Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits

Abstract

We study the quantum complexity class ${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$ 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 ${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$, 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: ${\mathsf{NC}^{0} \subsetneq \mathsf{AC}^{0} \subsetneq \mathsf{TC}^{0}}$, our result with Høyer and Špalek’s one implies the collapse of the hierarchy of the corresponding quantum ones: ${\mathsf{QNC}^\mathsf{0}_\mathsf{f}=\mathsf{QAC}^\mathsf{0}_\mathsf{f}=\mathsf{QTC}^\mathsf{0}_\mathsf{f}}$. 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 ${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$ and ${\mathsf{QTC}^\mathsf{0}_\mathsf{f}}$ circuits for implementing the same operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in ${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$, there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a ${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$ oracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a ${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$ circuit with gates for the quantum Fourier transform.