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.
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Takahashi, Y., Tani, S. Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits. comput. complex. 25, 849–881 (2016). https://doi.org/10.1007/s00037-016-0140-0
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DOI: https://doi.org/10.1007/s00037-016-0140-0