Quantum Computing. Hafiz Md. Hasan Babu

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      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.

      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
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