Modular Dissipation Analysis for QCA - Field-Coupled Nanocomputing: Paradigms, Progress, and Perspectives - page 357

Operational Decomposition: In this analysis step, we perform a spacetime decom-
position of the circuit. We first define clock zones and subzones. A clock zone
is a set of representational elements that are simultaneously subjected to the
same control operation in any given time step. Each clock zone may consist of
physically disjoint subsets of representational elements - called clock subzones -
that do not interact with one another directly as they change state. We denote
the u th clock zone as C ( u ) and the l th clock subzone of C ( u )as C l ( u ). We, then,
define a clock step which represents a time step during which a specified set
of control operations are applied to various clock zones. We denote the assign-
ment of control operation φ t to clock zone C ( u )as( C ( u ); φ t ), the v th clock step
φ v } v of control opera-

tions to all clock zones. The restoration processes that rethermalize the bath and
recharge the artifact's local particle reservoirs after the associated control opera-
tion drives these subsystems from their nominal states is also included in a clock
step. Hence, we can define a clock cycle as one period of the periodic sequence
φ = φ 1 φ 2 φ 3 ... of clock steps applied to the artifact to enable its operation.
This allows us to define the computational steps and cycle. The compu-
tational step c k , defined for the η th input x ( η ) in an input sequence ...x ( η −
1) x ( η ) x ( η +1) ... ,isthe k th of K clock steps required for evaluation of the output.
The computational cycle is then represented as Γ ( η ) = c 1 ...c k ...c K of the K
clock steps required to fully implement the logic operation for the η th input x ( η )
including the phase that loads x ( η ) into the artifact, the phases that evaluate
the output ( x ( η )), and the phases that transfer the output to the outside world
and erase all information about the input from the artifact. We denote c 1 as
the LOAD phase and c K as the phase in which all correlation between the com-
putational state of the artifact and the i th referent is lost. The computational
cycle Γ ( η ) may include clock steps from multiple clock cycles, and, in artifacts
that pipeline input data, Γ ( η ) may exclude clock steps that implement opera-
tions belonging only to other computational cycles (e.g. Γ ( η− 1) and Γ ( η ) ), i.e.
clock steps that do not affect representational elements whose states depend on
the η th input. Thus, the η th computational cycle includes only clock phases that
contribute directly to evaluation of output in the information processing artifact.

is specifically defined as an assignment φ v : { ( C ( u ); φ t )

Cost Analysis: We move on to the calculation of total dissipative cost associated
with one computational cycle based on information dynamics which involves
data zones and subzones. For the η th computational cycle, the k th data zone is
the set of representational elements that, at the completion of the k th compu-
tational step c k , hold information about the input data x ( η ). A data zone may
contain clock subzones belonging to multiple clock zones, and need not include
all subzones belonging to any given clock zone. It may consist of physically dis-
joint subsets of representational elements - called data subzones - which do not
interact with one another directly during some or all of the computational steps.
We denote the data zone associated with computational step c k as D ( c k ), and
the w th data subzone of D ( c k )as D w ( c k ). Note that, regardless of the circuit
implementation, there is one data zone defined at the end of computational
step of the computational cycle from c 1 to c K− 1 . By definition, there are no