Bayesian network based modeling and reliability analysis of quantum cellular

International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
INTERNATIONAL JOURNAL OF ELECTRONICS AND
0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 1, January- February (2013), © IAEME
COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 4, Issue 1, January- February (2013), pp. 131-145
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BAYESIAN NETWORK BASED MODELING AND RELIABILITY
ANALYSIS OF QUANTUM CELLULAR AUTOMATA
CIRCUITS

Dr.E.N.Ganesh
Professor ECE Dept, Rajalakshmi Institute of Technology,Chennai – 600124
TamilNadu, India
Email: enganesh50@yahoo.co.in

ABSTRACT

Quantum cellular automata (QCA) is a new technology in nanometer scale as one of
the alternatives to nano cmos technology, QCA technology has large potential in terms of
high space density and power dissipation with the development of faster computers with
lower power consumption. This paper considers the problem of reliability analysis of Simple
QCA circuits at layout level like QCA latches and NOT circuit. The tool used to tackle this
problem is Bayesian networks (BN) that derive from convergence of statistics and Artificial
Intelligence (AI). It consists of the representation of probabilistic causal relation between
variables of a system. Using this we have transformed QCA circuit in to Bayesian framework
to find the probability of getting correct output in terms of its polarization with respect to its
input configuration and temperature. Reliability analysis also discussed for finding the
defective cells in QCA circuit. This will increase overall efficiency of circuit and hence speed
of the circuit with lower power consumption.

Keywords: BN – Bayesian network, Quantum cellular automata, Reliability, Conditional
probability, Join probability distribution

1. INTRODUCTION

This paper considers the problem of reliability analysis of QCA circuit organized in
parallel and/or in serial given the reliability of the Input cells. The mathematical tool used to
tackle this problem is Bayesian networks. Reliability analysis of systems is very important in
order to be able to deliver errorless QCA cell at the output. A given QCA system is often
composed of many cells organized in serial and parallel, whose failure of one cell in serial

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may cause the failure of the whole system or at least reduce its performance. In terms of
reliability there are two types of organizations of cells, organization in serial and organization
in parallel. These organizations impact on the reliability of the resulting circuit. If a circuit
consists of n cells Ci, i=1,…n in serial, the system will be performing well if and only if each
component is performing well, so if Pi , is the reliability of the ith component, then the
reliability of the system Ps is given by
Ps = ∏i = 1 to n Pi (1)
On the contrary if a system consists of n cells Ci, i=1,…n in parallel, the system will perform
well if at least one of these cells perform well, so if Pi is the reliability of the ith component,
then the reliability of the system Pp is given by
Pp = 1 - ∏i = 1 to n ( 1 - pi) (2)
Eq. (1) and (2) constitute the fundamental relations for computing the reliability of a QCA
circuit because any system will consist of cells or groups of cells in serial and/or in parallel.
Quantum-dot Cellular Automata (QCA) is an emerging technology that offers a revolutionary
approach to computing at nano-level [1][2]. A dot can be visualized as well. Once electrons
are trapped inside the dot, it requires higher energy for electron to escape. The fundamental
unit of QCA is QCA cell created with four quantum Dots positioned at the vertices of a
square. [2] [3.]. Fig 1.a and 1.b below shows quantum cells with electrons occupying
opposite vertices.

1.a P = +1 (Binary 1) 1.b P = -1 (Binary0)
Fig1 QCA cells with four Quantum dots. [1][3][4][5]

This interaction forces between the neighboring cells able to synchronize their polarization.
Therefore an array of QCA cells acts as wire and is able to transmit information from one end
to another [6] [7][8][9][10]. Figure 2 and 3 Majority functions of QCA Cell.

