INTRODUCTION
The Quantum Cellular Automata (QCA) was first projected in 1993, unlike standard
computers in which information is transferred from one place to another by means
of electrical current, QCA transfers information by propagating a polarization
state (Zhang et al., 2004; Toth
and Lent, 2000; Orlov et al., 2003), QCA is
based on the encoding of binary information in the charge arrangement within
the quantum dot cells. Computational power is provided by the Coulombic interaction
between QCA cells (Graunke et al., 2005). The
interconnections between the cells are provided by the physics of celltocell
interaction due to the reorganization of electron positions (Toth
and Lent, 2000). Tentative experimental results signified that QCA may be
an extremely feasible option to CMOS. QCA cells and QCA logical devices have
been efficiently fabricated and tested (Zhirnov et al.,
2003). Nevertheless, a systematic methodical for constructing and designing
QCA circuits are crucial for the development in this field. Essential logical
devices and an adder have been designed by Bhanja and Sarkar
(2006). Such devices were simulate with a simulation software called AQUINAS.
Memory has been studied by Toth and Lent (1999) and
a complex SRAM cell has undergone successful simulation. Furthermore, a simple
shift register has also been constructed and simulated (Zhirnov
et al., 2003). Both of these design schematics take advantage of
architecture developed by the authors (Toth and Lent, 1999),
called the SQUARES architecture. The SQUARES architecture essentially consists
of cells that are 5 QCA cells wide and 5 QCA cells high. A library of different
QCA welldesigned devices such as a majority gate was then built up (with each
device\housed" in a square) and used to build the various schematics. This technique
resulted in successful simulations. The shortcoming to the SQUARES structural
design was that designs using it had less than optimal density (Bhanja
and Sarkar, 2006). Previous work on the QCA architecture was about studying
QCA cells. The few accessible QCA logical circuit designs such as the adder,
XOR gate and multiplexor, using the QCA the majority gate, were designed and
studied. For many QCA circuits simplified versions could be constructed with
QCA majority gates. It would be exceptionally valuable to create a CAD tool
that could translate a schematic containing conventional Boolean logic equations
into a schematic consisting entirely of QCA majority gates. Some accessible
tools such as Mentor Graphics' AutoLogic II achieve this task. It allows a schematic
created with general library components to be mapped to a specific technology
provided that a library for that technology exists. The intention was to create
such a QCA library with the hope that, once completed, this tool would take
as input any conventional schematic, Boolean equation, or VHDL code and generate
its minimized equivalent in QCA. An important comprehension of the lack of a
complete set of QCA design rules is essential for a complete and thorough CAD
program. As QCA is being investigated as an alternative to CMOS, an ultimate
goal should be to build complete microcomputers from QCA cells. This will not
be viable without a systematic way to manufacture and minimize logic functions
in a paradigm suitable for QCA. This is the main contribution of this research.
In this paper a JMap for the realization and minimization of logic functions
in terms of majority logic function is invented. Also, generalization of the
JMap for n variable logic functions will be introduced.
QUANTUMDOT CELLULAR AUTOMATA
Quantumdot Cellular Automata (QCA) are a computational nanotechnology. It
is based on a the basic QCA device QCA cell composed of four quantum dots arranged
in a square pattern as shown in Fig. 1. Four quantum dots
are situated to form a square (Srivastava and Bhanja 2007).
Quantum dots are small semiconductor islands with a diameter that is small
enough to make their charging energy greater than kBT (where kB is Boltzmann's
constant and T is the operating temperature). Precisely two mobile electrons
are loaded in the cell and can stir to different quantum dots in the QCA cell
by means of electron tunneling. The lines connecting the quantum dots in Fig.
1 represent tunneling paths. Coulombic repulsion will originate the electrons
to occupy only the corners of the QCA cell resulting in two specific polarizations.
The cell is charged with two excess electrons, which can be permitted, to tunnel
between the different quantum dots by a clocking mechanism. These electrons
tend to occupy diagonally opposed dots due to their Coulomb force repulsion.
The Coulomb force repulsion is also accountable for the transfer of information
from one cell to an adjacent cell.
For an isolated cell there are two energetically minimal equivalent arrangements of the two electrons in the QCA cell denoted cell polarization P = +1 and cell polarization P = 1. Cell polarization P = +1 represents a binary 1 while cell polarization P = 1 represents a binary 0. Quantumdot Cellular Automata (QCA) uses majority gate as a fundamental logic primitive.
When five QCA cells arranged in a cross pattern, they form a majority gate
as shown in Fig. 2. The fundamental QCA logical circuit is
the threeinput majority gate. Computation will be performed with the majority
gate by driving the device cell (central cell in Fig. 2) to
its lowest energy state.

