forward-only in-place query complexity $Θ(\sqrt N)=Θ(L^{1/4})$. Both lower bounds remain valid with a clean coherent bypass. The common lower bound uses an analysis-only recording replacement. In the replacement computation, tracing out the fixed random tag table after $T$ calls gives a sum of positive-semidefinite operator contributions, each depending on $h$ at no more than $T$ addresses. On such a set, the restrictions induced by random permutations and random Simon functions differ only if the set contains a hidden Simon pair, an event of probability $O(T^2/N)$.