Fig 2.and 3 Majority AND, OR gate [3] [4][5][6]

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Clocking is the requirement for synchronization of information flow in QCA circuits [11]
[12] [13]. It requires a clock not only to synchronize and control information flow but clock
actually provides power to run the circuit [14] [15]. Bayesian Networks (BN) derive from the
convergence of statistical methods that permit one to go from information (data) to
knowledge (probability laws, relationship between variables,…) with Artificial Intelligence
(AI) that permits computers to deal with knowledge (not only information), (see for example
[16]). The terminology BN comes from Thomas Bayes’s [17] 18th century work. Its actual
development is due to [17]. The main purpose of BN is to integrate uncertainty into expert
systems. BN consist in a graphical representation of the causality relation between a cause
and its effects. Figure 4 show that A is the cause and B its effect.

A B

Fig 4 Causality representation in BN

But as relation of causality is not strict, the next step is to quantify it by giving the probability
of occurrence of B when A is realized. So an BN consists of an oriented graph where nodes
represent variables, and oriented arcs represent the causality relation and a set of
probabilities. A rigorous definition of a BN is given in [18].Let us consider an acyclic
oriented graph g = (v,a) where v and a represents set of nodes and the arcs in the graph. A
trial E with whom there is associated a finite probability space and given n random variables (
Xi )1 < i < n, a and E defines Bayesian network, noted B = (G,P). There exists a bisection
between the nodes of G and var(Xi). The factorization property for this is

P(X1,X2 …….Xn) = Π P ( Xi / C(Xi)) (3)

Where C(Xi) depends on the set of causes(parents) of Xi. P (X1, X 2,……Xi) Eq.(3) is the
probability of simultaneous realization of variables X1, X2 ….Xi and P(X i / Yi) is the
conditionality probability. The main purpose of Bayesian networks is to propagate certain
knowledge of the state of one or more partitioned nodes through the network so that one shall
learn how the belief’s of the expert ion the Bayesian network will change, given B = (G,P)
and set of nodes it returns to compute P ( Xi / Y i ). Using the properties of chains, trees
networks and the properties of conditional probability, algorithms can be derived to
propagate certain knowledge in term of modifying the belief. BN is completely determined
by its structure and some parameters, namely a priori probabilities of nodes without parents
and conditional probabilities of intermediate nodes for different configurations of states of
their parents. The basic cell of the QCA circuit will be the component of which reliability
will be available. Clocked QCA circuits are considered here, the reliability of the cell can be
found in terms of probability of getting correct output of the output cell or group of cells in
that QCA circuit. Before going in detail about reliability analysis let us deal in detail about
simple QCA latch and transforming the latch into Bayesian network according to [19][ 20].

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2. BAYESIAN MODEL OF COMPUTATION

Consider a simple QCA latch of figure 5 drawn through QCA simulator from [21]. Figure 6
shows layout model Bayesian network of QCA latch from [22].

Figure 5 shows the simple QCA latch from [21]

Figure 6 shows the simple Bayesian model of QCA latch.

Each chance node in figure 7 represents a random variable of each cell in figure6 and directed
arc represents direct causal relation with the parent nodes. A link is directed to child node
from the parent node. Let us consider 5 QCA cells indexed in a manner of X1, X2, X 3, X 4
and X5. X1 be the input and X5 be the output cell. We use two state approximate model of
single QCA cell [20], Two state model can be derived from the quantum formulation based
on all possible configuration of pair of electron in a cell. [23] Each state can be observed in
one of possible state logical 0 ( x0) or logical 1 ( x1). The probability of observing a state is P
(Xi = xi), x denotes the states be in logical 0 or 1. Polarization of cell in terms of state
probabilities can be found from conditional and joint probabilities. The joint probability of
observing a set under steady state assignments for the cell can be determined from quantum
wave function which is cumbersome and requires quantum wave function calculations.
Instead as in [20] consider a joint wave function of two cells in terms of product of two
variables and representing the product as factored representation. By using Hocktree Fock
approximation as in [24] determine state probability, but by determining polarization as in
[25] the polarization can be determined from the given equation.

