Aaronson-Ambainis Conjecture Is True For Random Restrictions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914366434050048 |
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| author | Bhattacharya, Sreejata Kishor |
| author_facet | Bhattacharya, Sreejata Kishor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13952 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Aaronson-Ambainis Conjecture Is True For Random Restrictions Bhattacharya, Sreejata Kishor Computational Complexity 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. |
| title | Aaronson-Ambainis Conjecture Is True For Random Restrictions |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2402.13952 |