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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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