Aaronson-Ambainis Conjecture Is True For Random Restrictions
<|reference_start|>Aaronson-Ambainis Conjecture Is True For Random Restrictions: 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.<|reference_end|>
@article{bhattacharya2024aaronson-ambainis,
    title={Aaronson-Ambainis Conjecture Is True For Random Restrictions},
    author={Sreejata Kishor Bhattacharya},
    journal={arXiv preprint arXiv:2402.13952},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.13952},
    primaryClass={cs.CC}
}