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P = (ρ1 + ρ3 ) - (ρ2 + ρ4) / (ρ1 + ρ2 + ρ3 + ρ4 ) (4)

Where ρ represents the expectation values of charges of two cells, Using the above equation,
Eq.(4) polarization values can be found, expected values of polarization being directly
entered in the conditional probability table of a random node or a cell and hence using Genie
software the joint probability distribution of the circuit can be found. Using this method it is
possible to find the probability of getting output (polarization) for a given temperature and
input being found. Consider the QCA latch in figure 6, the joint probability function is
decompose in to product of individual conditional probability functions as

P ( X1, …..X5 ) = P(X5 / X4, X3). P( X4 / X3 , X2) …..P(X2 / X1) P(X1) (5)

But Eq.(5) hold for linear random nodes with easy message passing technique to find joint
distribution, we have consider not a linear cell instead a tree like structure. We have used here
[18] Joint tree method of message passing technique for propagating the polarization
information. Since we have considered as tree like structure, we used above method to form
clusters, we decompose the network in to clusters that form tree structures and treat the
variable in each cluster as compound variable that is capable of passing message to its
neighbor, the above network can be clustered in to three clusters X1,X2 and X3 as one
cluster, X2, X3, and X4 as second cluster and X4 and X5 as another cluster. The direct causes
or parent of a node depends on inferred causal ordering. The exact message passing scheme
depends on tree structure, whose nodes are clique of random variables. This tree of cliques is
obtained from the initial DAG structure via a series of transformations that preserve the
represented dependencies. These transformations are constructing moral and chordal graph
via constructing triangulated undirected graph. The moral graph is obtained by considering
DAG structure to a triangulated undirected graph structure called moral graph. Chordal graph
is obtained from moral graph by a process of triangulation. Triangulation is the process of
breaking all cycles in the graph to be composition of cycles over just 3 nodes by adding
additional links. There are many possible ways for achieving this. At one extreme, we can
add edges between every pair of nodes to arrive at final graph that is complete. When we
transform DAG to junction of cliques the preservation of parameters dependencies must be
taken care, here in this process automatically the dependencies are preserved. Figure 7,8, 9
shows the junction of tree clique for the example considered.

Figure 7 shows the moral graph of QCA Bayesian network

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Figure 8 chordal graph of QCA Bayesian network

Figure 9 Junction of cliques

Moral graph is obtained by drawing a line between x2 andx3, hence it forms a tree
structure and using minimum elimination of clique sets and running intersection property as
in [21], a three node clique set is formed which is a subtree from the tree, three set of clique is
formed using intersection property, [18]a chordal graph is the triangulation process of finding
junction clique set. Finally three sets Clique x1x2x3 , x2x3x4 and x4x5 with potential
function as shows in the figure 9. Clique sets c1,c2 and c3 are running with intersection
property, a junction tree between these clique is formed by connecting each predecessor to
the present clique set. A potential function is associated with each clique set formed from the
conditional probability of the variables in the set. Each potential function is determined by
the product of conditional probability functions mapped to that clique. Eq.(6) gives Product
of Cond .Probability.

C(x2x3) = ΠvuPa(x) ε x P ( v / Pa ( X)) (6)

The joint probability distribution is given by eq.(7)

P(x1…..x5) = Π c(xi) i = 1…5 (7)

Now the tree structure is useful for local message passing. Given any evidence, messages
consist of the updated probabilities of the common variables between two neighboring
cliques. We used average likelihood propagation algorithm for finding expected polarization
of output cell with respect to temperature and types of input. The probabilities are propagated
through the junction clique by local message passing. Messages are passed from leaf clique to
root clique, then again the present clique pass message to next clique and so on. Based on the
values of c the marginal are found z(yi) for each clique, when message passed first to second
clique, a scaling factor being sent to first clique to scale the marginal for moving to the next
clique, this way message being transmitted.

