Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
QuantumMob`Q3mini`
CNOT  |
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Details and Options
Examples
(8)
Basic Examples
(6)
Let us choose symbol S to refer to a quantum register of two (or more) qubits.
In[1]:=
Let[Qubit,S]
The first qubit is the control qubit and the second is the target qubit.
In[2]:=
cn=CNOT[S[1,$],S[2,$]]
Out[2]=
CNOT[{1},{}]
S
1
S
2
The last flavor index may be dropped since it is clear from the context.
$
In[3]:=
cn=CNOT[S[1],S[2]]
Out[3]=
CNOT[{1},{}]
S
1
S
2
In the quantum circuit model, the CNOT gate is displayed as follows.
In[4]:=
qc=QuantumCircuit[cn]
Out[4]=
In[1]:=
Let[Qubit,S]
Take the computational basis states as input.
In[2]:=
in=Basis[S@{1,2}](*Thisisashort-handformofBasis[S[{1,2,},$]].*)
Out[2]=
,,,
0
S
1
0
S
2
0
S
1
1
S
2
1
S
1
0
S
2
1
S
1
1
S
2
Here are the results of applying the CNOT gate on the computational basis states.
In[3]:=
out=qc**in(*Thisisequivalenttocn**in*)
Out[3]=
,,,
0
S
1
0
S
2
0
S
1
1
S
2
1
S
1
1
S
2
1
S
1
0
S
2
Compare the input and output states.
In[4]:=
Thread[inout]//TableForm
Out[4]//TableForm=
In[1]:=
Let[Qubit,S]
Consider a multi-control NOT gate.
In[2]:=
qc=QuantumCircuit[CNOT[S@{1,2},S[3]]]
Out[2]=
In[3]:=
op=ExpressionFor[qc]
Out[3]=
3
4
1
4
Z
S
1
Z
S
2
1
4
Z
S
1
X
S
3
1
4
Z
S
2
X
S
3
1
4
Z
S
1
Z
S
2
X
S
3
1
4
Z
S
1
1
4
Z
S
2
1
4
X
S
3
In[4]:=
mat=Matrix[qc];mat//MatrixForm
Out[4]//MatrixForm=
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
In[1]:=
Let[Qubit,S]
In[2]:=
qc=QuantumCircuit[CNOT[S@{1,2}{1,0},S[3]]]op=ExpressionFor[qc]
Out[2]=
Out[2]=
3
4
1
4
Z
S
1
Z
S
2
1
4
Z
S
1
X
S
3
1
4
Z
S
2
X
S
3
1
4
Z
S
1
Z
S
2
X
S
3
1
4
Z
S
1
1
4
Z
S
2
1
4
X
S
3
In[3]:=
new=QuantumCircuit[S[2,1],CNOT[S@{1,2},S[3]],S[2,1]]
Out[3]=
In[4]:=
qc-new//Elaborate
Out[4]=
0
In[1]:=
Let[Qubit,S]
The CNOT gate is closely related to the CZ (controlled-Z) and SWAP gates.
In[2]:=
cz=CZ[S[1],S[2]]swap=SWAP[S[1],S[2]]
Out[2]=
CZ[{,}]
S
1
S
2
Out[2]=
SWAP[,]
S
1
S
2
In[3]:=
Elaborate[cz]Elaborate[swap]
Out[3]=
1
2
1
2
Z
S
1
Z
S
2
1
2
Z
S
1
1
2
Z
S
2
Out[3]=
1
2
1
2
X
S
1
X
S
2
1
2
Y
S
1
Y
S
2
1
2
Z
S
1
Z
S
2
In[4]:=
Matrix[cz]//MatrixFormMatrix[swap]//MatrixForm
Out[4]//MatrixForm=
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | -1 |
Out[4]//MatrixForm=
1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 |
Compare the quantum circuit representations of the gates.
The CNOT, CZ, and SWAP gates are expanded automatically when necessary.