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.
Categories: cs.CC