## 2 Background Materials  
### QCA Basic Terminology  
Circuits are designed and implemented in QCA, based on QCA cells. As mentioned in QCA technology, communication and calculations are achieved via the interaction of coulombic repulsion of QCA cells and influence on neighboring cells, in order to influence the electrons enclosed in each cell, thus transmitting and spreading information [7]. As a result, based on the interaction of coulombic repulsion, QCA wire is used for the communication required in QCA, and digital gates are used to create its computing circuits. Figure 1a shows the QCA 90-degree cell in both logical 0 and 1 modes [7; 8; 9; 10]. Coulombic energy is used between neighboring cells for data transfer and QCA operations. QCA wire is created from an array of QCA cells and by stacking them together in such a way that, by entering an input with a logical value of \"1\" or \"0\", from the input cell to the output cell, the polarization of the cells will be the same. The input will bring all the cells to their polarity. Figure 1b shows the QCA 90-degree wire used to propagate the signal [8; 9]. Another type of coplanar wire is known as 45-degree wire. This wire is created by using a 45-degree cell and putting this type of cell together [7]. Figure 1c shows the QCA 45-degree cell. Figure 1d also shows the QCA 45-degree wire. As you can see, this type of QCA wire is created using 45-degree cells. In this type of wire,  
Figure 1: Schematics types of QCA cells (**a**) two structures logical values the QCA 90-degree cells, (**b**) QCA 90-degree wire, (**c**) two structures logical values the QCA 45-degree cells, (**d**) QCA 45-degree wire. logic are propagated in the odd cells, with the inverter the values of the input in the even cells. Since the 45-degree cells have less strength, due to the 45-degree rotation of the cell, the designs made with this type of cell have more implementation cost and less strength. Therefore, we have not used any 45-degree cells in our designs [7; 8; 9]. On the other hand, the QCA clock (Figure 2) was used to synchronize circuits [11; 12]. QCA clock has four phases, which are required for control signals and circuit synchronization. The confluence of two QCA wires is called a crossover. In order for QCA wires to not destroy each other's data at the crossovers of two wires, QCA crossovers are used. There are two different types of crossovers in QCA. The first type is known as multilayer crossovers. As seen in Figure 3a, in this type, one of the wires is passed through the intersection in one layer (first layer), and the other wire is passed in the third layer, so that the wires keep their data and prevent data destruction. We have not used this type of crossover, due to the high cost of implementation and large occupied area and volume [13]. Figure 3: QCA crossover, (**a**) multilayer crossovers, (**b**) coplanar crossovers by using rotation cells, and (**c**) coplanar crossover by using standard cells. Figure 2: The four-phase clock. The second type of QCA crossover is referred to as coplanar crossover. This type of crossover is created in two ways. In the first method, the coplanar crossover, is created using both 45\({}^{\circ}\) and 90\({}^{\circ}\) wire types. Figure 3b shows this type of crossover. As you can see, in this method, one of the wires is a 90\({}^{\circ}\) wire (standard wire), and the other wire is a 45\({}^{\circ}\) wire. As explained, in the 45\({}^{\circ}\) wire, in order for the input data to be received in the output cell, the number of cells must be odd. As a result, in this method, the number of cells must be odd. We have not used this type of crossover either, as we said before, due to the use of a 45\({}^{\circ}\) cell [14]. Another method of coplanar crossover design is only achieved by using standard wire. In this method, both wires are created using 90-degree cells, and non-adjacent clock phases are used at the crossover, so that the wires do not destroy each other's data.