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Generalized Pauli Operators: Implementation  |  |
Consider a system of -level quantum system (i.e., a ). The canonical basis of its Hilbert space is . Let and be the generalized Pauli operators, where and . Then, the Weyl-Heisenberg basis is the generating set of the generalized Pauli group, :={:a,b,c=0,1,2,…,d-1}.
d
{|0〉,|1〉,|2〉,…,|d-1〉}
Z:|z〉↦|z〉
z
ω
X:|z〉↦|z+1〉
z=0,1,2,…,d-1
ω:=exp(2π/d)
d
c
ω
a
X
b
Z
Here, we illustrate how to implement generalized Pauli operators of degree with a quantum register of n qubits.
d=
n
2
Q3 functions related to this topic.
Make sure that Q3 is loaded.
In[147]:=
Needs["QuantumMob`Q3`"]
Choose a symbol for the quantum register.
In[149]:=
Let[Qubit,S]
In[150]:=
$n=4;$N=Power[2,$n];SS=S[Range@$n,$]
Out[152]=
{,,,}
S
1
S
2
S
3
S
4
In[153]:=
xx=Table[CNOT[S@Range[k+1,$n],S[k]],{k,1,$n-1}];xx=Append[xx,S[$n,1]];qcX=QuantumCircuit@@xx
Out[155]=
In[156]:=
matX=Matrix[qcX];matX//MatrixForm
Out[157]//MatrixForm=
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
In[158]:=
in=Basis[S@{1,2,3,4}]
Out[158]=
,,,,,,,,,,,,,,,
0
S
1
0
S
2
0
S
3
0
S
4
0
S
1
0
S
2
0
S
3
1
S
4
0
S
1
0
S
2
1
S
3
0
S
4
0
S
1
0
S
2
1
S
3
1
S
4
0
S
1
1
S
2
0
S
3
0
S
4
0
S
1
1
S
2
0
S
3
1
S
4
0
S
1
1
S
2
1
S
3
0
S
4
0
S
1
1
S
2
1
S
3
1
S
4
1
S
1
0
S
2
0
S
3
0
S
4
1
S
1
0
S
2
0
S
3
1
S
4
1
S
1
0
S
2
1
S
3
0
S
4
1
S
1
0
S
2
1
S
3
1
S
4
1
S
1
1
S
2
0
S
3
0
S
4
1
S
1
1
S
2
0
S
3
1
S
4
1
S
1
1
S
2
1
S
3
0
S
4
1
S
1
1
S
2
1
S
3
1
S
4
In[159]:=
out=qcX**in
Out[159]=
,,,,,,,,,,,,,,,
0
S
1
0
S
2
0
S
3
1
S
4
0
S
1
0
S
2
1
S
3
0
S
4
0
S
1
0
S
2
1
S
3
1
S
4
0
S
1
1
S
2
0
S
3
0
S
4
0
S
1
1
S
2
0
S
3
1
S
4
0
S
1
1
S
2
1
S
3
0
S
4
0
S
1
1
S
2
1
S
3
1
S
4
1
S
1
0
S
2
0
S
3
0
S
4
1
S
1
0
S
2
0
S
3
1
S
4
1
S
1
0
S
2
1
S
3
0
S
4
1
S
1
0
S
2
1
S
3
1
S
4
1
S
1
1
S
2
0
S
3
0
S
4
1
S
1
1
S
2
0
S
3
1
S
4
1
S
1
1
S
2
1
S
3
0
S
4
1
S
1
1
S
2
1
S
3
1
S
4
0
S
1
0
S
2
0
S
3
0
S
4
In[160]:=
Thread[inout]//TableForm
Out[160]//TableForm=
0 S 1 0 S 2 0 S 3 0 S 4 0 S 1 0 S 2 0 S 3 1 S 4 |
0 S 1 0 S 2 0 S 3 1 S 4 0 S 1 0 S 2 1 S 3 0 S 4 |
0 S 1 0 S 2 1 S 3 0 S 4 0 S 1 0 S 2 1 S 3 1 S 4 |
0 S 1 0 S 2 1 S 3 1 S 4 0 S 1 1 S 2 0 S 3 0 S 4 |
0 S 1 1 S 2 0 S 3 0 S 4 0 S 1 1 S 2 0 S 3 1 S 4 |
0 S 1 1 S 2 0 S 3 1 S 4 0 S 1 1 S 2 1 S 3 0 S 4 |
0 S 1 1 S 2 1 S 3 0 S 4 0 S 1 1 S 2 1 S 3 1 S 4 |
0 S 1 1 S 2 1 S 3 1 S 4 1 S 1 0 S 2 0 S 3 0 S 4 |
1 S 1 0 S 2 0 S 3 0 S 4 1 S 1 0 S 2 0 S 3 1 S 4 |
1 S 1 0 S 2 0 S 3 1 S 4 1 S 1 0 S 2 1 S 3 0 S 4 |
1 S 1 0 S 2 1 S 3 0 S 4 1 S 1 0 S 2 1 S 3 1 S 4 |
1 S 1 0 S 2 1 S 3 1 S 4 1 S 1 1 S 2 0 S 3 0 S 4 |
1 S 1 1 S 2 0 S 3 0 S 4 1 S 1 1 S 2 0 S 3 1 S 4 |
1 S 1 1 S 2 0 S 3 1 S 4 1 S 1 1 S 2 1 S 3 0 S 4 |
1 S 1 1 S 2 1 S 3 0 S 4 1 S 1 1 S 2 1 S 3 1 S 4 |
1 S 1 1 S 2 1 S 3 1 S 4 0 S 1 0 S 2 0 S 3 0 S 4 |
In[161]:=
ang=2Pi/$N;zz=Table[Phase[ang*Power[2,$n-k],S[k,3]],{k,1,$n}];qcZ=QuantumCircuit[zz]
Out[163]=
In[164]:=
matZ=Matrix[N@qcZ]//IntegerChop;matZ//MatrixForm
Out[165]//MatrixForm=
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0.92388+0.382683 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0.707107+0.707107 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0.382683+0.92388 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | -0.382683+0.92388 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | -0.707107+0.707107 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | -0.92388+0.382683 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -0.92388-0.382683 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -0.707107-0.707107 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -0.382683-0.92388 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.382683-0.92388 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.707107-0.707107 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.92388-0.382683 |
In[166]:=
phs=Diagonal[matZ]//Normal//Arg;Mod[$N*phs/(2Pi),$N]//IntegerChop
Out[167]=
{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
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