Research Article | Open Access
Md. Abdullah-Al-Shafi, Ali Newaz Bahar, "An Architecture of 2-Dimensional 4-Dot 2-Electron QCA Full Adder and Subtractor with Energy Dissipation Study", Active and Passive Electronic Components, vol. 2018, Article ID 5062960, 10 pages, 2018. https://doi.org/10.1155/2018/5062960
An Architecture of 2-Dimensional 4-Dot 2-Electron QCA Full Adder and Subtractor with Energy Dissipation Study
Quantum-dot cellular automata (QCA) is the beginning of novel technology and is capable of an appropriate substitute for orthodox semiconductor transistor technology in the nanoscale extent. A competent adder and subtractor circuit can perform a substantial function in devising arithmetic circuits. The future age of digital techniques will exercise QCA as preferred nanotechnology. The QCA computational procedures will be simplified with an effective full adder and subtractor circuit. The deficiencies of variations and assembly still endure as a setback in QCA based outlines, and being capricious and inclined to error is the limitation of these circuits. In this study, a new full adder and subtractor design using unique 3-input XOR gate with cells redundancy is proposed. This designs can be utilized to form different expedient QCA layouts. The structures are formed in a single layer deprived of cross-wiring. Besides, this study is directed to the analysis of the functionality and energy depletion possessions of the outlined full adder and subtractor circuits. For the first time, QCADesigner-Energy (QD-E) version 2.0.3 tool is utilized to find the overall depleted energy. The attained effects with QCADesigner have verified that the outlined design has enhanced functioning in terms of intricacy, extent, and latency in contrast to the earlier designs. Moreover, the redundant form of full adder and subtractor has uncomplicated and robust arrangement competing typical styles.
Complementary Metal Oxide Semiconductor (CMOS) archetype based on current conventional transistors stays on Moores theory. This theory was intended in 1965 by Gordon Moore and forecasts that the figure of transistors positioned within a chip will multiply every 2 years. Thus, lessening transistor extent is desired to outline high-thickness, high-rapidity, and power-defeat circuits . Nowadays, numerous cohesive circuits have been prepared on an extent of 0.230-0.330 microns; however, when the dimension of the circuit extends to 0.50 microns or less, substantial borders for instance power depletion and drawbacks with lithography and layout intricacy ascend . Therefore, doubling the transistors number in a microchip every 2 years and doubling the clock rate every 3 years will not be potential. Therefore, to increase the functioning of arithmetic units, there is a requirement for additional archetypes . As inheritor to typical CMOS, scientists have proposed a scheme where computations are executed with quantum dots. This model was entitled quantum-dot cellular automata (QCA), and, same as nowadays processers that transfer information within electrical flow from one site to another, the conduction of polarization level initiates information transmission. In nanoscale, QCA as a novel device paradigm is appropriate for nanometer computer architectures. The vital element in QCA is a four-dot squared cell, and two unoccupied identical charges are integrated in that cell. The activity of dots is organized transversely by these electrons that are the upshot of Coulombic connection. Unlike CMOS archetype, the comparative structure of the charges rather than current is applied to encrypt the binary information in QCA. Minimal energy utilization, firm operation, and small magnitudes are the entities of QCA circuit [4–8]. The majority voter and inverter gate are the principal aspects of assembling QCA outlines . Assembly time fault phases and functioning time defect degrees in nanotechnologies are practicable , and the malfunctioning performance of QCA designs has to be depicted with the fault-tolerant model of QCA circuits. The implementation of full adder and subtractor as the major component of the arithmetic procedures can precisely influence the functioning of the entire architecture. Full adder and subtractor have a notable role in assembling Arithmetic and Logical Unit (ALU) that is a processing subsystem [11–14]. QCA technology has very minimal energy dissipation [4–6]. The focal role of this work is to organize a novel outline for full adder and subtractor circuit based on three-input XOR gate deprived of wire crossing. The proposed circuits are assembled by the number of cellblocks in QCA and they achieve the anticipated functions. In an abridged practice, the foremost impacts of this study are as follows:(i)designing an architecture for adder and subtractor by the minimal figure of cells, utilization area, and overall depleted energy,(ii)assessing the organized layout with other contemporary outlines based on XOR gate and without XOR gate in terms of cell numbers, intricacy, and the expended area.
