In an attempt to show that the acceptance probability of a quantum query algorithm making $q$ queries can be well-approximated almost everywhere by a classical decision tree of depth $\leq \text{poly}(q)$, Aaronson and Ambainis proposed the following conjecture: let $f: \{ \pm 1\}^n \rightarrow [0,1]$ be a degree $d$ polynomial with variance $\geq \epsilon$. Then, there exists a coordinate of $f$ with influence $\geq \text{poly} (\epsilon, 1/d)$. We show that for any polynomial $f: \{ \pm 1\}^n \rightarrow [0,1]$ of degree $d$ $(d \geq 2)$ and variance $\text{Var}[f] \geq 1/d$, if $\rho$ denotes a random restriction with survival probability $\dfrac{\log(d)}{C_1 d}$, $$ \text{Pr} \left[f_{\rho} \text{ has a coordinate with influence} \geq \dfrac{\text{Var}[f]^2 }{d^{C_2}} \right] \geq \dfrac{\text{Var}[f] \log(d)}{50C_1 d}$$ where $C_1, C_2>0$ are universal constants. Thus, Aaronson-Ambainis conjecture is true for a non-negligible fraction of random restrictions of the given polynomial assuming its variance is not too low.

翻译：为证明一个进行$q$次查询的量子查询算法的接受概率几乎处处可由深度$\leq \text{poly}(q)$的经典决策树良好逼近，Aaronson与Ambainis提出了如下猜想：设$f: \{\pm 1\}^n \rightarrow [0,1]$为方差$\geq \epsilon$的$d$次多项式，则存在$f$的某个坐标其影响值$\geq \text{poly}(\epsilon, 1/d)$。我们证明：对任意$d$次多项式$f: \{\pm 1\}^n \rightarrow [0,1]$（$d \geq 2$）且方差$\text{Var}[f] \geq 1/d$，若$\rho$表示存活概率为$\dfrac{\log(d)}{C_1 d}$的随机限制，则$$ \text{Pr} \left[f_{\rho} \text{ 存在一个坐标其影响值} \geq \dfrac{\text{Var}[f]^2 }{d^{C_2}} \right] \geq \dfrac{\text{Var}[f] \log(d)}{50C_1 d}$$其中$C_1, C_2>0$为通用常数。因此，当给定多项式的方差非过低时，Aaronson-Ambainis猜想对该多项式不可忽略比例的随机限制成立。