发表机构
UC Berkeley（加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出深度最优的量子编译方法，实现任意单量子比特门的常数深度合成，并证明有界宽度门下的对数对数深度下界，同时揭示浅层量子电路与Forrelation电路的新联系。
AI 中文摘要
我们实现了首个用于任意单量子比特门合成的常数深度电路。与以往方法不同，该构造是完全酉的，且无需预先提供的催化剂。对于任意常数 $\delta>0$，它使用 $O(\log^{1+\delta}(1/\varepsilon))$ 个干净的辅助量子比特、Hadamard 门和 $T$ 单量子比特门，$O(\log(1/\varepsilon))$ 宽度的广义 Toffoli 门，以及亚对数宽度的 Fan-Out 门，对任意单量子比特门进行 $\varepsilon$ 近似。我们进一步完全消除了 Fan-Out 门，表明仅使用 Hadamard、$T$ 和广义 Toffoli 门即可实现常数深度合成。当限制在标准的有界宽度门模型时，我们的构造深度为 $O(\log\log(1/\varepsilon))$，并且我们证明了匹配的 $\Omega(\log\log(1/\varepsilon))$ 深度下界。总体而言，我们确立了仅使用有界宽度门时 $\Theta(\log\log(1/\varepsilon))$ 深度是不可避免的，而允许甚至对数宽度的多量子比特门就足以实现常数深度合成。这些结果还揭示了浅层量子电路复杂度的新结构。我们给出了有界误差决策计算的保深度实数模拟，表明每个深度为 $d$ 的 QAC 电路都可以仅使用 Hadamard、$X$ 和广义 Toffoli 门在深度 $O(d)$ 内模拟。因此，任意单量子比特旋转和复数振幅不会增加 QAC 的有界误差决策能力，即使在常数深度下也是如此。特别是，这将长期存在的猜想 Parity$\notin$QAC$^0$ 简化为证明仅由 Hadamard、$X$ 和广义 Toffoli 门组成的电路的 Parity 下界。更一般地，这种实数规范形式揭示了标准浅层量子电路层级与具有受限预言机族的 Forrelation 电路层级之间的直接对应关系。
英文摘要
We achieve the first constant-depth circuit for arbitrary single-qubit gate synthesis. Unlike prior approaches, the construction is fully unitary and requires no pre-supplied catalyst. For any constant $δ>0$, it $\varepsilon$-approximates an arbitrary single-qubit gate using $O(\log^{1+δ}(1/\varepsilon))$ clean ancillae, Hadamard and $T$ single-qubit gates, $O(\log(1/\varepsilon))$-width generalized Toffoli gates, and sublogarithmic-width Fan-Out gates. We further eliminate Fan-Out entirely, showing that Hadamard, $T$, and generalized Toffoli gates alone suffice for constant-depth synthesis. When restricted to the standard bounded-width gate model, our construction has depth $O(\log\log(1/\varepsilon))$, and we prove a matching $Ω(\log\log(1/\varepsilon))$-depth lower bound. Overall, we establish that $Θ(\log\log(1/\varepsilon))$-depth is unavoidable with only bounded-width gates, yet allowing even logarithmic-width multi-qubit gates suffices to achieve constant-depth synthesis. These results also reveal new structure in shallow quantum circuit complexity. We give a depth-preserving real simulation of bounded-error decision computation, showing that every depth-$d$ QAC circuit can be simulated in depth $O(d)$ using only Hadamard, $X$, and generalized Toffoli gates. Thus arbitrary single-qubit rotations and complex amplitudes do not increase the bounded-error decision power of QAC, even at constant depth. In particular, this reduces the long-standing conjecture Parity$\notin$QAC$^0$ to proving a Parity lower bound against circuits consisting only of Hadamard, $X$, and generalized Toffoli gates. More generally, this real normal form exposes a direct correspondence between the standard shallow-depth quantum circuit hierarchy and a hierarchy of Forrelation circuits with restricted oracle families.
