Quantum Computing. Hafiz Md. Hasan Babu

multiplexers and one 2-to-1 quantum multiplexer. Figure 5.5 presents the 2n-to-1 multiplexer, and the properties of the 2n-to-1 quantum multiplexer are given in property 5.1.

      Figure 5.4. The quantum 8-to-1 multiplexer.

      Figure 5.5. Block diagram of 2n-to-1 multiplexers.

      Property 5.1. A quantum 2ⁿ-to-1 multiplexer can be designed with a 2n−1 gate which produces 2n+n−1 garbage outputs. It also requires a 5(2n−1) quantum cost and a delay of 5(2n−1)Δ, where n denotes the number of selection lines and Δ denotes the unit delay.

5.2 The quantum demultiplexer

      This section presents the design of the quantum demultiplexer. A demultiplexer (or DEMUX) is a device that takes a single input line and routes it to one of several digital output lines. A demultiplexer of 2ⁿ outputs has n select lines which are used to select the output line to which to send the input. A demultiplexer is also called a data distributor. The demultiplexer can be used to implement general purpose logic. By setting the input to true, the DEMUX behaves as a decoder. The reverse of a multiplexer is the demultiplexer.

5.2.1 The quantum 1-to-2 demultiplexer

      A 1-to-2 demultiplexer is the smallest unit of the architecture of a quantum demultiplexer. The characteristic function of a 1-to-2 demultiplexer is s0′D s0D on the different output line, as shown in table 5.2. A quantum Fredkin gate can be used as a 1-to-2 quantum demultiplexer as it can map the characteristic functions of a demultiplexer.

Table 5.2. Truth table of a 1-to-2 demultiplexer.

S	Y 1	Y 0
0	0	D
1	D	0

      Let, D be the inputs and S0 the select input of a 1-to-2 demultiplexer. When S0=0, then D input is transmitted to the second output Y0 and when S0=1, then the D input is transmitted to the third output Y1. Figure 5.6 shows the architecture of a quantum 1-to-2 demultiplexer using a quantum Fredkin gate. The quantum cost and delay of this quantum 1-to-2 demultiplexer are 5 and 5Δ,respectively. Moreover, the number of garbage outputs is one.

      Figure 5.6. The quantum Fredkin gate as a quantum 1-to-2 demultiplexer.

5.2.2 The quantum 1-to-4 demultiplexer

      The quantum 1-to-4 demultiplexer has two select lines, one data input, and four outputs. Figure 5.7 shows the design of a quantum 1-to-4 demultiplexer where Y0, Y1, Y2, and Y3 are the outputs, and S0 and S1 are the select lines. The bit combination of select lines controls the function of the 1-to-4 demultiplexer, as shown in table 5.3. Three quantum Fredkin gates are used in this design. Thus the quantum cost of the quantum 1-to-4 demultiplexer is 15 and the delay of the quantum 1-to-4 demultiplexer is 15Δ, while the number of garbage outputs is two.

      Figure 5.7. The quantum 1-to-4 demultiplexer.

Table 5.3. The truth table of a 1-to-4 demultiplexer.

S 1	S 0	Y 3	Y 2	Y 1	Y 0
0	0	0	0	0	D
0	1	0	0	D	0
1	0	0	D	0	0
1	1	D	0	0	0

5.2.3 The quantum 1-to-2ⁿ demultiplexer

      Figure 5.8 shows the design of a quantum 1-to-8 demultiplexer. As a consequence of the design of the quantum demultiplexer, a 1-to-2ⁿ quantum demultiplexer can be constructed using a 1-to-2ⁿ⁻¹ quantum demultiplexer and 2n−1 quantum Fredkin gates, which is shown in figure 5.9. The properties of the 1-to-2ⁿ quantum demultiplexer are given in property 5.2.

      Figure 5.8. The quantum 1-to-8 demultiplexer.

      Figure 5.9. Block diagram of a 1-to-2ⁿ demultiplexer.

      Property 5.2. A quantum 1-to-2ⁿ demultiplexer can be designed with 2n−1 gates which produce n garbage outputs. It also requires a 5(2n−1) quantum cost and a delay of 5(2n−1)Δ, where n denotes the number of selection inputs and Δ denotes the unit delay.

5.3 Summary

      In this chapter the quantum multiplexer (MUX) and demultiplexer (DEMUC) are presented and explained with the quantum circuit representation. In the multiplexer section, a 2-to-1 MUX, 4-to-1 MUX, and 8-to-1 MUX are described and generalized to the 2n-to-1 multiplexer. In the demultiplexer section, a 1-to-2 DEMUX, 1-to-4 DEMUX, and 1-to-8 DEMUX are described and generalized to the 1-to-2n demultiplexer. Moreover, the quantum cost and delay of the multiplexers and demultiplexers are provided.

Further reading

      [1] Haghparast M and Monfared A T 2017 Novel quaternary quantum decoder, multiplexer and demultiplexer circuits Int. J. Theor. Phys. 56 1694–707