Aaronson-Ambainis Conjecture Is True For Random Restrictions
Sreejata Kishor Bhattacharya
TL;DR
This work tackles the Aaronson-Ambainis conjecture, which links quantum-query complexity to classical decision-tree approximations for low-degree polynomials. The authors prove that for any degree $d$ polynomial $f:\{\pm1\ o [0,1]$ with $\mathrm{Var}[f]\ge 1/d$, a random restriction with survival probability $p=\dfrac{\log(d)}{C d}$ yields, with probability at least $\dfrac{\mathrm{Var}[f]\log(d)}{50 C d}$, a restricted function $f_{\rho}$ that has a coordinate with influence at least $\dfrac{\mathrm{Var}[f]^2}{d^{C}}$, establishing the conjecture for a non-negligible fraction of restrictions. The paper introduces an improved tail bound for low-degree functions under random restrictions and a junta-approximation framework showing that most $f_{\rho}$ depend on only $\mathrm{poly}(d)$ coordinates. By combining these results with variance considerations under restrictions, the authors obtain a robust, probabilistic confirmation of the conjecture in a broad regime and outline a pathway toward a full resolution via restriction-based strategies. This provides structural insight into how low-degree bounded polynomials behave under random restrictions and suggests practical avenues for bridging quantum and classical query complexities.
Abstract
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 $\geq ε$. Then, there exists a coordinate of $f$ with influence $\geq \text{poly} (ε, 1/d)$. 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 $ρ$ denotes a random restriction with survival probability $\dfrac{\log(d)}{C_1 d}$, $$ \text{Pr} \left[f_ρ \text{ has a coordinate with influence} \geq \dfrac{\text{Var}[f]^2 }{d^{C_2}} \right] \geq \dfrac{\text{Var}[f] \log(d)}{50C_1 d}$$ 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.