210
7 Constructions of QCCs
⎡
⎢
⎢
⎢
⎣
w 0 ζ
t
0 w 1 ζ
t
1 · · · w n−1 ζ
t
n−1
0
0 0
0
. . .
. . .
. . .
. . .
0
0 0
0
⎤
⎥
⎥
⎥
⎦
D,
where w = (w 0 , . . . , w n−1 ) is a vector such that GRS
⊥
k (ζ, w) = GRS n−k (ζ, v). The
code V 1 has parameters (n, n − k − 2, 2; 1, [d 1 ] f ≥ k + 1) q and V
⊥
1 has parameters (n, k + 2, 2; μ
⊥
1 , [d 1 ]
⊥
f ) q . Similarly, V 2 is an (n, t, 1; 1, [d 2 ] f ) q code and V
⊥
2
is an (n, n − t, 1; μ
⊥
2 , [d 1 ]
⊥
f ≥ t + 2) q code. Then there exists an [(n, n − t − k −
2, μ
∗
; 3, [d z ] f /[d x ] f )] q code, where (d z ) f ≥ t + 2 and (d x ) f ≥ k + 1.
Example 7.7.3 From Theorem 7.7.7, we can construct AQCCs with parameters
[(5, 1, μ
∗
; 3, [d z ] f ≥ 3/[d x ] f ≥ 2)] 5 , [(7, 1, μ
∗
; 3, [d z ] f ≥ 4/[d x ] f ≥ 3)] 7 , [(8, 1,
μ
∗
; 3, [d z ] f ≥ 5/[d x ] f ≥ 3)] 8 , [(17, 7, μ
∗
; 3, [d z ] f ≥ 7/[d x ] f ≥ 4)] 17 ,
[(17, 7, μ
∗
; 3, [d z ] f ≥ 6/[d x ] f ≥ 5)] 17 ,
[(17, 6, μ
∗
; 3, [d z ] f ≥ 7/[d x ] f ≥ 5)] 17 ,
[(17, 4, μ
∗
; 3, [d z ] f ≥ 9/[d x ] f ≥ 5)] 17 and so on.
Theorem 7.7.8 Let q ≥ 5 be a prime power. Assume that k ≥ 1 and n ≥ 5 are
integers such that n ≤ q and k ≤ n − 4. Choose an n-tuple ζ = (ζ 0 , . . . , ζ n−1 ) of
distinct elements of F q and an n-tuple v = (v 0 , . . . , v n−1 ) of nonzero elements of
F q . Then an [(n, n − t − k − 1, μ
∗
; 2, [d z ] f /[d x ] f )] q AQCC, where (d z ) f ≥ t + 2,
(d x ) f ≥ k + 1 and 1 ≤ t ≤ n − k − 1 can be constructed.
Proof Similar to that of Theorem 7.7.7.
Example 7.7.4 From Theorem 7.7.8, we obtain AQCCs with parameters [(5, 1,
μ
∗
; 2, [d z ] f ≥ 4/[d x ] f ≥ 2)] 5 , [(7, 2, μ
∗
; 2, [d z ] f ≥ 4/[d x ] f ≥ 3)] 7 , [(7, 2, μ
∗
; 2,
[d z ] f ≥ 5/[d x ] f ≥ 2)] 7 , [(7, 1, μ
∗
; 2, [d z ] f ≥ 5/[d x ] f ≥ 3)] 7 .
7.7.4 Discussion
Let C be an [(n, k, μ; γ, d f )] q quantum convolutional code. Recall that C is said to
be pure if does not exist errors of weight less than d f in the stabilizer of C. Let us
recall the quantum generalized Singleton bound (GQSB) for quantum convolutional
codes.
Theorem 7.7.9 ([7]) The free distance of an [(n, k, μ; γ, d f )] q F q 2 -linear pure convolutional stabilizer code is bounded by d f ≤
n−k
2
2γ
n+k
+ 1
+ γ + 1.
The parameters of our AQCCs are given by [(n, k, μ; γ, [d z ] f /[d x ] f )] q , where
(d x ) f = min{wt(C 1 \C
⊥
2 ), wt(C 2 \C
⊥
1 )} and (d z ) f = max{wt(C 1 \C
⊥
2 ), wt(C 2 \ C
⊥
1 )}.