The succeeding organization will be conferred in the remainder of the study. A concise synopsis of the QCA technology is exhibited in Section 2. Section 3 spotlights existing works on adder and subtractor. In Section 4, the proposed designs are formed in terms of cells fabrication and input and output positions. Simulation outcomes with relative assessment are characterized in Section 5, and for the first time, dissipated energy is estimated with QD-E which is organized in Section 6. Decisively, specific concluding explanations and analysis of potential focuses are organized in Section 7.
2. Primitive Premises of Quantum-Dot Cellular Automata
Logic gates, wires, and clocking are elementary QCA views. These assessments are exercised to comprehend QCA technology sounder. Four dots are involved in QCA cells and a top-level figure of polarized QCA cells is shown in Figure 1. Every single cell is assembled in a quadrilateral outline by four quantum dots. Two electrons are permitted to channel between neighboring dots and these electrons are utilized to direct the cell. The electrons are positioned in opposed spots because of their reciprocal electrostatic revulsion. Figure 1 presents two corresponding organizations of electrons in the QCA and the organizations are symbolized as polarization P = +1 and P = 1, correspondingly. With polarization P = -1 to denote logic “0” and P = +1 to denote logic “1”, binary figures can be recognized .
The entire logic functions are able to operate by ordering the groups of QCA cells. The polarization of adjoining cells is persuaded by Coulombic interfaces. The potential difficulties or barriers between the dots have outlined QCA architectures. The QCA architectures are clocked and coordinated through barriers . The communication of the cells and the performance of the entire architecture have been resolved by the QCA topology. The QCA cells are utilized to form all traditional logic circuits . The formation of the majority voter is an essential part of QCA circuits as exhibited in Figure 2(a).
In the majority voter, three input cells are unified. One of the cells is in inside and persists the majority polarization because it signifies the minimal energy level. The other cell is the output cell. If one input is fixed to 1, the majority gate functions like OR. Similarly, if one input is fixed to 0, the gate acts as AND of the two additional inputs. The inverter has different arrangements in QCA and measured as the alternative vital gate. The resilient outline of the inverter is shown in Figure 2(b). Two neighboring cells have contradictory polarities while the energy level is minimal based on the Coulombic revulsion. The majority voter and inverter are incorporated to achieve all logical circuits. QCA wire performs a significant task in circuit design. The electrostatic contact between cells initiates the binary signal to read from input to output in the wire. Two sorts of wires are presented in Figure 2(c). The representation of a 90-degree binary wire is pointed out in the figure and the inverter chain is operated to reverse the input cells decently by rotated cells as pointed out in the figure.
The combinational and successive circuits need four sequential and definite segments to operate timing in QCA. The clocking operates the information stream and provides the genuine potential in QCA . Four segments are Switch, Hold, Release, and Relax as integrated in QCA clock directed in Figure 3. The cell will be in a void position once the electrons are located in the central dots throughout relax level. Through Switch phase, inner barriers of dots gradually emerge and push the electrons between the edges. At the time of Hold phase, the barriers are active and the cell persists in polarization and influences its adjacent. Finally, in Release segment, the barriers are downward and the electrons are drawn into the central dots [16, 17].
The QCA circuits are conceived by three sorts of crossover methods, namely, coplanar, multilayer, and logical crossing. In coplanar crossing, one of the wires utilizes only cell type one, whereas the other utilizes just cell type two, resulting in the wires functioning autonomously on the uniform fabrication layer as presented in Figure 4(a). All logic crossing is implemented on a particular QCA layer through this process with no requirement for a complement in the conventional technologies. A drawback of coplanar crossing is excessive manufacturing sensitivity. The regular assembly is certainly broken by cells which are not in position and, incidentally, an adverse coupling between two wires is transpired .
A customary multilayer crossing is developed as presented in Figure 4(b). The signal change from a particular layer to another is disallowed because of the outsized perpendicular space of wires. This approach is further lenient of misplaced cells than the coplanar crossing; however, several active QCA layers are required to implement the technology. The operation of multilayer technology is still not reasonable, and the assignment of single layer cell involves extreme accuracy that is very confronting  so this work is organized so as to only utilize the coplanar crossing.
A special sort of coplanar wire crossing is performed by Shin Jeon and Yoo  that specified as logical crossing. In this structure, wire crossing via interfering of clocking phases is achieved as shown in Figure 4(c).
