Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits
Yasuhiro Takahashi, Seiichiro Tani · 2013
We study the quantum complexity class QNC0fof quantum operations implement able exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates. Our main result is that the quantum OR operation is in QNC0f, which is an affirmative answer to the question of Hoyer and Spalek. In sharp contrast to the strict hierarchy of the classical complexity classes: NC0⊊ AC0⊊ TC0, our result with Hoyer and Spalek's one implies the collapse of the hierarchy of the corresponding quantum ones: QNC0f= QAC0f= QTC0f. Then, we show that there exists a constant-depth sub quadratic-size quantum circuit for the quantum threshold operation. This allows us to obtain a better bound on the size difference between the QNC0fand QTC0fcircuits for implementing the same quantum operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in QNC0f, there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a QNC0foracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a QNC0fcircuit with gates for the quantum Fourier transform.