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 $O(n\log^{(d)}{n}\log^{(d+1)}{n})$ ancillae and $O(n\log^{(d)}{n})$ Fan-Out gates or $O(n\log^{(d)}{n})$ ancillae and $16d-10$ Global Tunable gates, where $d$ is any positive integer and $\log^{(d)}{n} = \log\cdots \log{n}$ is the $d$-times iterated logarithm. On the other hand, the implementation based on Boolean analysis requires $8d-6$ Global Tunable gates at the expense of $O(n^{1/(1-2^{-d})})$ ancillae.

翻译：本文探讨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方面的能力，并特别关注量子存储器设备。我们提出了两种实现均匀控制门的常数深度构造方案。这类门包括由 $|x\rangle|b\rangle\mapsto |x\rangle|b\oplus f(x)\rangle$（其中 $x\in\{0,1\}^n$，$b\in\{0,1\}$，$f$ 为布尔函数）定义的扇入门。我们的第一种构造基于计算控制寄存器 $|x\rangle$ 的独热编码，而第二种构造则基于布尔分析，并利用了 $f$ 的不同表示形式，例如其傅里叶展开。通过这些构造，我们为存储器容量为 $n$ 的只读与读写量子存储器设备——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，获得了常数深度电路。基于独热编码的实现方案需要 $O(n\log^{(d)}{n}\log^{(d+1)}{n})$ 个辅助量子比特和 $O(n\log^{(d)}{n})$ 个扇出门，或者需要 $O(n\log^{(d)}{n})$ 个辅助量子比特和 $16d-10$ 个全局可调门，其中 $d$ 为任意正整数，$\log^{(d)}{n} = \log\cdots \log{n}$ 表示 $d$ 次迭代对数。另一方面，基于布尔分析的实现方案需要 $8d-6$ 个全局可调门，但代价是需要 $O(n^{1/(1-2^{-d})})$ 个辅助量子比特。