QCA defects are divided into three arrangements, materializing through the fabrication of a circuit. Switching the cells from their anticipated positions is termed misalignment cells. The operation of the QCA circuit is not imitated by misalignment cells but, occasionally, there is an unanticipated result in a circuit. Another form of faults ensues once the cell is misplaced due to its defects. In this circumstance, the missed cell would have no impact on its adjacent and it influences the functionality of the design . The last one ensues while cells are rotated corresponding to the other cells in the array that is termed dislocation cells, and in this instance, the design does not operate. Figure 5(a) organizes an omitted cell layout in which the circuit stops functioning. As present in Figure 5(b), a circuit may have an unanticipated result once the displacement cell has 45-degree rotation angle. Figure 5(c) presents misalignment cell in a design. Evidently, the direction of the cell association is not substantial because of symmetry.
3. Related Designs
The most elementary constituent in the outline of the arithmetic block is a full adder, and the succeeding equations are frequently reflected in the pattern of a full adder.
The output Sum is the binary addition of the inputs A, B, and C where A, B are the unique two binary inputs coupled with C which is the Carry out of the least significant bits. Cop signifies the produced Carry from the sum of the binary inputs A, B, and C. Table 1 presents the logical table of a full adder. In latest nanotechnology, study on the architecture of nanoscale full adder is considerably enlarged particularly in the QCA [22, 23]. An adder designed with two majority voters is presented in . This design uses a five-input majority voter to produce Sum output. In , an outline of single-bit full adder with two inverters and three majority voters is presented. Both of the designs in [22, 23] use cross-wiring and thus are not robust. In , the design of presented full adder contains two foremost components, the three-input XOR gate and the three-input majority voter. In this layout, the orthodox wire crossing technique is used to conduct two input values individually.
A subtractor is a circuit that subtracts two bits with considering the outcome of lower substantial phase. The truth table for subtractor is presented in Table 2, where A, B, and C are corresponding inputs, and Diff and Borr are outputs. Boolean expressions for Diff and Borr are presented as follows.
In , a layout of full subtractor is proposed, which consists of 136 QCA cells with wire crossing. Another design is outlined in , where 62 cells are employed but with multilayer approach. Therefore, these designs are not robust. In the succeeding section, a new paradigm for the full adder and subtractor circuit based on the XOR gate  without cross-wiring is proposed.
4. Proposed Outline in QCA
To realize the performance of full adder and subtractor, circuit coulomb interaction is used. The geometry and accuracy of implantation are important to the appropriate operation of a device. As pointed out in Section 2, the blocks in the QCA are assembled with inverter and majority voter. Though, the most common majority voter has very inconsequential fault-tolerant performance against most of the common deficiencies. This section clarifies the outlined full adder and subtractor design. For this point, the structure of XOR gate is obtained from the design ; then define the architecture of the full adder and subtractor using QCA cells. The full adder structure has 28 cells. Three cells with labels A, B, and C are input cells. The output is termed as S and C, where S defines the Sum signal and C define the Carry signal. The remaining cells are labeled as device cells. The subtractor outline has 27 QCA cells. Three cells with labels A, B, and C are input cells. The output is entitled as Diff and Borr, where Diff state the Difference signal and Borr state the Borrow signal. Both of the architectures are conceived in a single layer and their latency is two clock zones, so the proposed designs are robust.
The schematic outline of the proposed full adder and subtractor design is shown in Figures 6(a) and 6(b), correspondingly. The full adder structure is comprised of one three-input XOR gate and one three-input majority voter.
Similarly, the subtractor design is comprised of one three-input XOR gate and one three-input majority voter with one inverter gate. As presented in Figure 7, the outlined designs are employed in a single layer. It uses only 90-degree QCA cells and does not use any coplanar wire crossing.
5. Testing of the Proposed Architectures with Comparative Assessment
In this section, QCADesigner ver. 2.0.3 is applied to simulate the outlined circuits. Simulation engines, features, and relative comparisons are explained in the remainder of this section.
5.1. Simulation Engine and Features
QCADesigner is a recognized simulation engine mostly used for estimating QCA circuits. Bistable approximation and coherence vector simulation are measured as two distinct simulation tools. The bistable approximation tool analyses the position of a particular cell with a time-independent method by kink energy procedure that estimates the charge of two cells taking reverse polarization; thus simulation period in this tool is condensed. The coherence vector methodology thinks through the time-reliant status of a cell in collaboration with the other cells through the similar kink energy procedure . Every cell in QCADesigner can perform in four ways, namely, input, output, fixed, and normal. The extent of the simple quantum cell is fixed at 18x18 with 5 nm thickness quantum dots. The center to center space was fixed at 20 nm for neighboring cells. The simulation tool is fixed to coherence vector form in QCADesigner engine with the simulation features being presented here: relaxation period 1.0e-015s, radius force 65 nm, relative permittivity 12.90, clock high-level 9.8e-22J, clock low-level, 3.8e-23J, separation of layer 11.50 nm, overall simulation period 7.0e-011s, and samples number 50000.