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Z(yi) = Σ c( xi) (8)

Z(yi) = Zi(yi) / Zj(yi) .Zj(yi) (9)

Eq. (8) gives summation of all the values of potential function to give marginal function,
Eq.(9) is rescaled depending on present junction clique and next junction clique scaling
function. Hence for each passing marginal function are rescaled and value is changed for next
clique. We used this technique for QCA latch cell for finding probability of getting correct
output at the output cell with respect temperature and inputs. Figure 10 a. shows the
probability of getting correct output of logic 1 and logic 0 with respect to input configurations
1 and 0. Figure 10.b and c shows probability of getting correct output in combinational and
Flip flop QCA circuits. The output value through Bayesian network gives nearly equal to
simulated value, hence Bayesian tools can be used for modeling nano circuits.

Fig.10 a. shows the Probability of getting correct output of QCA latch with respect to input
0 and 1.

Fig 10.b.Probability of getting correct ouput in Comb circuits through Baeysian networks

137


Fig 10.c.Probability of getting correct ouput in Flip flops circuits through Baeysian
networks

2.1 Simulation Results

The simulated values from QCA Designer circuit is taken as input to the bayseian network
and cryogenic temperature of 10 K is assumed throughout the Bayesian network. Applying
the algorithm and probability of getting correct output is calculated. Figure 10 a to c shows
the probability of getting correct ouput for latch, combinational and sequential circuits. It is
found that the value more or less the simulated value and the baysian algorithms used to
model the nano cicuits for device failures. The next section discuss about the reliability
issues of QCA circuits and therbyu the reducing the faults occuring in QCA circuits.

3. RELIABILITY ANALYSIS OF BAYESIAN STRUCTURE

QCA circuits are constructed by serial and parallel structures of QCA wires.
Reliability analysis can be done on this circuit in order to deliver errorless QCA cells in the
QCA circuit. The reliability analysis of these circuits depends on reliability of input cells so
that errorless QCA cell can be constructed. Bayesian networks used in chapter 5 are used for
reliability analysis of QCA circuits. The failure of a QCA cell in QCA circuit may cause the
failure of whole system and a single defective cell in parallel may reduce the overall
performance of a QCA circuit. If a circuit consists of n cells Ci, i=1,…n in serial, the system
will be performing well if and only if each component is performing well. So if Pi , is the
reliability of the ith component, then the reliability of the system Ps is given by equation 3.1.

Ps = π i =1ton Pi (3.1)

On the contrary if a system consists of n cells Ci, i=1,…n in parallel, the system will
perform well if at least one of these cells perform well, so if Pi is the reliability of the ith
component, then the reliability of the system Pp is given by equation 3.2.

PP = 1 − π i =1ton (1 − Pi ) (3.2)

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Equation 1 and 2 constitute the fundamental relations for computing the reliability of a QCA
circuit because any system will consists of cells or groups of cells in serial and/or in parallel
manner. Consider a simple QCA latch of figure 11 drawn through QCAdesigner tool. All
QCA circuits are transformed into Bayesian framework and simulated using Genie software.
Each cell in the QCA circuit is a node in Bayesian network. Each chance node has
Conditional probability table through which joint probability of entire circuit is found.

Figure 11.a QCA latch circuit drawn. Figure 11.b Bayesian using QCA designer
representation

Each chance node in figure 11 represents a random variable of each cell and directed arc
represents direct causal relation with parent nodes. The structure of the resulting BN for QCA
latch cell for reliability analysis consists of three groups of nodes, let Nc be the node without
parent say X1 Nint is the intermediate nodes consists of Nint,s serial intermediated node and
Npar,s parallel intermediate node. Here X3 and X4 are parallel intermediate nodes with X1 as a
parent node. X3 with X4, X2 with X4 are serial intermediate node and destination node as Nout
(X5). Figure 12 shows the framed Bayesian structure for reliability analysis.

