Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits
(1112.6063)Abstract
We study the quantum complexity class QNC0_f of quantum operations implementable exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates (called QNC0_f circuits). Our main result is that the quantum OR operation is in QNC0_f, 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 \subsetneq AC0 \subsetneq TC0, our result with Hoyer and Spalek's one implies the collapse of the hierarchy of the corresponding quantum ones: QNC0_f = QAC0_f = QTC0_f. Then, we show that there exists a constant-depth subquadratic-size quantum circuit for the quantum threshold operation. This implies the size difference between the QNC0_f and QTC0_f circuits for implementing the same quantum operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in QNC0_f, there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a QNC0_f oracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a QNC0_f circuit with gates for the quantum Fourier transform.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.