Fig. 1: 
QCA cell polarizations and representations of binary 1 and
binary 0 

Fig. 2: 
Majority gate consists of five quantum dots 
This happens when it assumes the polarization of the majority of the three
input cells. The device cell will always assume the majority polarization because
it is this polarization where electron repulsion between the electrons in the
three input cells and the device cell will be at a minimum.
THE PROPOSED NEW MINIMIZATION TECHNIQUE
In this study, a new technique for realization and minimization of logic functions
in the form of Majority of majority functions. To obtain our reduction, we first
create the KMap corresponding to the Boolean function we intend to represent.
Then, in each cell that has a Boolean 1 value we rename it 11. The cell with
Boolean 0 will be given the value 00. This will produce a new KMap that will
be pointed to as JMap denoting majority function logic minimization map that
mimics KMap but instead minimizing functions to the ANDOR plane, it minimized
in to the Majority of Majority plane.

Fig. 3: 
T shape that represent a majority function M (a’, b,
c) = a’b+bc+ca’ 

Fig. 4a: 
KMap of a Boolean function, F = a’b’c+ a’bc’+abc+abc’ 

Fig. 4b: 
JMap of a Boolean function F = a’b’c+ a’bc’+abc+abc’ 
We then express the Boolean in function in terms of majority of majority functions.
Minimization of such function will result in a majority function of majority
functions: M (M1, M2, M3).
Minimization of M will be done on the JMap by obeying the following rules:
• 
Cover any “11” at least twice 
• 
Cover a “00” at most once 
• 
The covering of the terms in the JMap is the same as in the KMap such
as covering adjacent minterms as much as you can as long as their number
are in the form 2^{n} 
• 
Cover the minterms in the form of T shape that represent a majority function 
• 
Overlapping is allowed, as long as rules 1 and 2 are enacted 
We now demonstrate the application of this method to various Boolean functions.
In Fig. 3, the T shape that represent a majority function
M (a’, b, c) = a’b+bc+ca’ is represented. The KMap and JMap
of the Boolean function F = a’b’c+a’bc’+abc+abc’ is
shown in Fig. 4a, b.
The innovative minimization JMap algorithm can produce different minimized
logic functions, its output is not unique it just like KMap it produces the
minimized function with the minterms covering as shown in Fig.
5, where the steps of the minimization of the function F is described and
the minimized function is: F = M (M (a’, b, c), M (a’, b, c’),
M (0, b, c’)) (Fig. 5d).

Fig. 5a: 
Step 1 of the minimization of the Boolean function F = a’b’c+a’bc’+abc+abc’ 

Fig. 5b: 
Step 2 of the minimization of F = a’b’c+ a’bc’+abc+abc’ 

Fig. 5c: 
Step 3 of the minimization of F = a’b’c+ a’bc’+abc+abc’ 

Fig. 5d: 
The steps of the minimization of F F = M (M (a’, b, c),
M (a’, b, c’), M (0, b, c’)) 
DESIGN OF A FULLADDER CELL IN QCA PARADIGM
In this study, we are going to design a full adder cell built in QCA model.
We are Considering the computation of the sum of a fulladder.

Fig. 6a: 
The KMap for the sum logic function of the full adder Sum 

Fig. 6b: 
The JMap for the full adder Sum 

Fig. 6c: 
The JMap for the Sum= M (M (X, Y’, C), M (X’Y’C),
C’) 

Fig. 6d: 
The full adder Sum Sum1 = M ((M (X, Y, C))’, M (X, Y,
C’), C) 

Fig. 6e: 
JMap minimization of the function Cout = XC+XY+YC 

Fig. 6f: 
Schematic diagram for the full adder Sum1 and Cout 
It will be represented in the KMap as shown in Fig. 6a.
While the utilization of the JMap technique on the Sum function is depicted
in Fig. 6bd. The minimization of the
carryout function is depicted in Fig. 6e.
The realization of the full adder cell using minimization depicted in Fig.
6d. will require three majority functions and an inverter as depicted in
Fig. 6f.
NEW ADDER DESIGN WITH FEWER QUANTUM DOTS
In this study, we are going to introduce a new adder design that can minimize the number of quantum dots required to realize the nbit propagation adders.
Equation 2 can be arranged as follows:
The ripple carry adder will be realized by Eq. 1 and 4.
Cell1 in the ripple adder will start y inverting Cin and it will produce S and
Cout’. The rest of the cells is realized using the previous Cout’
as the Cin’. The last cell in the ripple carry adder will invert its Cout’
to the final Cout.

