Quantum Gates on One Qubit

Christopher J. Ballance

Ville Bergholm, Juha J. Vartiainen, Mikko Möttönen +1 more

Uniformly controlled one-qubit gates are quantum gates which can be represented as direct sums of two-dimensional unitary operators acting on a single qubit. We present a quantum gate array which implements any $n$-qubit gate of this type using at most ${2}^{n\ensuremath{-}1}\ensuremath{-}1$ controlled-NOT gates, ${2}^{n\ensuremath{-}1}$ one-qubit gates, and a single diagonal $n$-qubit gate. To illustrate the versatility of these gates we then apply them to the decomposition of a general $n$-qubit gate and a state preparation procedure. Moreover, we study their implementation using only nearest-neighbor gates. We give upper bounds for the one-qubit and controlled-NOT gate counts for all the aforementioned applications. In all four cases, the proposed circuit topologies either improve on or achieve the previously reported upper bounds for the gate counts. Thus, they provide the most efficient method for general gate decompositions currently known.

Mario Mastriani

Farrokh Vatan, Colin P. Williams

By an explicit construction, we show that an arbitrary two-qubit gate can be implemented by using at most 16 elementary one-qubit gates and 3 CNOT gates. We show that this construction is optimal; in the sense that these numbers of gates is the minimal possible ones. Moreover, we show that if the two-qubit gate belongs to SO(4), then we need only 12 elementary one-qubit gates and 2 CNOT gates. 1

Svetoslav S. Ivanov, Nikolay V. Vitanov

Tonghao Xing, X. Wu, Guofu Xu

Nonadiabatic holonomic gates provide a promising way to realize high-fidelity quantum gates because of their robustness and high-speed implementation. Compared with quantum computation that consists of many nonadiabatic holonomic one-qubit and two-qubit gates, the nonadiabatic holonomic multiqubit gate, which can reduce the number of quantum gates, is more efficient to construct simpler and faster complicated quantum computation. We propose the simplest scheme to realize nonadiabatic holonomic three-qubit controlled gates. On the one hand, the parameters we set to realize the three-qubit controlled gates are all time independent, which makes it easy to control and operate the quantum systems. On the other hand, our scheme avoids the combination of the elementary gates and realizes the three-qubit controlled gates only by one-shot implementation.