3.5 Stabilizer Codes
35
where tr : F p m −→ F p is the trace map and ω = exp(2πi/ p) is a primitive pth root
of unity.
Remark 3.5.1 Note that the definitions of operators X (a) and Z (b) are natural
generalizations of the Pauli matrices X and Z , respectively, to q-ary alphabets. In
fact, X (a) acts by changing the vectors of the orthonormal basis, and Z (b) changes
the phase of the vectors of the basis.
We can now define the set of error operators.
Definition 3.5.4 Let X (a) and Z (b) be the operators defined above. The set
ε = {X (a)Z (b)|a, b ∈ F q }
is called the set of error operators.
In the sequence we define the concept of nice error basis.
Definition 3.5.5 Let β be a set of q
2 unitary matrices. We say that β is a nice error
basis if β satisfies the following conditions:
(1) I q ∈ β, where I q is the identity matrix of order q;
(2) if A, B ∈ β then AB is a scalar multiple of another element of β;
(3) if A, B ∈ β, with A = B, then Tr(A
† B) = 0, where Tr denotes the trace of the
matrix.
The set ε given in Definition 3.5.4 is a nice error basis.
Proposition 3.5.1 The set ε = {X (a)Z (b)|a, b ∈ F q } satisfies the three conditions
of Definition 3.5.5, i.e., ε is a nice error basis on C
q .
Exercise 3.5.2 Show Proposition 3.5.1.
In order to improve the understanding of the text we give here an example of a
nice error basis for q = 4. This is, in fact, the Example 2 of the paper by Ketkar et
al. [80].
Example 3.5.1 Let us consider the finite field with four elements F 4 = {0, 1, α, α}.
According to the notation adopted above, a basis for C
4 can be written as |0, |1,
|α and |α. Let
I 2 =
1 0
0 1
, σ X =
0 1
1 0
, and σ Z =
1 0
0 −1
.
By a simple computation we have
X (0) = I 2 ⊗ I 2 , X (1) = I 2 ⊗ σ X , X (α) = I 2 ⊗ I 2 , X (α) = σ X ⊗ σ X ,
Z (0) = I 2 ⊗ I 2 , Z (1) = σ Z ⊗ I 2 , Z (α) = σ Z ⊗ σ Z , X (α) = I 2 ⊗ σ Z .
35
where tr : F p m −→ F p is the trace map and ω = exp(2πi/ p) is a primitive pth root
of unity.
Remark 3.5.1 Note that the definitions of operators X (a) and Z (b) are natural
generalizations of the Pauli matrices X and Z , respectively, to q-ary alphabets. In
fact, X (a) acts by changing the vectors of the orthonormal basis, and Z (b) changes
the phase of the vectors of the basis.
We can now define the set of error operators.
Definition 3.5.4 Let X (a) and Z (b) be the operators defined above. The set
ε = {X (a)Z (b)|a, b ∈ F q }
is called the set of error operators.
In the sequence we define the concept of nice error basis.
Definition 3.5.5 Let β be a set of q
2 unitary matrices. We say that β is a nice error
basis if β satisfies the following conditions:
(1) I q ∈ β, where I q is the identity matrix of order q;
(2) if A, B ∈ β then AB is a scalar multiple of another element of β;
(3) if A, B ∈ β, with A = B, then Tr(A
† B) = 0, where Tr denotes the trace of the
matrix.
The set ε given in Definition 3.5.4 is a nice error basis.
Proposition 3.5.1 The set ε = {X (a)Z (b)|a, b ∈ F q } satisfies the three conditions
of Definition 3.5.5, i.e., ε is a nice error basis on C
q .
Exercise 3.5.2 Show Proposition 3.5.1.
In order to improve the understanding of the text we give here an example of a
nice error basis for q = 4. This is, in fact, the Example 2 of the paper by Ketkar et
al. [80].
Example 3.5.1 Let us consider the finite field with four elements F 4 = {0, 1, α, α}.
According to the notation adopted above, a basis for C
4 can be written as |0, |1,
|α and |α. Let
I 2 =
1 0
0 1
, σ X =
0 1
1 0
, and σ Z =
1 0
0 −1
.
By a simple computation we have
X (0) = I 2 ⊗ I 2 , X (1) = I 2 ⊗ σ X , X (α) = I 2 ⊗ I 2 , X (α) = σ X ⊗ σ X ,
Z (0) = I 2 ⊗ I 2 , Z (1) = σ Z ⊗ I 2 , Z (α) = σ Z ⊗ σ Z , X (α) = I 2 ⊗ σ Z .