Modular Dissipation Analysis for QCA - Field-Coupled Nanocomputing: Paradigms, Progress, and Perspectives - page 359

they do not interact with one another - the dissipative cost of information loss
from subzones in a given computational step is additive and the net dissipative
cost of erasure for all subzones erased during the k th computational step can be
lower bounded as

Δ E B k ≥

−k B T ln(2) ΔI C ( k w D ( k ) (2)

w

C ( k )
w

−ΔI C ( k w D ( k ) where is the amount of information about the initial state of clock

w

( k )
w

( k )
w

( k )
w

during the k th subzone C that is locally and irreversibly lost from C ↗D

computational step [ 2 ].

2.2 Modular Approach

The general approach outlined above enables isolation and quantification of irre-
versible information loss in digital circuits implemented in a wide variety of
nanocomputing paradigms. Full account is taken of interactions between neigh-
boring circuit structures that, in some paradigms, can render information era-
sure locally reversible (e.g. the

operation in QCA). Circuit
regions associated with irreversible information loss - which may be related to
logic gates or other functional circuit blocks in ways that are not obvious - need
not be known a priori to apply the general approach, allowing fundamental dis-
sipation bounds to be obtained for new circuit structures and/or in unfamiliar
nanocomputing paradigms where the appropriate “rules of thumb” have not yet
been identified. This generality comes at an analytical cost, however, as the iden-
tification of data zones and subzones is required on each computational step to
allow tracking of information flow and proper isolation of irreversible information
loss. This process is laborious for large circuits analyzed via the general approach.
The modular approach of the present work allows these same dissipation
bounds, obtained “holistically” in the general approach, to be obtained under
certain conditions through a simplified “reductionist” procedure of ( i ) decom-
posing a circuit into smaller fixed regions or zones, ( ii ) evaluating dissipation
bounds for those zones that are necessarily dissipative (hereafter “dissipation
zones”), and ( iii ) adding the bounds for each dissipation zone to obtain a dis-
sipation bound for the full circuit. This simplification is possible, however, only
after it has been established that a circuit decomposition that properly captures
cross-boundary interactions can be performed and the appropriate decomposi-
tion rules have been identified.
Suppose that such a decomposition is possible for a given circuit, that N diss
dissipation zones d n are identified, and that each dissipation zone is “used” once
for processing of each input to the full circuit. Where this is the case, the bound
on the total energy dissipated in processing one input over a computational cycle
simplifies to

erase with copy

N diss

Δ↕E B ≤ d n ΔE diss = (3)

n =1