5.2. Accuracy Study
The proposed designs are simulated and Figure 8 presents the outcomes. Furthermore, for the input set C B A = 000, 001, 010, 011, 100, 101, 110, 111 all separate outcomes in the output Sum are attained: 01110001, 00001111, 11001100, 11010100, 01010101, 00101011, 10101010, 01001101, 00010111, 00110011; similarly, all separate outcomes in the output Carry are attained: 11101000, 00110011, 00001111, 01001101, 10001110. For adder circuit, if inputs A, B, and C are zero, the output values for Sum (S) and Carry (C) are 0, separately. For inputs A=0, B=0, and C=1, the output values for Sum (S) and Carry (C) are 1 and 0, individually. Likewise, for full subtractor, if the input values are A=B=C=0, then the output values for Difference (Diff) and Borrow (Borr) are both 0, individually. For inputs values, A=B=0 and C=1, the output values for Difference (Diff) and Borrow (Borr) are both 1, respectively.
5.3. Intricacy Study of the Proposed Architectures
The intricacy of the outlined designs is presented in Table 3. The full adder layout is achieved with one XOR gate, 28 QCA cells with an extent of , and for subtractor layout one XOR gate, 27 QCA cells with an extent of .
5.4. Comparison Study of the Proposed Architectures
The designed full adder is compared with the preceding three designs based on XOR gate presented in [22, 23, 28]. Execution outcomes are specified in Table 4 and the table covers the total figure of cells used for design, covered extent, and latency as well as wire crossing. The proposed design achieved 69, 52, and 32% progress in total cell count compared to the designs presented in , , and , correspondingly. The proposed structure achieved 78, 54, and 50% progress in covered area compared to the designs organized in , , and , correspondingly. Besides, the number of clock zones has lessened by 60 and 50% contrasted to the designs organized in , , and , correspondingly. The outlined design does not employ any cross-wiring. Similarly, for subtractor design, a firm improvement is achieved. The proposed subtractor attained 48, 80, 88, and 90% improvement in total cell count compared to the designs organized in , , , and , respectively. An improvement of 23, 82, 77, and 93% is achieved in covered area compared to the designs presented in , , , and , respectively. Moreover, the number of clock zones has reduced by 71, 75, and 77% compared to the layouts presented in , , and , respectively. The subtractor design does not use any cross-wiring. The overall outcomes specify that the proposed architectures are superior to earlier outline in all of the factors.
6. Power Consumption of the Proposed Architectures
An extension of QCADesigner ver. 2.0.3 is QCADesigner-E (QD-E) of Konrad Walus . It performs the assessment of the power depletion of QCA logic circuits based on the study presented in . It works as a simulation block that is based on the Coherence Vector Simulation Engine. Besides, QD-E is completely well-suited to QCA layouts formed with the QCADesigner ver. 2.0.3. Tables 5 and 6 present the total power dissipated by the adder and subtractor design.
In the table, is the figure of complete energy transfers to the bath of all cells divided for each clock cycle, is the total energy moves between QCA cells and the clock divided for each clock cycle, and is for each clock cycle the figure of all errors of the cells energy assessment. The error follows from the theory that the total of all energy movements of a QCA cell is zero over an entire clock cycle, where error = - ( + ). As well, is the sum of total energy transfers to the bath through the entire simulation and is the associated error. is the average standards of the energy moves to the bath and the error for every clock cycle. is the figure of the energy moves to or from the clock throughout the entire simulation, whereas is the average energy movement through a clock cycle. To complete the entire simulation process, the adder takes 7 iterations and the subtractor takes 6 complete simulations.
Evaluation of the temperature influence on the output polarization of outlined designs is performed and the output polarization is apprehended at different temperature by QCADesigner tool. The average output polarization (AOP) for each QCA cell is evaluated from  and organized in Figure 9. Both architectures operate competently in the temperature range of 1-10 K, and the AOP for each cell is slightly changed in this measure. The AOP is measured in Joule (J).
