AI 中文总结
本文针对三类布尔函数的量子谕示，推导了电路规模、深度与辅助比特数间的近最优渐近权衡关系，相关界在对应辅助比特区间内除对数因子外最优，可应用于QROM实现等经典过程嵌入量子电路的场景。
AI 中文摘要
布尔函数的量子谕示是经典算法与量子算法之间的核心桥梁之一，但针对这类谕示的量子电路优化研究仍有待完善。本文针对三类布尔函数的量子谕示，提出了电路规模、电路深度和辅助比特数量之间的近最优权衡关系：\n输出规模为$b$的一般全布尔函数：在辅助比特数满足$1\le m\le\Theta\left(\frac{2^n}{n}\right)$时，规模为$\mathcal{O}\left(\frac{b2^n}{\log(n+m)}\right)$，深度为$\mathcal{O}\left(\frac{b2^n}{n+m}\right)$；\n有效支撑集规模为$d$、输出规模为$b$的部分布尔函数：在辅助比特数满足$\Theta\left(\log d\right)\le m\le \Theta\left(d\right)$时，规模为$\mathcal{O}\left(n\log d+bd\right)$，深度为$\mathcal{O}\left(\frac{n\log n\log d}{n+m}+\log n+\frac{d(\log d+b\log m)}{m}\right)$；\n真值输入规模为$d$的稀疏布尔函数：在辅助比特数满足$\Theta\left(\log n+\log d\right)\le m\le\Theta\left(\frac{nd}{\log d}\right)$时，规模为$\mathcal{O}\left(n^2\log d+\frac{nd}{\log(\log d+m/n)}\right)$，深度为$\mathcal{O}\left(\frac{n^2\log n\log d}{n+m}+\log n+\frac{nd}{m}\right)$。\n所有规模和深度界在相应的辅助比特数量区间内，除对数因子外均达到渐近最优。我们希望这些结果能应用于需要将经典过程嵌入量子电路的场景，例如QROM实现和量子算法设计。
英文摘要
Quantum oracle of Boolean functions is one of the central bridges between classical and quantum algorithms, but the study focusing at quantum circuit optimization of such oracle is yet closed. In this paper, we propose nearly optimal tradeoffs among circuit size, circuit depth and ancilla count, for quantum oracles of three kinds of Boolean functions: general total Boolean functions with output size $b$: with $1\le m\leΘ\left(\frac{2^n}{n}\right)$ ancilla, size $\mathcal{O}\left(\frac{b2^n}{\log(n+m)}\right)$, depth $\mathcal{O}\left(\frac{b2^n}{n+m}\right)$; partial Boolean functions of effective support size $d$ and output size $b$: with $Θ\left(\log d\right)\le m\le Θ\left(d\right)$ ancilla, size $\mathcal{O}\left(n\log d+bd\right)$, depth $\mathcal{O}\left(\frac{n\log n\log d}{n+m}+\log n+\frac{d(\log d+b\log m)}{m}\right)$; sparse Boolean functions of true input size $d$: with $Θ\left(\log n+\log d\right)\le m\leΘ\left(\frac{nd}{\log d}\right)$ ancilla, size $\mathcal{O}\left(n^2\log d+\frac{nd}{\log(\log d+m/n)}\right)$, depth $\mathcal{O}\left(\frac{n^2\log n\log d}{n+m}+\log n+\frac{nd}{m}\right)$. All the size and depth bounds are asymptotically optimal up to logarithmic factors in the corresponding ancilla count regions. We hope these results find applications in scenarios where classical procedures are needed to be embedded into quantum circuits, such as QROM implementation and quantum algorithm design.
