3.5 Stabilizer Codes
37
It is interesting to note that (a i , b i ) = (0, 0) if and only if (X (a i ), Z (b i )) = (I q , I q ).
Thus, from Definition 3.5.8, the weight wt(E) of E can be interpreted as the number
of nonidentity tensor components, as expected.
Definition 3.5.9 A quantum error-correcting code (QC) Q is a K -dimensional subspace of C
q
n . If Q has minimum distance d, then we say that Q is an ((n, K , d)) q
code. If K = q
k we write [[n, k, d]] q . The length n, the dimension K and minimum
distance d are the parameters of Q. The code Q is said to be pure to l if and only if
its stabilizer group does not contain non-scalar matrices of weight less than l; Q is
pure if and only if it is pure to its minimum distance.
Definition 3.5.10 We say that a quantum code Q has minimum weight d if it can
detect all errors in G n of weight less than d, but it cannot detect some error of weight
d.
A QC with minimum distance d corrects all errors of weight (d − 1)/2 or less.
We next recall two important definitions in group theory. The center Z (G n ) of the
group G n is the subgroup given by
Z (G n ) = {E ∈ G n |E F = F E, ∀ F ∈ G n }.
In words, Z (G n ) consists of the elements in G n that commute with all elements of
G n .
Let S be a subgroup of G n . The centralizer C G n (S) of S in G n is defined as
C G n (S) = {E ∈ G n |E F = F E, ∀ F ∈ S}.
Analogously, the elements of C G n (S) are the elements of G n that commute with
all elements of S. Further, let us consider S Z(G n ) as the group generated by S
and Z (G n ). The following result gives necessary and sufficient conditions for errordetection.
Theorem 3.5.1 Let S be a subgroup of G n such that S is the stabilizer group of a
stabilizer code Q of dimension greater than 1. A necessary and sufficient condition
in order to Q detects an error E ∈ G n is either E ∈ S Z(G n ) or E /
∈ C G n (S).
Exercise 3.5.3 Show Theorem 3.5.1.
There exist some well-known bounds with respect to the parameters of a quantum
code. The quantum Singleton bound will be utilized several times throughout this
book.
Lemma 3.5.1 Let C be an [[n, k, d]] q quantum code. Then the quantum Singleton
bound asserts that the parameters of C satisfy k + 2d ≤ n + 2. If C attains the
quantum Singleton bound, i.e., k + 2d = n + 2, then it is called a quantum maximum
distance separable (MDS) code.
37
It is interesting to note that (a i , b i ) = (0, 0) if and only if (X (a i ), Z (b i )) = (I q , I q ).
Thus, from Definition 3.5.8, the weight wt(E) of E can be interpreted as the number
of nonidentity tensor components, as expected.
Definition 3.5.9 A quantum error-correcting code (QC) Q is a K -dimensional subspace of C
q
n . If Q has minimum distance d, then we say that Q is an ((n, K , d)) q
code. If K = q
k we write [[n, k, d]] q . The length n, the dimension K and minimum
distance d are the parameters of Q. The code Q is said to be pure to l if and only if
its stabilizer group does not contain non-scalar matrices of weight less than l; Q is
pure if and only if it is pure to its minimum distance.
Definition 3.5.10 We say that a quantum code Q has minimum weight d if it can
detect all errors in G n of weight less than d, but it cannot detect some error of weight
d.
A QC with minimum distance d corrects all errors of weight (d − 1)/2 or less.
We next recall two important definitions in group theory. The center Z (G n ) of the
group G n is the subgroup given by
Z (G n ) = {E ∈ G n |E F = F E, ∀ F ∈ G n }.
In words, Z (G n ) consists of the elements in G n that commute with all elements of
G n .
Let S be a subgroup of G n . The centralizer C G n (S) of S in G n is defined as
C G n (S) = {E ∈ G n |E F = F E, ∀ F ∈ S}.
Analogously, the elements of C G n (S) are the elements of G n that commute with
all elements of S. Further, let us consider S Z(G n ) as the group generated by S
and Z (G n ). The following result gives necessary and sufficient conditions for errordetection.
Theorem 3.5.1 Let S be a subgroup of G n such that S is the stabilizer group of a
stabilizer code Q of dimension greater than 1. A necessary and sufficient condition
in order to Q detects an error E ∈ G n is either E ∈ S Z(G n ) or E /
∈ C G n (S).
Exercise 3.5.3 Show Theorem 3.5.1.
There exist some well-known bounds with respect to the parameters of a quantum
code. The quantum Singleton bound will be utilized several times throughout this
book.
Lemma 3.5.1 Let C be an [[n, k, d]] q quantum code. Then the quantum Singleton
bound asserts that the parameters of C satisfy k + 2d ≤ n + 2. If C attains the
quantum Singleton bound, i.e., k + 2d = n + 2, then it is called a quantum maximum
distance separable (MDS) code.