International Journal of Modern Physics B Vol. 27 (2013) 1350191 (15 pages) c  World Scientific Publishing Company DOI: 10.1142/S0217979213501919 A CLASS OF EFFICIENT QUANTUM INCREMENTER GATES FOR QUANTUM CIRCUIT SYNTHESIS XIAOYU LI *,†,‡ , GUOWU YANG *,§ , CARLOS MANUEL TORRES JR. † , DESHENG ZHENG * and KANG L. WANG † * College of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, P. R. China † Department of Electrical Engineering, University of California at Los Angeles Los Angeles, CA 90095, USA ‡ erin.xiaoyu.li@gmail.com § guowu@uestc.edu.cn Received 1 April 2013 Revised 16 September 2013 Accepted 23 September 2013 Published 17 October 2013 The quantum incrementer is one of the simplest quantum operators, which exhibits basic arithmetic operations such as addition, the propagation of carry qubits and the resetting of carry qubits. In this paper, three quantum incrementer gate circuit topologies are derived and compared based upon their total number of gates, the complexity of the circuits, the types of gates used and the number of carry or ancilla qubits implemented. The first case is a generalized n-qubit quantum incrementer gate with the notation of (n : 0). Two other quantum incrementer topologies are proposed with the notations of (n : n − 1 : RE) and (n : n − 1 : RD). A general method is derived to decompose complicated quantum circuits into simpler quantum circuits which are easier to manage and physically implement. Due to the cancelation of intermediate unitary gates, it is shown that adding ancilla qubits slightly increases the complexity of a given circuit by the order of 3n, which pales in comparison to the complexity of the original circuit of the order n 2 without reduction. Finally, a simple application of the generalized n-qubit quantum incrementer gate is introduced, which is related to quantum walks. Keywords : Quantum circuit synthesis; quantum incrementer gate; geometric quantum computation; quantum walks; ancilla qubits. PACS numbers: 03.67.Ac, 03.67.Lx, 03.65.Sq 1. Introduction Quantum logic synthesis 1–12 has been independently observed and studied in vari- ous aspects for many years. Remarkably, an essential question left to be answered is how to construct and generalize a single quantum gate to act as a basic element 1350191-1 Int. J. Mod. Phys. B Downloaded from www.worldscientific.com by "UNIV OF CALIFORNIA,LOS ANGELES" on 10/21/13. For personal use only.