# Efficient Design of Reversible Adder and Multiplier Using Peres Gates
## 2 Reversible Logic and Peres Gates  
Reversible logic stands out in the realm of digital circuit design due to its distinctive property of information preservation, allowing for computations where original input states can be uniquely reconstructed from the corresponding output states.  
This reversibility characteristic becomes particularly relevant in quantum computing, where the conservation of information is crucial for leveraging quantum superposition and entanglement. Peres gates are particularly valuable in reversible logic design because they facilitate the creation of circuits that adhere to the principles of information conservation. Peres gate [20] with three inputs and three outputs are shown in Figures 1-3. Both Figures 2 and 3 represent the Peres gate, but the difference between these two representations is that the Peres gate in Figure 2 has a gate count of 1, and the number of the two-qubit model is 4, whereas in Figure 3, where the Peres gate is constructed using one-three input Toffoli gate followed by one-two input Toffoli gate, the gate count is 2 and the number of the two-qubit model is 6 [21]. So, instead of one Peres gate, if we use two Toffoli gates to construct all the circuits discussed in this paper, then we need twice the number of Toffoli gates as compared to the Peres gate, and also, the quantum cost will further increase by 50% for the same. Similarly, comparing the quantum cost of the Fredkin gate with the Peres gate, there is an increase of 25% in the quantum cost of the Fredkin gate as compared to Peres gates, which further increases the quantum cost of the applications of adders and multipliers where we are replacing the Peres gates with the Fredkin gates [20]. All the circuits mentioned in this paper are constructed using the Peres gate shown in Figure 2.  
Figure 1: Peres gate.  
The truth table in Table 1 describes the relation between inputs and outputs. Output a is the same as input a; output b can be obtained by logically exoring the inputs a and b, whereas output c can be obtained as logical exoring of input c with logical ending of a and b, i.e., (ab xor c).  
In the context of the ripple carry adder (RCA) design, Peres gates are employed to perform reversible XOR operations, which are fundamental to binary addition. The ability to selectively apply NOT operations based on control bits makes Peres gates a versatile building block for constructing circuits with minimal energy dissipation, a critical consideration in quantum and reversible computing.  
The reversible nature of Peres gates ensures that the input states can be accurately reconstructed from the output states, aligning with the requirements of quantum computing algorithms. The controlled-NOT operation provided by Peres gates enables 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 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.