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1
×
10
18
ICHA
2pt Correlations (with NN)
2pt Correlations
Density Matrix
1
×
10
15
1
×
10
12
4000
3000
10
9
1
×
2000
1000
1
×
10
6
0
16
18
20
22
24
26
28
30
1
×
10
3
1
0
5
10
15
20
25
30
Number of Cells
Fig. 4.
The number of state variables that need to be solved as a function of the number
of cells in the circuit. Plots are shown for the different quantum mechanical treatments
discussed.
] is the commutator of
H
and
A
(
t
). Equa-
tion (
13
) is known as the Liouville equation of motion. By using this equation
and substituting in the Hamiltonian defined in Eq. (
3
), we can derive a set of
ordinary differential equations (ODE) describing the evolution of each of the
basis operators of our system, i.e.,
H
is the Hamiltonian and [
where
·
,
·
H, ʳ
x
(
j
)
,
d
dt
ʳ
x
(
j
)=
i
H, ʳ
y
(
j
)
,
d
dt
ʳ
y
(
j
)=
i
.
H, ʳ
x
(
j
)
ʳ
y
(
k
)
,
d
dt
ʳ
x
(
j
)
ʳ
y
(
k
)=
i
.
H, ʳ
z
(
j
)
ʳ
z
(
k
)
.
d
dt
ʳ
z
(
j
)
ʳ
z
(
k
)=
i
(14)
The dynamics of the coherence vector components can be obtained by
simply taking the expectation values of both sides of the equations listed
above, i.e.,
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