38
3 Quantum Error-Correcting Codes
More generally, we have the following result to stabilizer codes.
Lemma 3.5.2 (Quantum Singleton bound) ([80, Corollary 28]) The parameters of
an ((n, K , d)) q stabilizer code with K > 1 satisfy the inequality
K ≤ q
n−2d+2
.
Another bound much utilized for quantum codes is the quantum Hamming bound.
Although we do not utilize such bound in the quantum code construction presented
in this book, we present it here for completeness.
Lemma 3.5.3 (Quantum Hamming Bound) (see [38, Lemma 12]; see also [80, p.
9]) Let C be an ((n, K , d)) q pure stabilizer code. Then the parameters of C satisfy
the inequality:
q
n
K
≥
(d−1)/2
i=0
(q
2
− 1)
i
n
i
.
3.6 Calderbank–Shor–Steane Construction
The Calderbank–Shor–Steane (CSS for short) code construction is one of the most
interesting code constructions shown in the literature [121]. In fact, it is the first
construction method exhibited in the literature in the sense that one can derive families
of quantum codes by applying such construction, and not only few codes with specific
parameters. Such method utilizes two classical linear nested codes (or an Euclidean
self-orthogonal linear code) to address the problem of correcting phase and qubit flip
errors. The CSS codes form a subclass of the class of the stabilizer codes (see for
instance [80]). We next present a detailed construction of these codes.
The process starts with two binary linear codes C 1 and C 2 with parameters
[n, k 1 , d 1 ] and [n, k 2 , d 2 ], respectively, such that C 2 ⊂ C 1 and both C 1 and C
⊥
2 correct t errors. Based on these classical codes we define a quantum code CSS(C 1 , C 2 )
(this notation is not usual in the literature but we prefer to adopt it here to maintain
the notation of the textbook [121]) as follows.
Assume that c, c
∈ C 1 are two codewords of C 1 . Define the following relation
on C 1 : c ≈ c
⇐⇒ c − c
∈ C 2 . It is easy to see that ≈ is an equivalence relation on
C 1 . Moreover, it is not difficult to see that the equivalence class c determined by a
codeword c ∈ C 1 is equal to c = c + C 2 := {c + x|x ∈ C 2 }. Thus, the cosets c + C 2
form a partition of C 1 . Since we are working with quantum states (qubits or tensor
product of qubits) then we must adapt the notation. We then define the quantum state
|c + C 2 (already normalized) as
|c + C 2 :=
1
√ |C 2 |
x∈C 2
|c + x,
3 Quantum Error-Correcting Codes
More generally, we have the following result to stabilizer codes.
Lemma 3.5.2 (Quantum Singleton bound) ([80, Corollary 28]) The parameters of
an ((n, K , d)) q stabilizer code with K > 1 satisfy the inequality
K ≤ q
n−2d+2
.
Another bound much utilized for quantum codes is the quantum Hamming bound.
Although we do not utilize such bound in the quantum code construction presented
in this book, we present it here for completeness.
Lemma 3.5.3 (Quantum Hamming Bound) (see [38, Lemma 12]; see also [80, p.
9]) Let C be an ((n, K , d)) q pure stabilizer code. Then the parameters of C satisfy
the inequality:
q
n
K
≥
(d−1)/2
i=0
(q
2
− 1)
i
n
i
.
3.6 Calderbank–Shor–Steane Construction
The Calderbank–Shor–Steane (CSS for short) code construction is one of the most
interesting code constructions shown in the literature [121]. In fact, it is the first
construction method exhibited in the literature in the sense that one can derive families
of quantum codes by applying such construction, and not only few codes with specific
parameters. Such method utilizes two classical linear nested codes (or an Euclidean
self-orthogonal linear code) to address the problem of correcting phase and qubit flip
errors. The CSS codes form a subclass of the class of the stabilizer codes (see for
instance [80]). We next present a detailed construction of these codes.
The process starts with two binary linear codes C 1 and C 2 with parameters
[n, k 1 , d 1 ] and [n, k 2 , d 2 ], respectively, such that C 2 ⊂ C 1 and both C 1 and C
⊥
2 correct t errors. Based on these classical codes we define a quantum code CSS(C 1 , C 2 )
(this notation is not usual in the literature but we prefer to adopt it here to maintain
the notation of the textbook [121]) as follows.
Assume that c, c
∈ C 1 are two codewords of C 1 . Define the following relation
on C 1 : c ≈ c
⇐⇒ c − c
∈ C 2 . It is easy to see that ≈ is an equivalence relation on
C 1 . Moreover, it is not difficult to see that the equivalence class c determined by a
codeword c ∈ C 1 is equal to c = c + C 2 := {c + x|x ∈ C 2 }. Thus, the cosets c + C 2
form a partition of C 1 . Since we are working with quantum states (qubits or tensor
product of qubits) then we must adapt the notation. We then define the quantum state
|c + C 2 (already normalized) as
|c + C 2 :=
1
√ |C 2 |
x∈C 2
|c + x,