Figure 12 reliability analysis of QCA Latch using Bayesian network

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The transformation of QCA latch Bayesian structure to reliability analysis is by
considering each cell or node as serial or parallel (assuming all cells are in clock zone). This
analysis is useful to study the cell failure in terms of polarization and to identify the correct
QCA cell for the circuit failure. Once defined the structure of the BN, its parameters must be
determined: the parameters to be entered in to the Bayesian network are priori probabilities of
states of nodes without parents, that is nodes of type Nc and the conditional probabilities of
intermediate nodes, that is nodes of type Nint,s and Nint,p, knowing some configurations of the
states of their parents. Though more than two states can be considered for each node, here it
is considered that each node has only two states: failure (F) or no failure (NF). The
generalization to more than two states would not be very difficult. The parameters of the BN
consist of two types:
1. A priori probabilities of basic components given by their reliability or the complement of
their reliability.
2. Conditional probabilities of the intermediate nodes given the configurations of their
parents. A priori probabilities of nodes of type Nc are fully determined by the reliability of the
corresponding components.

4.0 RELIABILITY ALGORITHM OF QCA CIRCUITS.

If nc is the basic cell of Nc and pi is the reliability of the component Nc,
P ( N ci = NF ) = Pi and
i
P ( N = F ) = 1 − Pi , i = 1....nc
c (4.1)
Similarly for Nint,p and Nint,s Parameters can be defined as
P ( N int, p = NF / C ( N int, p )) = 0UN ∈ C ( N int, p ), N = Failure (4.2)
Else
P ( N int, p = NF / C ( N int, p )) = 1 (4.3)
and
P ( N int, p = F / C ( N int, p )) = 1UN ∈ C ( N int, p ), N = Failure (4.4)
Else
P ( N int, p = F / C ( N int, p )) = 0 (4.5)
P ( N int, s = NF / C ( N int, s )) = 1UN ∈ C ( N int, s ), N = NoFailure (4.6)
Else
P ( N int, s = NF / C ( N int, s )) = 0 (4.7)
And
P ( N int, s = F / C ( N int, s )) = 1UN ∈ C ( N int, s ), N = NoFailure (4.8)
Else
P ( N int, s = F / C ( N int, s )) = 1 (4.9)

Equations 6.3 to 6.11 gives conditional probability defined for intermediate nodes with conditions
failure (F) and nofailure (NF). The polarization is defined for each cell, for example
P (Ci = NF ) = 0.9
(4.10)
P (Ci = F ) = 0.1, i = 1....4.

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Full polarization value (logic 1) is 0.9 to +1 and -0.9 to -1, partially polarized values are 0.5
to 0.8 or -0.5 to 0-0.8, and logic 0 polarization values are -0.5 to + 0.5. Let X1 cell be the Nc
node, interaction between X2 and X4 be G1 as intermediate node, interaction between X3
and X4 be G2, second node be either G1 or G2 of Nint,p, third intermediate node be Nint,s
and finally destination or output node Nout. Now the Bayesian network defined over QCA
latch according to above statements as shown in figure 13, assume logic 1 as input decision
and searching for utility at the output node in terms of its probability.

Figure 13 a Decision and utility nodes for evaluating logic1 probability at the output node
which is 0.832 with intermediate node CPT is shown in table 4.1.

Table 4.1 Intermediate node G1 node CPT

Table 4.1 shows the intermediate CPT table, if the entries in the CPT are changed slightly
from 0.9 to 0.6 less polarized that leads to decrease the probability then intermediated nodes
has to be checked. Next is to examine the cell which is less polarized by G1 or G2 nodes.

Figure 14 Decision and utility nodes for evaluating logic1 probability at the output node
which is 0.822, decreased than in figure 13.

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Table 4.1 and 4.2 gives G1 and G2 CPT, the failure of the whole system means that
component X4 and cell G1 will surely fail. The element responsible for the failure of G1 is X3
and X4 with 71.00% of chances against G2 with 29.00% of chances. So the system critical
elements, in terms of reliability, are G1( X3 and X4) and X4 . One can simulate many other
configurations to find the error occurred in terms of probability of output.

