Modular Dissipation Analysis for QCA - Field-Coupled Nanocomputing: Paradigms, Progress, and Perspectives - page 355

concrete nanocomputing paradigm by bringing physical law directly to bear on
the underlying computational strategy that defines the paradigm.
For any given nanocomputing circuit operated under a certain clocking
scheme, a fundamental lower bound on the dissipative cost of executing the
circuit's computational function is determined via a two-stage process: abstrac-
tion and analysis . In the abstraction phase, an idealized physical abstraction
of the circuit and its operation is constructed. This abstract description is con-
structed so it captures the essential functional features of the underlying com-
putational strategy, implemented precisely as envisioned in the computational
paradigm. (We call this “paradigmatic operation.”) In the analysis stage, a
physical-information-theoretic analyses of the abstract circuit and its surround-
ings is performed for the time intervals relevant to each computational step
in the circuit's computational cycle. These step-specific analyses yield a lower
bounds on the amount of energy that is unavoidably dissipated into the circuit's
local environment under paradigmatic operation, including both the dissipa-
tion required to execute logically irreversible operations and other unavoidable
paradigm-dependent “overhead” costs e.g. particle supply costs required to main-
tain the computational “working substance” in transistor-based paradigms). The
dissipation bounds obtained for each computational step are finally summed
over all steps in the computational cycle to lower bound the input-averaged
energy cost of each computation performed by the circuit. These two phases are
described in further detail below.

Abstraction. The abstraction is composed of two main stages. First, we create
a physical abstraction of the circuit and its surroundings in a globally closed and
isolated universe. Second, in the process abstraction, we present assignments to
identify local physical operations each of which is decomposed into a control and
a restoration operations. The circuit's interaction with the other information-
bearing subsystems and the bath is described by the control operations. The
restoration operations represent the coupling between the remote environment
and the bath. The control and restoration operations provide the circuit evolu-
tion required to implement computation. We now discuss the abstraction proce-
dure in detail.
Physical Abstraction: The physical abstraction of the circuit and its surround-
ings is depicted schematically in Fig. 1 . The upper half of the figure represents the
computationally relevant domain. This domain includes an information process-
ing artifact which is the computing circuit of interest and computationally sup-

porting subsystems such as external registers and adjacent circuit stages, as well as
an input referent

A

that holds a physical instantiation of the input data that will
be processed by the artifact. The lower half represents the environmental domain
consisting of a heat bath

R

, which is the part of the environment that is in direct
thermal contact with the artifact and nominally at temperature T . The greater
environment

B

¯

includes heat reservoirs that “rethermalize” the bath and anything
else that is required to ensure that the universe is globally closed. Constructing a
globally closed universe as presented above enables us to assume it evolves unitar-
ily via Schrodinger's equation. The global unitarity, together with identification

B