150
6 Asymmetric Quantum Codes
Proof Let us consider the direct sum codes C 1 ⊕ C 3 = [n + n
∗
, k 1 + k 3 , min{d 1 ,
d 3 }] q and C 2 ⊕ C 4 = [n + n
∗
, k 2 + k 4 , min{d 2 , d 4 }] q . Since the inclusions C 2 ⊂ C 1
and C 4 ⊂ C 3 hold, it follows that the inclusion C 2 ⊕ C 4 ⊂ C 1 ⊕ C 3 also holds. We
know that a parity check matrix of the code (C 2 ⊕ C 4 )
⊥ is given by
G 2 ⊕ G 4 =
G 2 0
0 G 4
.
Thus, the minimum distance of (C 2 ⊕ C 4 )
⊥ equals min{d
⊥
2 , d
⊥
4 }. Therefore, applying
the CSS construction to the codes C 1 ⊕ C 3 and C 2 ⊕ C 4 and (C 2 ⊕ C 4 )
⊥ , we obtain
an [[n + n
∗
, (k 1 + k 3 ) −(k 2 + k 4 ), d
z /d
x ]] q AQECC, where d
z ≥ min{d 1 , d 3 } and
d
x ≥ min{d
⊥
2 , d
⊥
4 }. We are done.
Theorem 6.7.4 also holds in a more general setting, as states the following result.
Theorem 6.7.5 Assume that there exist two stabilizer codes with parameters ((n 1 ,
K 1 , d
(1)
z /d
(1)
x )) q and ((n 2 , K 2 , d
(2)
z /d
(2)
x )) q . Then there exists an ((n 1 + n 2 , K 1 K 2 ,
d
∗
z /d
∗
x )) q code, where d
∗
z = min{d
(1)
z , d
(2)
z } and d
∗
x = min{d
(1)
x , d
(2)
x }.
Proof The proof follows the same line of [80, Lemma 73]. We only show the result
in the case of X -weight because the proof for Z -weight is analogous. Note that if
((n 1 , K 1 , d
(1)
z /d
(1)
x )) q and ((n 2 , K 2 , d
(2)
z /d
(2)
x )) q are stabilizer codes with orthogonal
projectors P 1 and P 2 , respectively, and stabilizer S 1 and S 2 , respectively, then P 1 ⊗ P 2
is an orthogonal projector onto a K 1 K 2 -dimensional subspace Q
⊕ of C
q
(n 1 +n 2 ) . The
stabilizer of Q
⊕ is given by
S
⊕
= {E 1 ⊗ E 2 |E 1 ∈ S 1 , E 2 ∈ S 2 }.
Assume that an error F 1 ⊗ F 2 ∈ G n 1 ⊗ G n 2 is not detectable. Hence, it follows
that F 1 ∈ C G n 1 (S 1 ) and F 2 ∈ C G n 2 (S 2 ). Moreover, either F 1 /
∈ S 1 Z (G n 1 ) or F 2 /
∈
S 2 Z (G n 2 ), otherwise F 1 ⊗ F 2 would be detectable. Thus, from [80, Lemma 11],
either F 1 or F 2 is not detectable, so wt X (F 1 ⊗ F 2 ) is, at least, min{d
(1)
x , d
(2)
x }. Therefore, the result follows.
6.7.3 Puncturing Codes
The technique of puncturing codes is well known in the literature as in the classical
[67, 114] as well as in the quantum case [25, 80, 133]. In this subsection, we show
how to construct AQECCs by puncturing classical codes.
Let C be an [n, k, d] q code. Then we write C
P i to denote the punctured code in
the coordinate i. Recall that the dual of a punctured code is a shortened code (see
[67]).
We are now ready to show how to construct new stabilizer codes by applying
puncturing techniques in the corresponding classical linear codes.
6 Asymmetric Quantum Codes
Proof Let us consider the direct sum codes C 1 ⊕ C 3 = [n + n
∗
, k 1 + k 3 , min{d 1 ,
d 3 }] q and C 2 ⊕ C 4 = [n + n
∗
, k 2 + k 4 , min{d 2 , d 4 }] q . Since the inclusions C 2 ⊂ C 1
and C 4 ⊂ C 3 hold, it follows that the inclusion C 2 ⊕ C 4 ⊂ C 1 ⊕ C 3 also holds. We
know that a parity check matrix of the code (C 2 ⊕ C 4 )
⊥ is given by
G 2 ⊕ G 4 =
G 2 0
0 G 4
.
Thus, the minimum distance of (C 2 ⊕ C 4 )
⊥ equals min{d
⊥
2 , d
⊥
4 }. Therefore, applying
the CSS construction to the codes C 1 ⊕ C 3 and C 2 ⊕ C 4 and (C 2 ⊕ C 4 )
⊥ , we obtain
an [[n + n
∗
, (k 1 + k 3 ) −(k 2 + k 4 ), d
z /d
x ]] q AQECC, where d
z ≥ min{d 1 , d 3 } and
d
x ≥ min{d
⊥
2 , d
⊥
4 }. We are done.
Theorem 6.7.4 also holds in a more general setting, as states the following result.
Theorem 6.7.5 Assume that there exist two stabilizer codes with parameters ((n 1 ,
K 1 , d
(1)
z /d
(1)
x )) q and ((n 2 , K 2 , d
(2)
z /d
(2)
x )) q . Then there exists an ((n 1 + n 2 , K 1 K 2 ,
d
∗
z /d
∗
x )) q code, where d
∗
z = min{d
(1)
z , d
(2)
z } and d
∗
x = min{d
(1)
x , d
(2)
x }.
Proof The proof follows the same line of [80, Lemma 73]. We only show the result
in the case of X -weight because the proof for Z -weight is analogous. Note that if
((n 1 , K 1 , d
(1)
z /d
(1)
x )) q and ((n 2 , K 2 , d
(2)
z /d
(2)
x )) q are stabilizer codes with orthogonal
projectors P 1 and P 2 , respectively, and stabilizer S 1 and S 2 , respectively, then P 1 ⊗ P 2
is an orthogonal projector onto a K 1 K 2 -dimensional subspace Q
⊕ of C
q
(n 1 +n 2 ) . The
stabilizer of Q
⊕ is given by
S
⊕
= {E 1 ⊗ E 2 |E 1 ∈ S 1 , E 2 ∈ S 2 }.
Assume that an error F 1 ⊗ F 2 ∈ G n 1 ⊗ G n 2 is not detectable. Hence, it follows
that F 1 ∈ C G n 1 (S 1 ) and F 2 ∈ C G n 2 (S 2 ). Moreover, either F 1 /
∈ S 1 Z (G n 1 ) or F 2 /
∈
S 2 Z (G n 2 ), otherwise F 1 ⊗ F 2 would be detectable. Thus, from [80, Lemma 11],
either F 1 or F 2 is not detectable, so wt X (F 1 ⊗ F 2 ) is, at least, min{d
(1)
x , d
(2)
x }. Therefore, the result follows.
6.7.3 Puncturing Codes
The technique of puncturing codes is well known in the literature as in the classical
[67, 114] as well as in the quantum case [25, 80, 133]. In this subsection, we show
how to construct AQECCs by puncturing classical codes.
Let C be an [n, k, d] q code. Then we write C
P i to denote the punctured code in
the coordinate i. Recall that the dual of a punctured code is a shortened code (see
[67]).
We are now ready to show how to construct new stabilizer codes by applying
puncturing techniques in the corresponding classical linear codes.