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2 Coherent Cell-Cell Dynamics
By ignoring correlations and coherence, the ICHA can often fail to predict the
correct ground state and dynamics of QCA systems. However, as previously
discussed, including both coherence and correlation results in a non-polynomial
expansion of the basis set - making solving large systems computationally
intractable. The problem lies in the fact that the ICHA uses only the single
cell operators (the Pauli spin matrices) to derive its solutions, which on their
own, do not carry enough information to accurately predict the ground state of
most QCA systems. The full-basis two-state approximation, on the other hand,
uses all
N
cell operators when arriving at its solution, which provides us with
more information than we need. Thus, what we seek is an intermediate treat-
ment which uses only the bare-minimum number of operators required to obtain
an accurate ground state solution for a QCA system. Attempting to develop
such a treatment in the Schrodinger picture will prove cumbersome, and thus
it will benefit us to formulate our problem in the Heisenberg picture, in which
the operators incorporate a dependency on time, but the state vectors are time-
independent. In the Heisenberg picture, we can separate the state variables of our
system into groups corresponding to the state of the individual cells (single-cell
operators) and into groups corresponding to the two-cell, three-cell, etc., opera-
tors [
40
]. We can then remove any higher-order operators deemed negligible to
the system's evolution, and thus reduce the overall computational requirements
for solving the ground state of a QCA system. This method is known as the
Coherence Vector formulation, and allows us to easily compute the dynamics of
a QCA system as well.
The coherence vector can be constructed by first recognizing that the density
operator in Hilbert space of dimension
n
, like any Hermitian operator, can be
represented by the
s
=
n
2
(
n
)[
47
]. For an
N
cell system (i.e., a system with dimension
n
=2
N
), the density operator
ʻ
can be represented as [
47
]:
−
1 generating (basis) operators of
SU
s
1
1
2
N
2
N
1+
i
ˆ
i
,
ʻ
=
(6)
i
=1
where
ˆ
i
represents the
i
th
generating operator of
SU
(2
N
), and the elements
i
are defined by
ˆ
i
≥
ʻ
ˆ
i
},
i
=
≤
=Tr
{
(7)
and are the components that form the coherence vector. For any system of
dimension
n
=2
N
,the
ˆ
i
basis operators take the form
ˆ
i
=
ʛ
(1)
ʛ
(2)
ʛ
(
N
)
,
↗
↗···↗
(8)
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