Modelling Techniques for Simulating Large QCA Circuits - Field-Coupled Nanocomputing: Paradigms, Progress, and Perspectives - page 268

all QCA simulations performed to date, and are currently used to treat QCA lay-
outs in QCADesigner. While the use of the 2-state and relaxation-time approx-
imations have been validated in previous literature, the ICHA, which neglects
any correlations or entanglement between cells, is known to arrive at the incor-
rect ground state for many circuits. In spite of its shortcomings, the ICHA is
widely-used due to its relative simplicity and low computational overhead.
Fortunately, other methods for solving the dynamical behaviour of QCA
circuits exist, whose computational overhead exists somewhere in the solution
space spanned by the ICHA and a full quantum mechanical treatment. Using
the so-called coherence-vector formalism, the state variables of a QCA system
are separated into groups corresponding to the state of individual cells, and into
groups corresponding to the two-cell, three-cell, etc., operators. In doing so, we
are able to remove those groups from our calculations that do not contribute
to the system's dynamics in any meaningful way, and thus reduce the overall
computational requirements of treating a QCA circuit. As it turns out, only
the 2 nd -order operators (i.e., two-cell correlations) were large enough to play a
significant role in the dynamics, and thus allowing all higher-order correlations
terms to be either neglected or approximated using 2 nd -order operators. In doing
so, the complexity of solving for the dynamics of an N -cell QCA circuit reduces
from O (4 N )to O ( N 2 ). Further reductions in the complexity are also possible if
only nearest-neighbour interactions are considered. In such cases, the complexity
drops to O ( N ), which is on par with that of the ICHA. The Liouville equation
of motion is used to calculate the time-dependance for each of the system's
remaining operators. The collection of these equations form a co-dependent set of
ODEs which can be solved using standard ODE solvers. And lastly, interactions
between the cells and the environment are accounted for via the use of the
relaxation-time approximation which drives the QCA system towards its steady-
state. These methods for solving large QCA systems have been shown to offer
accurate results with relatively low computational overhead.
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