## 1 Introduction  
The International Technology and Roadmap for Semiconductors (ITRS) points out major problems with existing CMOS technology, such as high-power loss, threshold voltages, thermal runaway, and high leakage current, and states that it is approaching the end of its semiconductor nano-system roadmap [1]. [PERSON]'s law, which states that the number of components on a single chip doubles approximately every two years [2], no longer applies in nanoscale devices, and various quantum mechanical effects dominate device physics [3]. In addition, as CMOS-VLSI micro technology reduces the size of transistors disproportionately, large energy and information losses are emerging as major problems. Information loss is a major problem in irreversible digital computation systems, and there is a large and growing demand for nanoscale computation systems that can minimize heat dissipation [4]. Therefore, the design of reversible circuits is a key structural engineering challenge for solving the problem of information loss while minimizing energy dissipation. [PERSON] demonstrated that 1-bit information loss results in energy dissipation of \(k_{\mathrm{B}}\)TIn2 joules, where \(k_{\mathrm{B}}=1.38\times 10^{-23}\) JK\({}^{-1}\) is the Boltzmann constant and \(T\) is the temperature in Kelvin [5]. At room temperature (\(T=300\) K), the heat release occurring during a binary transaction is 0.017 eV and is considered physically irreversible, and this microscopic physical state can be restored to what it was before the process occurred. [PERSON] demonstrated the validity that energy loss of \(k_{\mathrm{B}}\)TIn2 joules in an irreversible circuit can be recovered in a reversible circuit [6]. Recently, with the rapid rise of quantum computing [7], quantum logic gates are attracting attention as logic gates that can replace existing digital circuits in quantum circuit calculation models. The Toffoli gate, developed in 1980, is a universal gate that can implement any desired Boolean function as a reversible circuit [8], and it can be realized by five two-qubit quantum gates [9]. Along with this, various universal reversible gates, such as the Fredkin gate, Feynman gate, and Peres, were developed [10; 11; 12]. Since then, various reversible gates, such as RUG [13], RQCA [14], URG [15], TR [16], and PQR [17], have been continuously developed and implemented using QCA [18]. Quantum-dot cellular automata (QCA), proposed by [PERSON] and [PERSON], has emerged as an alternative to overcome the problems of existing CMOS and implements existing reversible circuits with ultra-low power consumption. The dissipated energy is measured based on the Hamiltonian matrix, using the Hartree-Fock approximation in relation to the Coulomb repulsion between QCA cells, as shown in (1) [19]. \[H= \begin{bmatrix}\frac{-E_{k}}{2}\dot{\sum_{i}C_{i}f_{i,j}}&-\gamma\\ -\gamma&\frac{E_{k}}{2}\dot{\sum_{i}C_{i}f_{i,j}}\end{bmatrix} = \begin{bmatrix} \frac{-E_{k}}{2}\big{(}C_{j-1}+C_{j+1}\big{)}&-\gamma\\ -\gamma&\frac{E_{k}}{2}\big{(}C_{j-1}+C_{j+1}\big{)}\end{bmatrix}  \tag{1}\]  
where \(E_{k}\) is the energy cost of two neighboring cells with opposite polarization, called kink energy, \(C_{i}\) denotes the polarization of the \(i\)-th neighboring cell, and \(f_{i,j}\) denotes the geometrical factor identifying the electrostatic interaction between cells \(i\) and \(j\) due to the geometrical distance. This kink energy is related to the energy cost of two cells with the opposite polarization. \(\gamma\) denotes the electron tunneling energy inside the cell, which is controlled by the clock. The nonadiabatic power estimation model was used to estimate the power loss or energy dissipation of the cell [20; 21]. The expected value of the Hamiltonian at each time instant is given by  
\[E=\langle H\rangle=\frac{\hbar}{2}\overrightarrow{\Gamma}\cdot\overset{ \rightarrow}{\lambda} \tag{2}\]  
where \(\overrightarrow{\Gamma}\) is the 3-D energy vector, and \(\overset{\rightarrow}{\lambda}\) is the coherence vector.