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Quantum-dot cellular automata (QCA) offers a new computing paradigm for nanotechnology. The basic logic elements of this technology are the majority voter (MV) and the inverter (INV). However, an experimental evaluation has shown that MV is not efficiently used during technology mapping by existing logic-synthesis tools. In this paper, we propose the design and characterization of a novel complex, yet very small, QCA logic gate: the and-or-inverter (AOI) gate. The paper presents a detailed simulation-based analysis of the AOI gate, as well as the study of QCA defects and their effects at the logic level. The AOI implements a universal logic gate; all elementary gates can be implemented by the AOI gate. Moreover, many two-level logic functions can be directly implemented by a single AOI gate. The AOI gate performs quite favorably, in terms of digital logic synthesis. Unlike MV, this gate is efficiently used by existing logic-synthesis tools. Our experimental data on synthesis of complex designs show that using the AOI gate instead of MV, results in up to 23.9% logic area savings, while improving the overall delay.