# Efficient Design of Reversible Adder and Multiplier Using Peres Gates
## 3 Design Methodology  
This section details the design methods used for the development of the Peres gate and different adder and multiplier circuits, including the Vedic multiplier using reversible \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{3}{c}{**Input Variables**} & \multicolumn{3}{c}{**Output Variables**} \\ \hline
**a** & **b** & **c** & **a** & **b** & **c** \\ \hline
1 & 1 & 1 & 1 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Truth table for Peres gate. Figure 3: Peres gate quantum circuit using two Toffoli gates. Figure 2: Peres gate quantum circuit. logic. The design of the ripple carry adder, carry-save adder, look ahead carry adder, carry skip adder, and carry bypass adder [22; 23; 24] using Peres gates follows a systematic and staged approach leveraging the reversible nature of Peres gates to achieve an efficient and coherent circuit. The methodology involves the incorporation of Peres gates in a cascading fashion to perform reversible XOR operations on corresponding bits of the input numbers and carry-outs from the preceding stages. Reversible logic gates enable the creation of a coherent and efficient RCA design, contributing to the exploration of reversible logic in practical applications. This section sets the stage for the subsequent discussion on the design methodology where the unique characteristics of Peres gates are leveraged to construct a reversible ripple carry adder, carry-save adder, and reversible multipliers using these two adders. The use of reversible logic, particularly through Peres gates, presents an exciting avenue for the advancement of computing paradigms, including quantum and emerging reversible computing technologies. The design of the ripple carry adder and carry-save adder using Peres gates follows a stage-wise approach:  
* First Bit (LSB): A Peres gate is employed to perform a reversible XOR operation on the least significant bits of the numbers to be added and the input carry. This stage produces the sum bit and a carry-out for the next stage. * Intermediate Bits (1 to 2*n\(-\)1): Peres gates are used in a cascading fashion where each gate performs a reversible XOR operation on the corresponding bits of the input numbers and the carry-out from the previous stage. This process is repeated for the intermediate bits. * Final Bit (MSB): The most significant bits of the input numbers and the carry-out from the previous stage are fed into a Peres gate to produce the final sum bit and carry-out. The output of the RCA is the concatenation of the sum bits (S [2*n\(-\)1:0]). The final carry-out (Cout_n) serves as an overflow indicator, providing information about whether the addition operation resulted in a carry beyond the designated bit size. Peres gates, which belong to the class of three input quantum gates, have shown promise in implementing various logical functions, and we focused on utilizing Peres gates to design 4 \(\times\) 4 reversible multipliers [25; 26; 27; 28; 29; 30; 31; 32]. Peres gates introduced by [PERSON] in 1984 have been recognized for their universality, meaning they can be used to construct any reversible circuit [32]. This characteristic makes Peres gates an attractive choice for designing complex reversible circuits such as multipliers. The 2 \(\times\) 2 and 4 \(\times\) 4 reversible multipliers are constructed using a combination of Toffoli gates and Peres gates. The simplest form of multiplier circuit formed as a 1-bit multiplier is with a single Peres gate with multiplicand as a0 and multiplier as b0, and the third input given to the Peres gate will be the product result as the output given by c0, as shown in Figure 4. The product obtained at the output is 0 when multiplicand a0 is 0 or multiplier b0 is 0, and the maximum value of the product is 1 when both a0 and b0 are at logic 1. In Figure 5, variable a is considered to be a multiplicand, and variable b is a multiplier.