Table 4.2 CPT of node G2 due to X2 and X4 QCA cells
(0.71 – 1, 0.3 -0)
Cell X2 Failure No - Failure
Cell X4 F NF F NF
Failure 1 0 0 0
No 0 1 1 1
failure

Table 4.3 CPT of node G1 due to X3 and X4 QCA cells
( 0.71 -1, 0.3 -0)
Cell X3 Failure No - Failure
Cell X4 F NF F NF
Failure 1 1 1 0
No 0 0 0 1
failure

4.1 QCA NOT circuit with CLOCK Zone

A QCA circuit organized in parallel is more reliable than one which is organized in serial.
Serial QCA circuits require clocking for group of cells whereas a parallel QCA cell (circuit
with more layers in parallel) which has clock zones running for more no of layers. One way
to improve the reliability of a QCA circuit is to put two or more layers in parallel instead of
one layer. But this process will increase the complexity of the circuit. An example of a
system composed of QCA cells is given in Figure 6.5. The system has two main parallel
branches, one branch consisting only of the components C4 and the other one consisting of
component C1 in serial with a group formed by C2 and C3 in parallel.

Figure 15.QCA not circuit with serial and parallel layers a input and y output

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C1 C2

C3

C4

Figure 16 Equivalent Component structure for reliability analysis of QCA not circuit

Table 4.4 CPT of node G2
Cells c2 Failure No - Failure
Cells c3 F NF F NF
Failure 1 0 0 0
No 0 1 1 1
failure

Table 4.5 CPT of node G1
Cells G2 F NF F NF
Failure 1 1 1 0
No 0 0 0 1
failure

Table 4.6 CPT of intermediates node.
Cells G1 F NF F NF
Failure 1 0 0 0
Nofailure 0 1 1 1

Figure 15 is QCA NOT circuit and 16 gives the reliability transformation of finding the
components in 15. QCA NOT circuit has two parallel branches and one loop in one of the
parallel branches. As discussed in the previous section, C1 has group of cells in upper parallel
arm, C2 and C3 form the smaller loop in upper parallel arms and C4 form the lower parallel
arm. Let us define the components (group of cells)
• C1, C2, C3 and C4 are nodes of type Nc;
• G2 is a node of type Nint,s formed by regrouping the component C1 act in parallel
components C2 and C3;
• G1 is a node of type Nint,s formed by regrouping the component C1 and the subsystem
G2;
• Finally the subsystem G1 and the component C4 act in parallel on the system so that
this one is a node of type Nint,p.

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Table 4.4, 4.5, 4.6 gives Conditional Probability of cells of G1 and G2 intermediate nodes.
The failure of the whole system means that component C4 and subsystem G1 will surely fail.
The element responsible for the failure of G1 is C1 with 91.74% of chances against G2 with
9.17% of chances. So the system critical elements, in terms of reliability, are C1 and C4. The
group of cells in C1 and C4 have to be examined to find particular defect cell in terms of its
probability of getting correct polarization or not. The same way analysis can be carried out
for QCA system to find defective cells.
The reliability of QCA cell can be found using this Bayesian network of transforming
QCA circuits to its Bayesian framework. The QCA latch circuit failure is due to one of the
intermediate nodes as its probability decreases when it is a defect one, similarly QCA not
circuit also is used to find reliability of group of cells in serial and parallel arms. This can be
extended for simulating errorless QCA systems. The extension of this work is to assume the
type of defects or device level uncertainties in QCA circuits and by using different algorithm
simulating its Bayesian framework to find defective cells in terms of its polarization.

5. CONCLUSION

Bayesian network can be used to find the probability based modelling of QCA cells in
terms of its temperature and input configurations. We have simulated the probability based
Bayesian modelling of finding the correct output of simple latch circuit. We discussed
reliability analysis of QCA latch and Not circuit, we found the probability decreases as we
move to higher nodes when present cell or node is a defect one, reliability of the cell can be
found using this Bayesian network of transforming qca circuits to its Bayesian framework.
We discussed qca latch circuit failure is due to one of the intermediate nodes as its probability
decreases when it is a defect one, we analysed QCA not circuit also to find reliability of
group of cells in serial and parallel arms. This can be extended for simulating errorless QCA
systems. We have not considered the type of defects occur for particular cell. The extension
of this work is to assume the type of defects or device level uncertainties in QCA circuits and
by using different algorithm simulating its Bayesian framework.

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