Fig. 7a: 
The minimized JMap for the Cout’=M (X’, Y’,
C’) 

Fig. 7b: 
The minimized JMap for Sum3 

Fig. 8: 
Schematic diagram for the full adder Sum3 and Cout’ 
Table1: 
Implementation on diffrent gates using our methodology 

The majority gate realization is depicted in Fig. 7 and
8.
RESULTS AND DISCUSSION
In this study, we are comparing our methodology to the technique demonstrated
by the authors in (Zhang et al., 2004). We minimized
all the 13 functions that the authors demonstrated in their paper. The comparison
is depicted in Table 1, where it can be concluded that our
technique is producing the same results but with some advantageous features,
the first feature is the ease of use of JMap and the second feature is that
JMap can give alternative solutions if more than one exists. The third and
most important feature is that JMap constitutes a basis for Algebra based on
majority of majority minimization. The fourth feature is that the JMap technique
can be expanded to more than three variables.
XJMAP: PROPOSED EXTENDED PROCEDURE TO BUILD MAJORITY EXPRESSIONS
Here, we suggest an extended procedure method to build simplified majority expressions for a given Boolean function using JMap. The method uses the subsequent algorithm to produce a majority gate expression of the threevariable Boolean function.
Algorithm 1: 
XJMap 

Algorithm 2: 
Construct XJMap 

An Example of XJMap will be depicted by executing the minimization of the
following function:
The corresponding JMap is depicted in Fig. 9a. There are
four covering comprising the minimal cover. The two covers, B and A are overlapped,
but not overlapped with the other covers. The other two covers A′B′C′
and A′B′C′ are overlapped together, but not with the first
two covers. Therefore, we have to extend the technique to use the XJMap technique.
This JMap will be divided into two j maps, that are depicted in Fig.
9b. The minimization of the XJMap will be the minimization of each JMap
separately, then the Majority of at most three j maps together are combined
to form nested Majority of Majority functions.

Fig. 9a: 
JMap of F = ABC+A’B’C’ 

Fig. 9b: 
XJMap of executing the minimization of the following function:
F = ABC+A’B’C’ 
The minimum cover will be as follows: M(M(A,B,0), M(C’,M(A’B’C),
0),1)
EXTENSION TO FOUR AND FIVE VARIABLES BOOLEAN FUNCTIONS
We are going to extend XJMap for 4 and 5 variables Boolean functions.
Algorithm 3: 
XJMap45 variables 

Algorithm 4: 
Construct XJ45map 

An Example of XJ45map will be depicted by executing the minimization of the following function: F=ABCD+A’B’C’

Fig. 10a 
:J45map executing the minimization of the following function:
F = ABCD+A’B’C’ 

Fig. 10b: 
XJ45map The final solution will be as follows: F = M (C’,
M (A’B’D’), 0), M (AB,CD,0), 1) 
The originated J45 map and the corresponding XJ4 map are depicted in Fig.
10. The final solution will be as follows:
F=M (C’, M (A’B’D’),
0), M (AB, CD, 0), 1) 
The QCA logic circuits that are shown in Fig. 6, 8
and 9 were simulated using QCAD esigner. The simulation procedure
is depicted in the following steps as defined in (Zhang
et al., 2004).
Steps:
• 
The first step is to generate the layout of the QCA circuit 
• 
The second step is to setup the circuit clocking 
• 
The third step is to set up the vector table simulation to simulate the
circuit 
The QCA cells are made of 2 nm quantum dots. The cells are separated by 10
nm. This design requires only 36% of the area used by the original design f
these circuits withut the Majority of Majority minimization techniques. While
maintaining the same clocking performance. Also the design for the gates in
Fig. 6 and 9 achieved decrease in area
by 40 and 35% from the original designs. The decrease in area requirement is
consistent with the theoretical reasoning.
CONCLUSION
This research is an effort to investigate the potential of a nanotechnology
(QCA) as being a viable alternative to CMOS VLSI. In this study we are concentrated
particularly on logic design issues. Namely, a solution for potential systematic
extensible technique for QCA logic realization and minimization was developed.
Most importantly, this solution can be applied and scaled to more complicated
designs. Emerging technologies specifically, Quantumdot Cellular Automata (QCA)
uses majority gate as a fundamental logic primitive instead of the ANDOR combination
gates. We report an intuitive systematic methodology JMap minimization technique
for reduction of 3variable Boolean functions into a simplified majority representation.
The main contribution of this work is the general and systematic way the new
approach can be applied. All attempts in the literature were complex and lacked
the systematic approach we are offering (Toth and Lent,
2000).