量子快速与经典繁琐的问题
The Quantumly Fast and the Classically Forrious
AI总结：
本文通过新的构造方法，证明了极值Forrelation问题需要Ω(2^{0.4999n})个经典查询，与推测的最优下界一致。
摘要：
我们研究极值Forrelation问题，其中，给定对布尔函数f和g的oracle访问权，并承诺满足either forr(f,g)=1或forr(f,g)=-1，必须在尽可能少的oracle查询次数下确定哪一种情况成立。众所周知，这个问题可以用一个量子查询解决；然而，Girish和Servedio（TQC 2025）最近表明该问题需要~Ω(2^{n/4})个经典查询，并推测最优下界是~Ω(2^{n/2})。通过完全不同的构造，我们改进了他们的结果，证明了下界为Ω(2^{0.4999n})，这与推测的下界在指数上只相差一个任意小的常数。
英文摘要：
We study the extremal Forrelation problem, where, provided with oracle access to Boolean functions $f$ and $g$ promised to satisfy either $\operatorname{forr}(f,g)=1$ or $\operatorname{forr}(f,g)=-1$, one must determine (with high probability) which of the two cases holds while performing as few oracle queries as possible. It is well known that this problem can be solved with one quantum query; yet, Girish and Servedio (ITCS 2026) recently showed this problem requires $\widetildeΩ(2^{n/4})$ classical queries, and conjectured the optimal bound to be $\widetildeΘ(2^{n/2})$. By generalizing their construction, we build on their result and prove a non-adaptive lower bound of $Ω(2^{(1/2- o(1))n})$, which matches the conjectured lower bound up to a vanishing constant in the exponent.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
