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two-dimensional subspace of the two-state basis vectors. For suciently low tem-
peratures, we can therefore expect the state of a cell or a line to be well described
within the two-state approximation. However, while the two-state approximation
offers significant computational savings over the full basis quantum mechanical
treatment described in the previous section, the exponential growth in the basis
set still limits its application to systems containing only a small number of inter-
acting cells. A thirty-cell system, for instance, would yield over a billion basis
vectors. In practice, QCA circuits will likely require millions of cells. A sim-
ple 4-bit processor, for example, was designed using almost 30,000 cells [
44
].
Thus, further approximations are required in order to reduce the computational
complexity of modeling large QCA systems.
1.3
Intercellular Hartree Approximation
In the full quantum mechanical treatments discussed above, arrays of QCA cells
are being solved for directly while retaining the full many-electron degrees of
freedom. The inclusion of all of these additional variables contributes to the
exponential growth in the basis set as the system grows with the number of cells,
N
. Thus, in order to mitigate this growth, some quantum degrees of freedom
must be removed during calculations. One such approach is to use the intercel-
lular Hartree approximation (ICHA), for which the effects arising from quantum
correlations and exchange are fully neglected [
37
,
38
]. In this treatment, an
N
-cell
system is decoupled into a set of
single-cell subsystems that are assumed to
interact classically through the expectation values of their polarization without
any quantum mechanical coherence between neighboring cells. Using this approx-
imation in conjunction with the two-state approximation, it is only necessary to
diagonalize
N
2
N
Hamiltonian. This
allows large circuits, which would otherwise be intractable using a full quantum
mechanical model, to be simulated on a classical computer. The Hamiltonian (in
the polarization basis) for a single cell,
i
, is simply,
2 Hamiltonians as opposed to one 2
N
N
2
×
×
2
j∈Ω
−ʷ
i
ʳ
x
+
1
H
i
=
E
i,
k
P
j
ʳ
z
,
(4)
where
P
j
.
Because the solutions of one cell's Hamiltonian define parameters that enter its
neighbours' Hamiltonians, the system of coupled Schrodinger equations must
be solved iteratively to obtain self-consistency. If a circuit evolves adiabatically
and cells remain at the ground state, then this problem simplifies even further
by recognizing that the polarization of any cell,
i
, can be evaluated numerically
using [
45
],
is the polarization of cell
j
, and is found by evaluating,
P
j
=
−≤ʳ
z
≥
2
γ
j
E
i,j
k
1
P
j
P
i
=
1+
2
γ
j
2
,
(5)
E
i,
k
P
j
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