Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
QuantumMob`Q3mini`
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Details and Options
Examples
(11)
Basic Examples
(10)
In[1]:=
expr=Pauli[{1,1}]+Pauli[{2,2}]+JPauli[{3,3}]
Out[1]=
X⊗X+Y⊗Y+JZ⊗Z
This is the matrix representation in the standard basis.
In[2]:=
mat=Matrix[expr];mat//MatrixForm
Out[2]//MatrixForm=
J | 0 | 0 | 0 |
0 | -J | 2 | 0 |
0 | 2 | -J | 0 |
0 | 0 | 0 | J |
This converts the matrix representation back to the operator expression.
In[3]:=
new=ExpressionFor[mat]Elaborate[new]
Out[3]=
JZ⊗Z+2⊗+2⊗
+
X
-
X
-
X
+
X
Out[3]=
X⊗X+Y⊗Y+JZ⊗Z
In[1]:=
ket=Ket[{0,0}]+I*Ket[{0,1}]-I*Ket[{1,0}]
Out[1]=
|0,0〉+|0,1〉-|1,0〉
This is the column vector representation of the state vector in the standard basis.
In[2]:=
vec=Matrix[ket];vec//Normal
Out[2]=
{1,,-,0}
This converts the column vector back to the state-vector expression.
In[3]:=
new=ExpressionFor[vec]new-ket
Out[3]=
|0,0〉+|0,1〉-|1,0〉
Out[3]=
0
In[1]:=
Let[Qubit,S]
Consider an operator expression for labeled qubits.
In[2]:=
op=1+S[1,2]+S[1,3]**S[2,1]
Out[2]=
1++
Z
S
1
X
S
2
Y
S
1
This is the matrix representation of the above operator in the standard basis for qubits S[1,$] and S[2,$] involved in the expression. In many cases, automatically extract the relevant qubits.
In[3]:=
mat=Matrix[op];mat//MatrixForm
Out[3]//MatrixForm=
1 | 1 | - | 0 |
1 | 1 | 0 | - |
| 0 | 1 | -1 |
0 | | -1 | 1 |
This converts the matrix representation back to an operator expression.
In[4]:=
new=ExpressionFor[mat,S@{1,2}]Elaborate[new]
Out[4]=
1++-+
Z
S
1
+
S
2
Z
S
1
-
S
2
+
S
1
-
S
1
Out[4]=
1++
Z
S
1
X
S
2
Y
S
1
As you note above, returns the expression in terms of the raising and lowering operators. If you want it in terms of the Pauli X and Y operators, use .
ExpressionFor
Elaborate
In[5]:=
expr=Elaborate[new]
Out[5]=
1++
Z
S
1
X
S
2
Y
S
1
In[6]:=
RaisingLoweringForm[expr]
Out[6]=
1++-+
Z
S
1
+
S
2
Z
S
1
-
S
2
+
S
1
-
S
1
In[7]:=
op-new//Elaborate
Out[7]=
0
In[1]:=
ket=(3Ket[]+4IKet[S[1]1,S[2]1])/5//KetRegulate
Out[1]=
1
5
0
S
1
0
S
2
1
S
1
1
S
2
In the standard basis for qubits, S[1,$] and S[2,$], it has the following column vector representation.
In[2]:=
vec=Matrix[ket];vec//MatrixForm
Out[2]//MatrixForm=
3 5 |
0 |
0 |
4 5 |
This converts the column vector back to the state-vector expression.
In[3]:=
new=ExpressionFor[vec,S@{1,2}]ket-new//Simplify
Out[3]=
3
5
0
S
1
0
S
2
4
5
1
S
1
1
S
2
Out[3]=
0
In[1]:=
bra=Dagger[ket]vec=Matrix[bra];vec//MatrixForm
Out[1]=
1
5
0
S
1
0
S
2
1
S
1
1
S
2
Out[1]//MatrixForm=
3 5 |
0 |
0 |
- 4 5 |
Let us now consider a system of qudits.
Here is a state vector.
The column vector representation of the state vector in the standard basis is given as follows.
Consider an operator on a system of qudits.
This is its matrix representation in the standard basis.
This converts the matrix representation back to an operator expression.
It looks different from the original operator. However, if you take into account the closure relation, then the two expressions are the same.
Spin objects with spin=1/2 are essentially the same as qubits.
Here is an operator we want to examine.
This is its matrix representation in the standard basis.
This converts the matrix representation back to the operator expression.
Here is a state vector.
The column vector representation is as follows.
This converts the column vector representation back to the state vector.
Let us now consider a bigger spin.
As the Hilbert space associated with fermions is finite dimensional, one can find an operator expression for fermions from a matrix representation in a systematic way.
For example, consider a tight-biding Hamiltonian.
As the conversion of the matrix representation into the operator expression involves the Jordan-Wigner transformation, it is a bit slower for fermions.
Compare the above evaluation with the one for the Pauli operators (for labeled qubits).
As it does not involve the Jordan-Wigner transformation, the evaluation is faster than for Fermions.