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Computer Science
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Quantum Computation
Lindblad Master Equation
Solving the quantum Liouvillian master equation
Example Notebook
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W
e
s
o
l
v
e
t
h
e
L
i
n
d
b
l
a
d
m
a
s
t
e
r
e
q
u
a
t
i
o
n
:
∂
t
ρ
=
-
i
[
H
,
ρ
]
+
∑
i
γ
i
L
i
ρ
†
L
i
-
1
2
†
L
i
L
i
,
ρ
w
i
t
h
ρ
t
h
e
d
e
n
s
i
t
y
m
a
t
r
i
x
,
L
i
t
h
e
j
u
m
p
o
p
e
r
a
t
o
r
a
n
d
γ
i
i
t
s
c
o
r
r
e
s
p
o
n
d
i
n
g
d
a
m
p
i
n
g
r
a
t
e
.
Install the QuantumFramework paclet:
I
n
[
1
]
:
=
P
a
c
l
e
t
I
n
s
t
a
l
l
[
"
W
o
l
f
r
a
m
/
Q
u
a
n
t
u
m
F
r
a
m
e
w
o
r
k
"
]
O
u
t
[
1
]
=
Load the paclet:
I
n
[
2
]
:
=
N
e
e
d
s
[
"
W
o
l
f
r
a
m
`
Q
u
a
n
t
u
m
F
r
a
m
e
w
o
r
k
`
"
]
Get the initial state:
I
n
[
3
]
:
=
ρ
0
=
Q
u
a
n
t
u
m
S
t
a
t
e
[
{
C
o
s
[
π
/
8
]
,
E
x
p
[
π
/
4
]
S
i
n
[
π
/
8
]
}
]
O
u
t
[
3
]
=
Q
u
a
n
t
u
m
S
t
a
t
e
P
u
r
e
s
t
a
t
e
Q
u
d
i
t
s
:
1
T
y
p
e
:
V
e
c
t
o
r
D
i
m
e
n
s
i
o
n
:
2
Set the
Ω
,
γ
and N parameters in finite temperature amplitude damping mechanism:
I
n
[
4
]
:
=
Ω
=
5
0
;
γ
=
1
;
n
=
3
;
Set up the Hamiltonian, list of jump operators and their corresponding rates:
I
n
[
5
]
:
=
H
=
Ω
/
2
Q
u
a
n
t
u
m
O
p
e
r
a
t
o
r
[
"
Z
"
]
;
L
s
=
Q
u
a
n
t
u
m
O
p
e
r
a
t
o
r
/
@
{
"
J
-
"
,
"
J
+
"
}
;
γ
s
=
γ
{
n
+
1
,
n
}
;
With H a quantum operator as Hamiltonian,
L
i
the jump operators, and
γ
i
the corresponding rates, determine the state:
I
n
[
6
]
:
=
ρ
t
=
Q
u
a
n
t
u
m
E
v
o
l
v
e
[
H
,
L
s
,
γ
s
,
ρ
0
,
{
t
,
0
,
1
}
]
O
u
t
[
6
]
=
Q
u
a
n
t
u
m
S
t
a
t
e
P
a
r
a
m
e
t
r
i
c
s
t
a
t
e
Q
u
d
i
t
s
:
1
T
y
p
e
:
M
a
t
r
i
x
D
i
m
e
n
s
i
o
n
:
2
Plot the evolution:
I
n
[
7
]
:
=
S
h
o
w
[
Q
u
a
n
t
u
m
S
t
a
t
e
[
"
U
n
i
f
o
r
m
M
i
x
t
u
r
e
"
]
[
"
B
l
o
c
h
P
l
o
t
"
]
,
P
a
r
a
m
e
t
r
i
c
P
l
o
t
3
D
[
E
v
a
l
u
a
t
e
@
R
e
@
ρ
t
[
t
]
[
"
B
l
o
c
h
V
e
c
t
o
r
"
]
,
{
t
,
0
,
1
}
]
]
O
u
t
[
7
]
=
O
n
e
c
a
n
a
l
s
o
i
n
c
o
r
p
o
r
a
t
e
t
h
e
r
a
t
e
s
i
n
t
o
t
h
e
d
e
f
i
n
i
t
i
o
n
o
f
o
p
e
r
a
t
