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be solved before QCA devices can be implemented. Thus, studying the dynamics
of QCA systems has remained mainly theoretical up until now.
A great deal of theoretical and modelling work has been done on the topic of
QCA [ 37 - 41 ]. This research has aimed to capture the qualitative and quantitative
characteristics of QCA cells and of arrays of cells. Because of the diculties
inherent in solving the complete quantum mechanical problem, a number of
simplifying assumptions are typically made. These include a reduction of the
Hilbert space to two states per cell [ 39 ], treatment of intercell interactions via
a mean-field approach [ 37 , 38 ], and finally an assumption of exponential energy
relaxation [ 26 , 40 , 42 ]. In this chapter, we provide a thorough review of these
approximations and discuss the current methods for modelling the dynamics of
QCA circuits.
Simulation of QCA Systems. In the proposed QCA implementations, a
QCA cell will feature 4-6 quantum dots and two mobile electrons [ 1 , 20 , 43 ].
A schematic diagram of a single QCA cell is shown in Fig. 1 (a). This figure
shows a cell consisting of four quantum dots arranged in a square pattern.
The two basis states of the QCA cell considered in this work is also shown.
While cells containing five or six dots have shown to improve the behaviour of
QCA devices [ 20 , 28 , 35 ], it greatly increases the numerical complexity of the cell
model, and thus we limit the scope of this chapter to four-dot cells.
Fig. 1. Schematic of the basic four-site cell. (a) The geometry of the cell. The inter-dot
distance is designated by d . (b) Coulombic repulsion causes the electrons to align along
the diagonals of the cell. These two bistable states result in cell polarizations of P =+1
and P = 1.
 
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