Recently, Chia, Chung and Lai (STOC 2020) and Coudron and Menda (STOC 2020) have shown that there exists an oracle $\mathcal{O}$ such that $\mathsf{BQP}^\mathcal{O} \neq (\mathsf{BPP^{BQNC}})^\mathcal{O} \cup (\mathsf{BQNC^{BPP}})^\mathcal{O}$. In fact, Chia et al. proved a stronger statement: for any depth parameter $d$, there exists an oracle that separates quantum depth $d$ and $2d+1$, when polynomial-time classical computation is allowed. This implies that relative to an oracle, doubling quantum depth gives classical and quantum hybrid schemes more computational power. In this paper, we show that for any depth parameter $d$, there exists an oracle that separates quantum depth $d$ and $d+1$, when polynomial-time classical computation is allowed. This gives an optimal oracle separation of classical and quantum hybrid schemes. To prove our result, we consider $d$-Bijective Shuffling Simon's Problem (which is a variant of $d$-Shuffling Simon's Problem considered by Chia et al.) and an oracle inspired by an "in-place" permutation oracle.
翻译:最近,Chia, Chung and Lai (STOC 2020) 和 Coudlior 和 Menda (STOC 2020) 都表明, 存在一个极值 $\ mathcal{O} 美元, 例如 $mathsf{BQP} mathcal{O}\ neq (mathsf{BP ⁇ BQQQQQQQQQQQQ}}) 和 Coudlion 和 Menda (STOC 2020) 。 事实上, Chia 等人 证明了一个更强烈的语句: 对于任何深度参数 $dd$, 存在一个将量值深度与$和$ 2d+1美元分开的神谕。 这意味着相对于一个极值, 翻倍的量度深度能让经典和量子混合的混合计划有更多的计算能力。 在任何深度参数 $d* 和 $d+1 $ 美元 。 事实上, Chia 都证明了一个分量值 和 $ 美元, 当混合古度计算为美元时, 的量值和 美元, 将量值 的量值 折算为 。