A comparison of power dissipation among proposed and existing designs is presented in Table 7. The outlined circuits dissipated very low power compared to existing designs.
New full adder and subtractor based on XOR gate are organized in this work. The proposed architectures are used to acquire the high-performance logic element and it gets robustness and adeptness through the attained outcomes. The outlined single layer architectures are expressively more robust than the typical full adder and subtractor. This study has exhibited the new full adder and subtractor layout that can be applied to a competent building unit for higher blocks to design more complex circuit in the future. For the first time, QD-E is utilized in this work to estimate total power dissipation of QCA architectures. An n-bit adder is acquired by connecting the designed QCA full adder; in this respect, the employed hardware is reduced by occupying the regular clocking configuration and parallel arrangement of the original procedure.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
- S. R. Heikalabad, A. H. Navin, M. Hosseinzadeh, and T. Oladghaffari, “Midpoint memory: A special memory structure for data-oriented models implementation,” Journal of Circuits, Systems and Computers, vol. 24, no. 5, 2015.
- W. Yu, B. Zhang, C. Liu et al., “Impact of Si cap, strain and temperature on the hole mobility of (s)Si/sSiGe/(s)SOI quantum-well p-MOSFETs,” Microelectronic Engineering, vol. 113, pp. 5–9, 2014.
- D. Fey, M. Schneider, J. Jahns, and H. Knuppertz, “Optical multiplexing techniques for photonic Clos networks in High Performance Computing Architectures,” The Journal of Supercomputing, vol. 62, no. 2, pp. 620–632, 2012.
- M. Abdullah-Al-Shafi, A. N. Bahar, M. A. Habib et al., “Designing single layer counter in quantum-dot cellular automata with energy dissipation analysis,” Ain Shams Engineering Journal, 2017.
- Md. Abdullah-Al-Shafi and A. N. Bahar, “Energy Optimized and Low Complexity 2-Dimensional 4 Dot 2 Electron Flip-Flop and Quasi Code Generator in Nanoscale,” Journal of Nanoelectronics and Optoelectronics, vol. 13, no. 6, pp. 856–863, 2018.
- M. Abdullah-Al-Shafi and A. N. Bahar, “Ultra-efficient design of robust RS flip-flop in nanoscale with energy dissipation study,” Cogent Engineering, vol. 4, no. 1, 2017.
- M. Abdullah-Al-Shafi, A. N. Bahar, F. Ahmad, and K. Ahmed, “Performance evaluation of efficient combinational logic design using nanomaterial electronics,” Cogent Engineering, vol. 4, no. 1, 2017.
- M. Abdullah-Al-Shafi and A. N. Bahar, “A new approach of presenting binary to grey and grey to binary code converter in majority voter-based QCA nanocomputing,” Journal of Computational and Theoretical Nanoscience, vol. 14, no. 5, pp. 2416–2421, 2017.
- A. N. Bahar, M. S. Uddin, M. Abdullah-Al-Shafi, M. M. R. Bhuiyan, and K. Ahmed, “Designing efficient QCA even parity generator circuits with power dissipation analysis,” Alexandria Engineering Journal, 2016.
- T. Wei, K. Wu, R. Karri, and A. Orailoglu, “Fault tolerant Quantum Cellular Array (QCA) design using Triple Modular Redundancy with Shifted Operands,” in Proceedings of the 2005 Asia and South Pacific Design Automation Conference, ASP-DAC 2005, pp. 1192–1195, January 2005.
- H. Cho and E. E. Swartzlander, “Adder designs and analyses for quantum-dot cellular automata,” IEEE Transactions on Nanotechnology, vol. 6, no. 3, pp. 374–383, 2007.
- K. Navi, M. H. Moaiyeri, R. F. Mirzaee, O. Hashemipour, and B. M. Nezhad, “Two new low-power full adders based on majority-not gates,” Microelectronics Journal, vol. 40, no. 1, pp. 126–130, 2009.
- A. N. Bahar, F. Ahmad, N. M. Nahid, M. Kamrul Hassan, M. Abdullah-Al-Shafi, and K. Ahmed, “An optimal design of conservative efficient reversible parity logic circuits using QCA,” International Journal of Information Technology, pp. 1–10, 2018.
- A. N. Bahar, R. Laajimi, M. Abdullah-Al-Shafi, and K. Ahmed, “Toward Efficient Design of Flip-flops in Quantum-Dot Cellular Automata with Power Dissipation Analysis,” International Journal of Theoretical Physics, pp. 1–10, 2018.
