论文标题
在完全界限的块形形式和量子算法的经典模拟中的影响
Influence in Completely Bounded Block-multilinear Forms and Classical Simulation of Quantum Algorithms
论文作者
论文摘要
Aaronson-bambainis猜想（计算理论'14）说，布尔式超公管上的每个低度界限多项式都具有影响力的变量。这个猜想，如果是的，则意味着每一个$ d $ Query量子算法的接受概率几乎可以通过$ \ mathrm {poly}（poly}（d）$ - 查询古典算法的$ \ mathrm {poly}（poly}（poly}（poly}（poly}）），几乎可以在几乎所有地方（即几乎所有输入）进行良好的接受概率。我们证明了一个猜想的特殊情况：在每个完全有限的程度 - $ d $ block-multinear形式都具有恒定的差异，总是存在一个具有影响至少$ 1/\ mathrm {poly}（d）$的变量。从某种意义上说，这种多项式表征了量子查询算法的接受概率，如Arunachalam，Briët和Palazuelos（Sicomp '19）所示。作为推论，我们为特定类别的量子算法获得了有效的经典模拟，包括$ k $ fold-fold folderation。我们的主要技术结果依赖于与自由概率理论的联系。
The Aaronson-Ambainis conjecture (Theory of Computing '14) says that every low-degree bounded polynomial on the Boolean hypercube has an influential variable. This conjecture, if true, would imply that the acceptance probability of every $d$-query quantum algorithm can be well-approximated almost everywhere (i.e., on almost all inputs) by a $\mathrm{poly}(d)$-query classical algorithm. We prove a special case of the conjecture: in every completely bounded degree-$d$ block-multilinear form with constant variance, there always exists a variable with influence at least $1/\mathrm{poly}(d)$. In a certain sense, such polynomials characterize the acceptance probability of quantum query algorithms, as shown by Arunachalam, Briët and Palazuelos (SICOMP '19). As a corollary we obtain efficient classical almost-everywhere simulation for a particular class of quantum algorithms that includes for instance $k$-fold Forrelation. Our main technical result relies on connections to free probability theory.
