# Consequences of $BQP \subseteq P/poly$?

While Adleman's theorem shows, that $\mathsf{BPP} \subseteq \mathsf{P}/\text{poly}$, I'm not aware of any literature investigating the possible inclusion of $\mathsf{BQP} \subseteq \mathsf{P}/\text{poly}$. What complexity-theoretic consequences would such an inclusion have?

Adleman's theorem is sometimes called "the progenitor of derandomization arguments." $\mathsf{BPP}$ is believed to be derandomizable, whereas there is no evidence that the "quantumness" of $\mathsf{BQP}$ could somehow be removed. Is this possible evidence that $\mathsf{BQP}$ is unlikely to be in $\mathsf{P}/\text{poly}$ ?

• 3rd level will do. Both $\mathrm{BPP}\subseteq\mathrm P/\mathrm{poly}$ and Karp–Lipton relativize, so first $\mathrm{BPP}^\mathrm{NP}/\mathrm{poly}=\mathrm{P}^\mathrm{NP}/\mathrm{poly}$, and second, if $\Sigma^P_2\subseteq\mathrm{(BP)P}^\mathrm{NP}/\mathrm{poly}$, then $\Sigma^P_3=\Pi^P_3$. – Emil Jeřábek supports Monica Aug 9 '14 at 21:19
• (And various known strengthenings of KL also relativize for that matter, in particular the assumption above actually collapses PH to $S^P_3\subseteq\mathrm{ZPP^{NP^{NP}}}\subseteq\Sigma^P_3\cap\Pi^P_3$, except I’ve never seen $S^P$ with a subscript other than 2, so it’s likely nonstandard notation.) – Emil Jeřábek supports Monica Aug 9 '14 at 21:45