Quantum-dot cellular automata (QCA) is an advanced nanotechnology. It is applied to delineate nanoscale technology-based logic circuits. It can potentially be replaced the complementary metal oxide semiconductor (CMOS) technology. This paper proposes an optimal and single clocked multiplexer (Mux) circuit, which is made with the help of 12 number of QCA cells in QCA nanotechnology. The proposed Mux circuit is designed in such a way that it can be used easily for the design of the demultiplexer (DeMux) and data flip-flop (DFF) circuits. The proposed Mux is easily converted to DeMux by exchanging the input and output terminals only. The effectiveness of the proposed Mux, DeMux, and DFF is examined with the designs that are similar and available in literature using the QCA Designer-E and QCA Pro tools. The design of the proposed Mux is 89.06% fault-tolerant and has decreased the quantum cost by 62.50% as compared to best reported design. Energy measurement plays a key role when designs are operating at nanoscale level. Energy approximation is done with the help of the QCA Pro tool. The proposed designs are more energy efficient compared to the existing works.