Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
QuantumMob`Q3mini`
Rotation  |
|
| | ||||
|
| | ||||
|
| | ||||
|
| | ||||
|
| | ||||
|
| | ||||
Details and Options
Examples
(5)
Basic Examples
(5)
This gives the rotation operator in terms of the Pauli operators.
In[1]:=
op=Rotation[ϕ,2]
Out[1]=
CosI-YSin
ϕ
2
ϕ
2
This gives the matrix representation of the above rotation operator.
In[2]:=
mat=Matrix[op]//ExpToTrig;mat//MatrixForm
Out[2]//MatrixForm=
Cos ϕ 2 | -Sin ϕ 2 |
Sin ϕ 2 | Cos ϕ 2 |
For a quick access to the rotation matrix, you can use .
TheRotation
In[3]:=
TheRotation[ϕ,2]//MatrixForm
Out[3]//MatrixForm=
Cos ϕ 2 | -Sin ϕ 2 |
Sin ϕ 2 | Cos ϕ 2 |
This is a way to specify a rotation operator on multiple qubits.
In[4]:=
Rotation[{a,2},{b,3}]//Garner
Out[4]=
CosCosI⊗I-CosY⊗ISin-CosI⊗ZSin-Y⊗ZSinSin
a
2
b
2
b
2
a
2
a
2
b
2
a
2
b
2
You can specify an arbitrary axis other than the coordinates axes.
In[1]:=
Rotation[ϕ,{1,1,0}]
Out[1]=
CosI-+Sin
ϕ
2
X
2
Y
2
ϕ
2
Construct an rotation operator on multiple qubits.
In[2]:=
op=Rotation[{ϕ,{1,0,0}},{ϕ,{0,1,0}}]
Out[2]=
2
Cos
ϕ
2
2
Sin
ϕ
2
1
2
1
2
The above is equivalent to the following as the rotation axes happen to coincide with the coordinate axes.
In[3]:=
new=Rotation[{ϕ,1},{ϕ,2}]
Out[3]=
2
Cos
ϕ
2
2
Sin
ϕ
2
1
2
1
2
In[4]:=
opnew
Out[4]=
True
Let us consider rotations on qubits.
In[1]:=
Let[Qubit,S]Let[Real,ϕ]
Consider a rotation around the z-axis.
In[2]:=
op=Rotation[ϕ,S[1,3]]
Out[2]=
Exp-ϕ
1
2
Z
S
1
In[3]:=
Dagger@Rotation[ϕ,S[1,3]]
Out[3]=
Expϕ
1
2
Z
S
1
In[4]:=
Elaborate[op]
Out[4]=
Cos-Sin
ϕ
2
Z
S
1
ϕ
2
In[5]:=
Matrix[op,S@{1,2}]//MatrixForm
Out[5]//MatrixForm=
- ϕ 2 | 0 | 0 | 0 |
0 | - ϕ 2 | 0 | 0 |
0 | 0 | ϕ 2 | 0 |
0 | 0 | 0 | ϕ 2 |
You can specify an arbitrary rotation axis.
In[6]:=
op=Rotation[ϕ,{1,1,1},S[1,$]]
Out[6]=
Exp-ϕ++
1
2
X
S
1
3
Y
S
1
3
Z
S
1
3
In[7]:=
Dagger[op]
Out[7]=
Expϕ++
1
2
X
S
1
3
Y
S
1
3
Z
S
1
3
In[8]:=
Elaborate[op]
Out[8]=
Cos---
ϕ
2
Sin
X
S
1
ϕ
2
3
Sin
Y
S
1
ϕ
2
3
Sin
Z
S
1
ϕ
2
3
In[9]:=
Matrix[op,S@{1,2}]//ExpToTrig//MatrixForm
Out[9]//MatrixForm=
Cos ϕ 2 Sin ϕ 2 3 | 0 | - (1+)Sin ϕ 2 3 | 0 |
0 | Cos ϕ 2 Sin ϕ 2 3 | 0 | - (1+)Sin ϕ 2 3 |
(1-)Sin ϕ 2 3 | 0 | Cos ϕ 2 Sin ϕ 2 3 | 0 |
0 | (1-)Sin ϕ 2 3 | 0 | Cos ϕ 2 Sin ϕ 2 3 |
In[10]:=
QuantumCircuit[op]
Out[10]=
In[11]:=
Let[Real,ϕ]qc=QuantumCircuit[Rotation[ϕ,S[1,3]],Rotation[ϕ,S[2,1]]]
Again, you can specify an arbitrary rotation axis by a vector along the axis.