Modular Dissipation Analysis for QCA - Field-Coupled Nanocomputing: Paradigms, Progress, and Perspectives - page 360

where Δ↕E B ≤ d n is the amount of energy dissipated in the n -th dissipation zone on
each use. Lower bounds on the Δ↕E B ≤ d n , which are fixed by the amount ΔI d n
of information irreversibly lost from the dissipation zones on each use, would be
obtained from dissipation analyses performed individually for each dissipation zone.
The advantages of this approach would, of course, be questionable if new
decomposition rules had to be sought and codified anew for each dissipation
zone in every circuit that is to be analyzed. A more productive strategy would
be to tie circuit decomposition rules to design rules rather than to individual cir-
cuits, so a single set of decomposition rules applies to all circuits realized within a
given nanocomputing paradigm that are designed according the specified design
rules. If this is possible, it would enable the analytical simplification promised by
the modular approach while imposing only mild constraints on the allowed space
of circuit structures: Adherence to specified design rules is a modest and familiar
constraint from a circuit design perspective. Below we show that decomposition
rules can indeed be formulated for Landauer-clocked QCA circuits that are con-
structed according to example design rules that reflect common design practice,
and we demonstrate that simplified modular analysis based on these rules yields
bounds identical to those of the general approach for a concrete QCA circuit.

3 Illustrative Example: A QCA Half Adder

As a vehicle for exploring and demonstrating the modular approach, we consider
the Landauer-clocked QCA half adder circuit depicted in Fig. 2 . The cell layout,
timing diagram, and logic diagram are all shown, with color coding correspond-
ing to clocking zones. This circuit, which is designed so it is free of wire crossings,
is composed of 135 cells that make up five majority gates (each with one fixed
input), two inverters, and required interconnects. The cells in each clock zone
cycle periodically through switch, hold, release and then relax phases to propa-
gate input information through the circuit via Landauer clocking [ 8 ]. New input
data is loaded into the circuit every time the adder cells adjacent to the input
cells A and B go through the switch phase of the clock cycle. The corresponding
outputs are available in the adder cells adjacent to the output cells S and C two
full clocking cycles later, and are loaded into the output cells when these clock
zones are in the subsequent hold phase.
We sketch dissipation analysis of this circuit via the general approach in
Sect. 3.1 . In Sect. 3.2 , we present the set of QCA circuit design rules obeyed
by this adder design 1 , establish corresponding decomposition rules that enable
modular dissipation analysis of any circuit that adheres to these design rules,
and employ these decomposition rules in a simple modular dissipation analysis

1 Studies of a similar QCA adder circuit, including detailed dissipation analysis via the
general approach, have been presented elsewhere [ 3 , 5 ]. Differences in the adder of
these previous studies and the adder of this work (Fig. 2 ) stem from reconciliation
of the previous design with the QCA design rules presented in Sect. 3.2 . Statement
of these design rules is deferred to Sect. 3.2 since the procedures used for dissipation
analysis via the general approach are independent of them.