Reversible and DNA Computing. Hafiz M. H. Babu

of quantum gates Energy of a basic quantum gate

      Example 1.7

Figure 1.5 shows the quantum realization of the reversible HNG gate. From this figures, it is seen that the quantum realization of reversible HNG gate requires total six quantum gates. So, the power of the reversible HNG gate is (6 142.3) meV = 853.8 meV, where the number of quantum gates of HNG circuit is 6.

1.9 Area

      The area of a reversible gate is defined by the feature size. This size varies according to the number of quantum gates. The size of the basic quantum gates ranges from 50–300 Å. The Angstrom (Å) is a unit equal to m (one ten‐billionth of a meter), or 0.1 nm. Its symbol is the Swedish letter Å. So, the area of a reversible gate can be defined as follows:

      Area = Number of quantum gates Size of a basic quantum gate

Figure 1.5 The quantum representation of reversible HNG gate.

      Example 1.8

Figure 1.5 shows the quantum realization of the reversible HNG gate. From this figures, it is seen that the quantum realization of reversible HNG gate requires total six quantum gates. So, the area of the reversible HNG gate is ((50 6) Å – (300 6)Å) = (300 Å – 1800 Å), where the number of quantum gates of HNG circuit is 6.

1.10 Hardware Complexity

      The hardware complexity of a reversible logic circuit specifies the total number of Ex‐OR operations, NOT operations, and AND operations used in the circuit. Consequently, the hardware complexity can be determined using the following equation:

      (1.10.1)

      where

        = Hardware complexity (total logical operations)

        = A two input EX‐OR gate logical operation

        = A two input AND gate logical operation

        = A NOT gate logical operation

      Example 1.9

Figure 1.6 shows the block diagram of a reversible Fredkin (FRG) gate. The figure describes that there is only one NOT operation, two EX‐OR operations, and four AND operations. So, the hardware complexity of the reversible FRG gate is .

1.11 Quantum Gate Calculation Complexity

      The quantum gate calculation complexity of the quantum representation of a reversible circuit specifies the total number of quantum gates (NOT gates, CNOT gates, and controlled‐V (controlled‐) gates) used in the quantum representation of a reversible circuit. Consequently, the quantum gate calculation complexity can be determined using the following equation:

      (1.11.1)

      where

        = Quantum gate calculation complexity

        = A quantum NOT gate

        = A quantum CNOT gate

        = A quantum controlled‐V (controlled‐) gate

Figure 1.6 Block diagram of the reversible FRG gate.

Figure 1.7 Quantum representation of a reversible FRG gate.

      Example 1.10

Figure 1.7 shows the quantum representation of a reversible Fredkin (FRG) gate. The figure describes that there is only one NOT operation, four quantum CNOT operations, and three quantum controlled‐V (controlled‐) operations. So, the quantum gate calculation complexity of the reversible FRG gate is .

1.12 Fan‐Out

      Fan‐out is a term that defines the maximum number of inputs in which the output of a single logic gate can be fed. The fan‐out of any reversible circuit is 1.

      Example 1.11

      The fan‐out of any reversible circuit is 1.

1.13 Self‐Reversible

      A gate is said to be self‐reversible if its dual combination is the same as itself.

      Example 1.12

In Figure 1.8, there are two Toffoli gates that are in the cascading form. If the outputs of the first Toffoli gate are fed to the input of the second Toffoli gate, then the output of the second Toffoli gate is equal to the input of the first Toffoli gate. Here the outputs of first gate are P, Q, and R, where P = A, Q = B, and R = AB C. Then the outputs of second gate are X, V, and Z, where X = A, Y = B, and Z = AB AB C = 0 C = C.

1.14 Reversible Computation

In a reversible circuit, correct output is found by applying correct input instance