ANALYTIC ONE-BIT AND CNOT GATE CONSTRUCTIONS OF GENERAL n-QUBIT CONTROLLED GATES

TLDR

General n‑qubit controlled unitary gates are frequently used in quantum information processing tasks. The authors aim to extend prior construction methods by deriving two analytic expressions for general n‑qubit controlled unitary gates using one‑qubit and two‑qubit CNOT gates. They present one exponential‑in‑qubit and one polynomial‑in‑qubit expression, efficient for up to ten qubits and for larger systems respectively. These constructions generalize earlier results and provide scalable implementations for larger qubit counts. The study builds on Barenco et al.’s 1995 Phys.

Abstract

General n-qubit controlled unitary gates are frequently used in quantum information processing tasks. Barenco, Bennett, Cleve, Di Vincenzo, Margolus and Shor [Phys. Rev. A52 (1995) 3457] have given the general construction methods, and explicit results for up-to-four-qubits controlled unitary gates. We extended their calculation and gave two analytic expressions for the construction of general n-qubit controlled unitary gates in terms of one-qubit and two-qubit CNOT gates. There are two expressions – one is exponential in the qubit number which is efficient for up to ten qubits, and the other is polynomial in the qubit number, which is efficient for more than ten qubits.