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Commutation Relations for Qubits
Let be the angular momentum operator. Then, is the rotation operator around the axis n by angle ϕ. This implies that the angular momentum operators satisfy the following transformation rules.
J:=(,,)
x
J
y
J
z
J
U=exp(-ϕJ·n)
U=(n,ϕ)
ν
J
†
U
∑
μ
μ
J
R
μν
where is the three-dimensional rotation matrix around axis n by angle ϕ.
R(n,ϕ)
As a simple example of applications of Q3, here we illustrate the fundamental commutation relations among the Pauli operators.
Represents the exponential function of operators (noncommutative variables) | |
Represents the rotation operator | |
Represents the Euler rotation operator |
Q3 functions related to the fundamental commutation relations of the Pauli operators on qubits.
In[1]:=
Needs["QuantumMob`Q3`"]
Throughout this document, symbol S will be used to denote qubits and Pauli operators on them. Similarly, symbol c will be used to denote complex-valued coefficients.
In[2]:=
Let[Qubit,S]Let[Complex,c]
Method Using
Consider a rotation around the z-axis.
In[204]:=
Let[Real,ϕ]U=Rotation[ϕ,3]
Out[205]=
Cos-Sin
ϕ
2
0
σ
z
σ
ϕ
2
We are going to transform the Pauli operators under conjugation by U.
In[200]:=
in={Pauli[1],Pauli[2],Pauli[3]}
Out[200]=
{,,}
x
σ
y
σ
z
σ
According to the fundamental commutation relations, the following two expressions are equivalent.
In[206]:=
out=U**in**Dagger[U]
Out[206]=
{Cos[ϕ]+Sin[ϕ],Cos[ϕ]-Sin[ϕ],}
x
σ
y
σ
y
σ
x
σ
z
σ
In[207]:=
new=in.RotationMatrix[ϕ,{0,0,1}]
Out[207]=
{Cos[ϕ]+Sin[ϕ],Cos[ϕ]-Sin[ϕ],}
x
σ
y
σ
y
σ
x
σ
z
σ
In[208]:=
out-new
Out[208]=
{0,0,0}
Method Using
S[⋯]**S[⋯]**⋯
Consider a rotation around the y-axis.
In[213]:=
U=Rotation[ϕ,S[2]]Elaborate[U]
Out[213]=
Rotation[ϕ,]
y
S
Out[214]=
Cos-Sin
ϕ
2
y
S
ϕ
2
We are going to transform the Pauli operators under conjugation by U.
In[215]:=
in=S[All]
Out[215]=
{,,}
x
S
y
S
z
S
According to the fundamental commutation relations, the following two expressions are equivalent.
In[216]:=
out=Rotation[ϕ,S[2]]**SS**Rotation[-ϕ,S[2]]
Out[216]=
{Cos[ϕ]-Sin[ϕ],,Cos[ϕ]+Sin[ϕ]}
x
S
z
S
y
S
z
S
x
S
In[217]:=
new=SS.RotationMatrix[ϕ,{0,1,0}]
Out[217]=
{Cos[ϕ]-Sin[ϕ],,Cos[ϕ]+Sin[ϕ]}
x
S
z
S
y
S
z
S
x
S
In[218]:=
out-new
Out[218]=
{0,0,0}
 |
|
|
""