We explore the power of the unbounded Fan-Out gate and the Global Tunable gates generated by Ising-type Hamiltonians in constructing constant-depth quantum circuits, with particular attention to quantum memory devices. We propose two types of constant-depth constructions for implementing Uniformly Controlled Gates. These gates include the Fan-In gates defined by $|x\rangle|b\rangle\mapsto |x\rangle|b\oplus f(x)\rangle$ for $x\in\{0,1\}^n$ and $b\in\{0,1\}$, where $f$ is a Boolean function. The first of our constructions is based on computing the one-hot encoding of the control register $|x\rangle$, while the second is based on Boolean analysis and exploits different representations of $f$ such as its Fourier expansion. Via these constructions, we obtain constant-depth circuits for the quantum counterparts of read-only and read-write memory devices -- Quantum Random Access Memory (QRAM) and Quantum Random Access Gate (QRAG) -- of memory size $n$. The implementation based on one-hot encoding requires either $O(n\log{n}\log\log{n})$ ancillae and $O(n\log{n})$ Fan-Out gates or $O(n\log{n})$ ancillae and $6$ Global Tunable gates. On the other hand, the implementation based on Boolean analysis requires only $2$ Global Tunable gates at the expense of $O(n^2)$ ancillae.

翻译：我们研究了无界扇出门和由伊辛型哈密顿量生成的全局可调门在构建常数深度量子电路中的能力，特别关注量子存储器件。我们提出了两种类型的常数深度构造方法来实现均匀控制门。这些门包括由$|x\rangle|b\rangle\mapsto |x\rangle|b\oplus f(x)\rangle$定义的扇入门，其中$x\in\{0,1\}^n$，$b\in\{0,1\}$，且$f$是一个布尔函数。第一种构造基于计算控制寄存器$|x\rangle$的独热编码，而第二种构造基于布尔分析，利用了$f$的不同表示（如傅里叶展开）。通过这些构造，我们获得了存储大小为$n$的只读和读写存储器件（量子随机存取存储器(QRAM)和量子随机存取门(QRAG)）的量子对应物的常数深度电路。基于独热编码的实现需要$O(n\log{n}\log\log{n})$个辅助比特和$O(n\log{n})$个扇出门，或$O(n\log{n})$个辅助比特和6个全局可调门。另一方面，基于布尔分析的实现仅需2个全局可调门，但需要$O(n^2)$个辅助比特。