Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits

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.