We study the quantum complexity class $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 of quantum operations implementable exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates. Our main result is that the quantum OR operation is in $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0, which is an affirmative answer to the question posed by Høyer and Špalek. In sharp contrast to the strict hierarchy of the classical complexity classes: $${\mathsf{NC}^{0} \subsetneq \mathsf{AC}^{0} \subsetneq \mathsf{TC}^{0}}$$NC0⊊AC0⊊TC0, our result with Høyer and Špalek’s one implies the collapse of the hierarchy of the corresponding quantum ones: $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}=\mathsf{QAC}^\mathsf{0}_\mathsf{f}=\mathsf{QTC}^\mathsf{0}_\mathsf{f}}$$QNCf0=QACf0=QTCf0. Then, we show that there exists a constant-depth subquadratic-size quantum circuit for the quantum threshold operation. This allows us to obtain a better bound on the size difference between the $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 and $${\mathsf{QTC}^\mathsf{0}_\mathsf{f}}$$QTCf0 circuits for implementing the same operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0, there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 oracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 circuit with gates for the quantum Fourier transform.

中文翻译：

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等深精确量子电路层次结构的崩溃

我们研究了量子运算的量子复杂性类 $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0，可通过具有无限扇出门的恒定深度多项式大小的量子电路完全实现. 我们的主要结果是量子 OR 运算在 $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 中，这是对 Høyer 和 Špalek 提出的问题的肯定回答。与经典复杂性类的严格层次结构形成鲜明对比： $${\mathsf{NC}^{0} \subsetneq \mathsf{AC}^{0} \subsetneq \mathsf{TC}^{0}}$$ NC0⊊AC0⊊TC0，我们与 Høyer 和 Špalek 的结果意味着相应量子层次结构的崩溃：$${\mathsf{QNC}^\mathsf{0}_\mathsf{f}=\mathsf{QAC }^\mathsf{0}_\mathsf{f}=\mathsf{QTC}^\mathsf{0}_\mathsf{f}}$$QNCf0=QACf0=QTCf0。然后，我们表明，对于量子阈值操作，存在一个恒定深度的亚二次尺寸量子电路。这使我们能够更好地限制 $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 和 $${\mathsf{QTC}^\mathsf 之间的大小差异{0}_\mathsf{f}}$$QTCf0 电路用于实现相同的操作。最后，我们证明，如果量子傅立叶变换模一个素数在 $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 中，则存在多项式时间精确经典算法对于使用 $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 oracle 的离散对数问题。这意味着，在一个似是而非的假设下，存在一个经典的难题，该问题可以通过带有门的 $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 电路完全解决量子傅立叶变换。