In this context, if one puts the constraint of pure codes, the free distance (d x ) f with
respect to qudit-flip errors satisfies the GQSB. However, much research remains to
7 Constructions of QCCs
⎡
⎢
⎢
⎢
⎣
w 0 ζ
t
0 w 1 ζ
t
1 · · · w n−1 ζ
t
n−1
0
0 0
0
. . .
. . .
. . .
. . .
0
0 0
0
⎤
⎥
⎥
⎥
⎦
D,
where w = (w 0 , . . . , w n−1 ) is a vector such that GRS
⊥
k (ζ, w) = GRS n−k (ζ, v). The
code V 1 has parameters (n, n − k − 2, 2; 1, [d 1 ] f ≥ k + 1) q and V
⊥
1 has parameters (n, k + 2, 2; μ
⊥
1 , [d 1 ]
⊥
f ) q . Similarly, V 2 is an (n, t, 1; 1, [d 2 ] f ) q code and V
⊥
2
is an (n, n − t, 1; μ
⊥
2 , [d 1 ]
⊥
f ≥ t + 2) q code. Then there exists an [(n, n − t − k −
2, μ
∗
; 3, [d z ] f /[d x ] f )] q code, where (d z ) f ≥ t + 2 and (d x ) f ≥ k + 1.
Example 7.7.3 From Theorem 7.7.7, we can construct AQCCs with parameters
[(5, 1, μ
∗
; 3, [d z ] f ≥ 3/[d x ] f ≥ 2)] 5 , [(7, 1, μ
∗
; 3, [d z ] f ≥ 4/[d x ] f ≥ 3)] 7 , [(8, 1,
μ
∗
; 3, [d z ] f ≥ 5/[d x ] f ≥ 3)] 8 , [(17, 7, μ
∗
; 3, [d z ] f ≥ 7/[d x ] f ≥ 4)] 17 ,
[(17, 7, μ
∗
; 3, [d z ] f ≥ 6/[d x ] f ≥ 5)] 17 ,
[(17, 6, μ
∗
; 3, [d z ] f ≥ 7/[d x ] f ≥ 5)] 17 ,
[(17, 4, μ
∗
; 3, [d z ] f ≥ 9/[d x ] f ≥ 5)] 17 and so on.
Theorem 7.7.8 Let q ≥ 5 be a prime power. Assume that k ≥ 1 and n ≥ 5 are
integers such that n ≤ q and k ≤ n − 4. Choose an n-tuple ζ = (ζ 0 , . . . , ζ n−1 ) of
distinct elements of F q and an n-tuple v = (v 0 , . . . , v n−1 ) of nonzero elements of
F q . Then an [(n, n − t − k − 1, μ
∗
; 2, [d z ] f /[d x ] f )] q AQCC, where (d z ) f ≥ t + 2,
(d x ) f ≥ k + 1 and 1 ≤ t ≤ n − k − 1 can be constructed.
Proof Similar to that of Theorem 7.7.7.
Example 7.7.4 From Theorem 7.7.8, we obtain AQCCs with parameters [(5, 1,
μ
∗
; 2, [d z ] f ≥ 4/[d x ] f ≥ 2)] 5 , [(7, 2, μ
∗
; 2, [d z ] f ≥ 4/[d x ] f ≥ 3)] 7 , [(7, 2, μ
∗
; 2,
[d z ] f ≥ 5/[d x ] f ≥ 2)] 7 , [(7, 1, μ
∗
; 2, [d z ] f ≥ 5/[d x ] f ≥ 3)] 7 .
7.7.4 Discussion
Let C be an [(n, k, μ; γ, d f )] q quantum convolutional code. Recall that C is said to
be pure if does not exist errors of weight less than d f in the stabilizer of C. Let us
recall the quantum generalized Singleton bound (GQSB) for quantum convolutional
codes.
Theorem 7.7.9 ([7]) The free distance of an [(n, k, μ; γ, d f )] q F q 2 -linear pure convolutional stabilizer code is bounded by d f ≤
n−k
2
2γ
n+k
+ 1
+ γ + 1.
The parameters of our AQCCs are given by [(n, k, μ; γ, [d z ] f /[d x ] f )] q , where
(d x ) f = min{wt(C 1 \C
⊥
2 ), wt(C 2 \C
⊥
1 )} and (d z ) f = max{wt(C 1 \C
⊥
2 ), wt(C 2 \ C
⊥
1 )}.
In this context, if one puts the constraint of pure codes, the free distance (d x ) f with
respect to qudit-flip errors satisfies the GQSB. However, much research remains to