- P. D. Tougaw and C. S. Lent, “Logical devices implemented using quantum cellular automata,” Journal of Applied Physics, vol. 75, no. 3, pp. 1818–1825, 1994.
- C. H. Bennett, “Logical reversibility of computation,” IBM Journal of Research and Development, vol. 17, no. 6, pp. 525–532, 1973.
- R. Landauer, “Irreversibility and heat generation in the computing process,” IBM Journal of Research and Development, vol. 5, no. 3, pp. 183–191, 1961.
- K. Walus and G. A. Jullien, “Design tools for an emerging SoC technology: Quantum-dot cellular automata,” Proceedings of the IEEE, vol. 94, no. 6, pp. 1225–1243, 2006.
- A. Gin, P. D. Tougaw, and S. Williams, “An alternative geometry for quantum-dot cellular automata,” Journal of Applied Physics, vol. 85, no. 12, pp. 8281–8286, 1999.
- S.-H. Shin, J.-C. Jeon, and K.-Y. Yoo, “Design of wire-crossing technique based on difference of cell state in quantum-Dot cellular automata,” International Journal of Control and Automation, vol. 7, no. 4, pp. 153–164, 2014.
- A. Roohi, R. F. DeMara, and N. Khoshavi, “Design and evaluation of an ultra-area-efficient fault-tolerant QCA full adder,” Microelectronics Journal, vol. 46, no. 6, pp. 531–542, 2015.
- S. Angizi, E. Alkaldy, N. Bagherzadeh, and K. Navi, “Novel robust single layer wire crossing approach for Exclusive OR Sum of Products logic design with Quantum-dot Cellular Automata,” Journal of Low Power Electronics, vol. 10, no. 2, pp. 259–271, 2014.
- D. Abedi, G. Jaberipur, and M. Sangsefidi, “Coplanar Full Adder in Quantum-Dot Cellular Automata via Clock-Zone-Based Crossover,” IEEE Transactions on Nanotechnology, vol. 14, no. 3, pp. 497–504, 2015.
- F. Ahmad, G. M. Bhat, H. Khademolhosseini, S. Azimi, S. Angizi, and K. Navi, “Towards single layer quantum-dot cellular automata adders based on explicit interaction of cells,” Journal of Computational Science, vol. 16, pp. 8–15, 2016.
- H. Dallaki and M. Mehran, “Novel subtractor design based on quantum-dot cellular automata (qca) nanotechnology,” International Journal of Nanoscience and Nanotechnology, vol. 11, no. 4, pp. 257–262, 2015.
- M. Hayati and A. Rezaei, “Design of novel efficient adder and subtractor for quantum-dot cellular automata,” International Journal of Circuit Theory and Applications, vol. 43, no. 10, pp. 1446–1454, 2015.
- K. Walus, T. J. Dysart, G. A. Jullien, and R. A. Budiman, “QCADesigner: a rapid design and simulation tool for quantum-dot cellular automata,” IEEE Transactions on Nanotechnology, vol. 3, no. 1, pp. 26–31, 2004.
- S. R. Heikalabad, M. N. Asfestani, and M. Hosseinzadeh, “A full adder structure without cross-wiring in quantum-dot cellular automata with energy dissipation analysis,” The Journal of Supercomputing, vol. 74, no. 5, pp. 1994–2005, 2018.
- S. Karthigai Iakshmi, G. Athisha, M. Karthikeyan, and C. Ganesh, “Design of subtractor using nanotechnology based QCA,” in Proceedings of the 2010 IEEE International Conference on Communication Control and Computing Technologies, ICCCCT 2010, pp. 384–388, October 2010.
- E. Ganesh, L. Kishore, and M. Rangachar, “Implementation of quantum cellular automata combinational and sequential circuits using majority logic reduction method,” International Journal of Nanotechnology and Applications, vol. 2, no. 1, pp. 89–106, 2008.
- J. Timler and C. S. Lent, “Power gain and dissipation in quantum-dot cellular automata,” Journal of Applied Physics, vol. 91, no. 2, pp. 823–831, 2002.
- V. Pudi and K. Sridharan, “Efficient design of a hybrid adder in quantum-dot cellular automata,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 19, no. 9, pp. 1535–1548, 2011.
Copyright © 2018 Md. Abdullah-Al-Shafi and Ali Newaz Bahar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.