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data zones at the completion of step
c
K
. As a computation progress through
steps in the computational cycle, the data zone will change its size and topology
while propagating from input to output. Data subzones will generally split and
merge throughout a computation, generally changing in number from step to
step.
The total information loss is the sum contributions from information lost to
the bath in each computational step, with “information loss” for the
k
th
step
defined as the amount of information about the state of each subzone in the
(
k −
1)
th
data zone that is not in the state of the corresponding subzone in the
k
th
data zone, summed over all data subzones.
We assume that data subzones that do not interact with one another essen-
tially “act alone” for the purposes of dissipation calculations, as the interactions
that erase information about the prior state will necessarily be local. This high-
lights the importance of correctly identifying and classifying the relevant physical
interactions that occur throughout the computational cycle, as identification of
the data subzones - and thus the appropriate level of analysis for obtaining
dissipation bounds - depends on the nature of these interactions.
The dissipation bounds are then obtained as follows. In any given computa-
tional step, any information lost from
A
that is not completely transferred to
¯
A
results in local energy dissipation into the bath. Information is “lost” from
data subzone
D
w
(
c
k−
1
) during computational step
c
k
if, at the conclusion of
c
k
,
the initial states of erased clock subzones of
D
w
(
c
k−
1
) cannot uniquely inferred
from the final states of clock subzones
C
l
(
u
)
∗ D
(
c
k
) that interacted directly with
D
w
(
c
k−
1
) during
c
k
. The total dissipative cost of one computational step is taken
to be the sum of contributions from all informationally lossy data-subzone-to-
data-subzone transitions, and the cost of processing one input is then the sum of
contributions from each computational step. Based on these assumptions, only
one of the clock subzones contributes to the dissipative cost associated with
each data subzone in the
k
-th step. For the
w
-th data subzone and
k
-th step,
this clock zone is denoted
(
k
)
w
. The total dissipative cost of one computational
step is obtained by summing the contributions from all informationally lossy
data-subzone-to-data-subzone transitions, and he cost of processing one input
is obtained by summing of contributions from each computational step. This is
to say that the total energy cost is additive at the subzone level, i.e. the energy
costs associated with these individual data-subzone information losses can be
“cashed out” individually and summed over a full cycle to get
C
K
K
k−
1
Δ↕E≤
Δ↕E≤
k
=
Δ↕E≤
=
(1)
k
=1
k
=1
wⓦ{w}
k−
1
where
represent the average energy transferred to the bath in the
k
th
computational step and over the full cycle, respectively. This again assumes
that data subzones that do not interact with one another for the purposes of
dissipation calculations, as their environmental interactions will be local. With
this assumption - that the
↕E≤
k
and
Δ↕E≤
(
k
)
w
C
interact locally with the bath since by definition
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