## 1 Introduction  
In today's world, digital systems, and especially computer systems, play an important role in human life, and they are widely used. As a result, scientists and researchers have focused all of their efforts, both orbital and economic, on the development, improvement, and optimization of these systems. Moreover, the miniaturization of day-to-day electronic devices and components and the lack of responsiveness of common technology (CMOS), due to the appearance of problems at the nanoscale, have led to the acquisition of nanoscale devices. These devices have led to the coming existence of new nanoscale technologies. This field of electronics is called nanoelectronics or nanotechnology. Among these technologies, quantum-dot cellular automata (QCA) is one of them, which has introduced a new landscape with unique features. QCA technology is considered as a new communication and computing technology due to its unique features, such as small size, high operating speed, and ultra-low power consumption. This technology has drawn the attention of many researchers and scientists. As a result, QCA can be considered as a potential solution to the scale limitations of complementary metal-oxide-semiconductor (CMOS) technology [1]. As previously mentioned, QCA technology is a novel nanoscale computational paradigm that may significantly enhance electronic circuit design technology [2]. It uses nano-level interactions to implement digital logic with low energy consumption and high clocking frequency. The advancements in the fabrication [3; 4] and simulation [5; 6] of QCA circuits have enabled researchers to design complete computational circuits [7]. Moreover, adders are one of the most basic and main computational circuits of digital logic, and as the most common and most widely used arithmetic digital circuits, they are the subject of many research studies [8]. However, with the appearance of new technologies to operate these technologies, it is necessary to create well-known tools needed for use in more complex applications, which, in the digital field, leads to the design of basic and main logic gates. The optimization of logic gates is very important because the improvement and optimization of these gates, due to their many uses in larger and more complicated circuits of the cause, improve and optimize these circuits. One of the most important and widely used logic gates is the XOR gate. The improvement and optimization of the XOR gate are very important. Adders are constituent and also one of the main and basic blocks of many VLSI systems, such as microprocessors and other various processors. One of the factors leading to optimizing the relevant blocks is the presentation of new designs in proportion to the development of new technologies. Designing a full adder with a simple structure and limited power consumption can play an essential role in simplifying digital circuits. There are certain factors for a better design of full adder circuits. The most important criterion is the usage in more complex and larger circuits. Other criteria include low power consumption, high speed, and simplex circuits; these criteria themselves can be considered as components of the usage criterion. Among the most important items used are full adders, which are used in more complex and larger computing circuits and communication devices. Given that mass media is portable and fully available today, the main concern is an increase in battery life, which reduces the need for recharging. In other words, the main concern is reducing power consumption. Increasing the performance of the full adder is directly related to improving orbital parameters and improving system performance. Thus, many researchers are looking for ways to reduce power consumption. Adders are widely used in mobile digital devices, and they are the main element in many computing circuits. Nowadays, with the increasing use of electronic devices, the need for the optimal design of adders has become very important.