o
r
s
a
s
γ
i
L
i
a
n
d
h
a
v
e
o
n
l
y
Q
u
a
n
t
u
m
E
v
o
l
v
e
[
H
,
{
L
1
,
L
2
,
…
}
,
Q
u
a
n
t
u
m
S
t
a
t
e
[
…
]
,
{
t
,
t
i
,
t
f
}
]
:
I
n
[
8
]
:
=
ρ
t
2
=
Q
u
a
n
t
u
m
E
v
o
l
v
e
H
,
γ
s
L
s
,
ρ
0
,
{
t
,
0
,
1
}
O
u
t
[
8
]
=
Q
u
a
n
t
u
m
S
t
a
t
e
P
a
r
a
m
e
t
r
i
c
s
t
a
t
e
Q
u
d
i
t
s
:
1
T
y
p
e
:
M
a
t
r
i
x
D
i
m
e
n
s
i
o
n
:
2
Plot the evolution:
I
n
[
9
]
:
=
S
h
o
w
[
Q
u
a
n
t
u
m
S
t
a
t
e
[
"
U
n
i
f
o
r
m
M
i
x
t
u
r
e
"
]
[
"
B
l
o
c
h
P
l
o
t
"
]
,
P
a
r
a
m
e
t
r
i
c
P
l
o
t
3
D
[
E
v
a
l
u
a
t
e
@
R
e
@
ρ
t
2
[
t
]
[
"
B
l
o
c
h
V
e
c
t
o
r
"
]
,
{
t
,
0
,
1
}
]
]
O
u
t
[
9
]
=
One can also create the Liouvillian
(
d
ρ
/
d
t
=
ℒ
ρ
):
I
n
[
1
0
]
:
=
ℒ
=
Q
u
a
n
t
u
m
O
p
e
r
a
t
o
r
[
"
L
i
o
u
v
i
l
l
i
a
n
"
[
H
,
L
s
,
γ
s
]
]
O
u
t
[
1
0
]
=
Q
u
a
n
t
u
m
O
p
e
r
a
t
o
r
U
n
k
n
o
w
n
m
a
p
D
i
m
e
n
s
i
o
n
:
2
→
2
O
r
d
e
r
:
{
1
}
→
{
1
}
Evolution super-operator:
ρ
t
=
E
x
p
[
ℒ
t
]
ρ
0
I
n
[
1
1
]
:
=
e
x
p
ℒ
=
Q
u
a
n
t
u
m
E
v
o
l
v
e
[
I
ℒ
,
N
o
n
e
,
{
t
,
0
,
1
}
]
;
ρ
t
3
[
t
_
]
=
e
x
p
ℒ
[
t
]
[
ρ
0
]
O
u
t
[
1
1
]
=
Q
u
a
n
t
u
m
S
t
a
t
e
U
n
k
n
o
w
n
s
t
a
t
e
Q
u
d
i
t
s
:
1
T
y
p
e
:
M
a
t
r
i
x
D
i
m
e
n
s
i
o
n
:
2
Plot the evolution:
I
n
[
1
2
]
:
=
S
h
o
w
[
Q
u
a
n
t
u
m
S
t
a
t
e
[
"
U
n
i
f
o
r
m
M
i
x
t
u
r
e
"
]
[
"
B
l
o
c
h
P
l
o
t
"
]
,
P
a
r
a
m
e
t
r
i
c
P
l
o
t
3
D
[
E
v
a
l
u
a
t
e
@
R
e
@
e
x
p
ℒ
[
ρ
0
]
[
"
B
l
o
c
h
V
e
c
t
o
r
"
]
,
{
t
,
0
,
1
}
]
]
O
u
t
[
1
2
]
=
Since the Liouvillian is time-independent, one can also do this:
I
n
[
1
3
]
:
=
ρ
t
4
[
t
_
]
=
E
x
p
[
ℒ
t
]
[
ρ
0
]
O
u
t
[
1
3
]
=
Q
u
a
n
t
u
m
S
t
a
t
e
U
n
k
n
o
w
n
s
t
a
t
e
Q
u
d
i
t
s
:
1
T
y
p
e
:
M
a
t
r
i
x
D
i
m
e
n
s
i
o
n
:
2
Plot the evolution:
I
n
[
1
4
]
:
=
S
h
o
w
[
Q
u
a
n
t
u
m
S
t
a
t
e
[
"
U
n
i
f
o
r
m
M
i
x
t
u
r
e
"
]
[
"
B
l
o
c
h
P
l
o
t
"
]
,
P
a
r
a
m
e
t
r
i
c
P
l
o
t
3
D
[
E
v
a
l
u
a
t
e
@
R
e
@
ρ
t
4
[
t
]
[
"
B
l
o
c
h
V
e
c
t
o
r
"
]
,
{
t
,
0
,
1
}
]
]
O
u
t
[
1
4
]
=
See Also
Wolfram/QuantumFramework
Publisher Information
Contributed by:
Wolfram Research, Quantum Computation Framework Team
