Reversible Logic Based Design and Test of Field Coupled Nanocomputing Circuits - Field-Coupled Nanocomputing: Paradigms, Progress, and Perspectives - page 135

approach to QCA computing; Sect. 8 presents the proposed multiplexer conserv-
ative QCA gate; Sect. 9 discusses design methodology for non-reversible testable
design based on MX-cqca gate; and Sect. 10 provides some discussions and
conclusions.

2 Background

A conservative logic gate is a multiple-output logic element in which the number
of 1s at the inputs is equal to that of the corresponding outputs. According to
[ 18 , 61 ], a conservative logic circuit can be considered as a directed graph whose
nodes are conservative logic gates, and the edges are wires of arbitrary lengths.
The fanout at the output is not allowed in conservative logic circuits. A con-
servative logic network can be reversible in nature if the one-to-one mapping is
maintained between the inputs and the outputs, while it will be irreversible
in nature if one-to-one mapping is not preserved. Researchers in [ 27 , 60 , 61 ]
have proved that: (i) in the event of unidirectional stuck-at-faults in a conserva-
tive logic network, either the number of 1s in its output set will differ from the
number of 1s in its input set, or the output set is correct; (ii) in a conservative
logic network the two vector test set, all 1s and all 0s, provide 100 % coverage for
unidirectional stuck-at faults. Any stuck-at-1 fault in the conservative logic cir-
cuit can be detected by setting all inputs to 0s followed by subsequent checking
of the outputs for the presence of any 1s. Any stuck-at-0 faults can be detected
by setting all inputs to 1s followed by subsequent checking of outputs for the
presence of any 0s. The comprehensive proofs can be referred in [ 27 , 60 , 61 ].

2.1 Conservative Reversible Fredkin Gate

The Fredkin gate is a popularly used reversible conservative logic gate, first
proposed by Fredkin and Toffoli in [ 18 ]. The Fredkin gate shown in Fig. 1 (a) can
be described as a mapping (c,

i 0 , i 1 )to( o 0 =c, o 1 =c ₓ i 0 +c i 1 , o 2 =c i 0 +c ₓ i 1 ),

where c, o 2 are the outputs, respectively.

Table 1 shows the truth table for the Fredkin gate which demonstrates that
Fredkin gate is reversible and conservative in nature, that is, it has unique input
and output mapping and also has the same number of 1s in the outputs as in
the inputs. Fredkin gate is also called as controlled swap gate as it can swap two
input bits

i 0 , i 1 are the inputs and o 0 , o 1 and

c 0 = 1. The controlled swap operation of the Fredkin gate
is illustrated in Fig. 1 (b), (c).

i 0 , i 1 when

2.2 Basics of QCA Computing

In this work, the conservative logic gates are implemented in the QCA nanotech-
nology, thus we are also providing the introductory material on QCA computing.
A QCA cell is a coupled dot system in which four dots are at the vertices of a
square. The cell has two extra electrons that occupy the diagonals within the
cell due to electrostatic repulsion. The cell polarization P measures the charge