3.6 Calderbank–Shor–Steane Construction
41
which is similar to a bit flip type of error. To correct it, we proceed analogously to
the correction of e b , i.e., we introduce an ancilla and apply the parity check matrix
G 2 for C
⊥
2 (which is a generator matrix for the code C 2 ) as done previously, to obtain
G 2 e p and to correct the error e p , obtaining the quantum state
1
√
2 n /|C 2 |
w∈C
⊥
2
(−1)
c·w
|w.
Applying again the Hadamard gate to each qubit results in the original state
|c + C 2 :=
1
√ |C 2 |
x∈C 2
|c + x.
Therefore, the resulting quantum code CSS(C 1 , C 2 ) has parameters [[n, k 1 − k 2 ]]
and can correct arbitrary errors up to t qubits.
After some years of research, the CSS construction was generalized to nonbinary
alphabets, as we can see in the following lemma.
Lemma 3.6.1 ([25, 80]) Let q be a prime power. Let C 1 and C 2 denote two classical
linear codes both over the field F q , with parameters [n, k 1 , d 1 ] q and [n, k 2 , d 2 ] q ,
respectively, such that C 2 ⊂ C 1 . Then there exists an [[n, K = k 1 −k 2 , D]] q CSS
quantum code, where D = min{wt (c) : c ∈ (C 1 \C 2 ) ∪ (C
⊥
2 \C
⊥
1 )}.
Remark 3.6.1 As it was said previously, the CSS codes shown in Lemma 3.6.1
are constructed over nonbinary alphabets. In fact, the original version of the CSS
code construction (as we have presented in this subsection) was presented for binary
alphabets. For nonbinary alphabets, a quantum state is called quantum digits or qudits
(for short).
Remark 3.6.2 In the following section (Sect. 4) we present carefully the background
concerning linear codes mentioned in Lemma 3.6.1. We prefer not to present Theory
of Linear Codes before in order to maintain the quantum theory without interruption.
41
which is similar to a bit flip type of error. To correct it, we proceed analogously to
the correction of e b , i.e., we introduce an ancilla and apply the parity check matrix
G 2 for C
⊥
2 (which is a generator matrix for the code C 2 ) as done previously, to obtain
G 2 e p and to correct the error e p , obtaining the quantum state
1
√
2 n /|C 2 |
w∈C
⊥
2
(−1)
c·w
|w.
Applying again the Hadamard gate to each qubit results in the original state
|c + C 2 :=
1
√ |C 2 |
x∈C 2
|c + x.
Therefore, the resulting quantum code CSS(C 1 , C 2 ) has parameters [[n, k 1 − k 2 ]]
and can correct arbitrary errors up to t qubits.
After some years of research, the CSS construction was generalized to nonbinary
alphabets, as we can see in the following lemma.
Lemma 3.6.1 ([25, 80]) Let q be a prime power. Let C 1 and C 2 denote two classical
linear codes both over the field F q , with parameters [n, k 1 , d 1 ] q and [n, k 2 , d 2 ] q ,
respectively, such that C 2 ⊂ C 1 . Then there exists an [[n, K = k 1 −k 2 , D]] q CSS
quantum code, where D = min{wt (c) : c ∈ (C 1 \C 2 ) ∪ (C
⊥
2 \C
⊥
1 )}.
Remark 3.6.1 As it was said previously, the CSS codes shown in Lemma 3.6.1
are constructed over nonbinary alphabets. In fact, the original version of the CSS
code construction (as we have presented in this subsection) was presented for binary
alphabets. For nonbinary alphabets, a quantum state is called quantum digits or qudits
(for short).
Remark 3.6.2 In the following section (Sect. 4) we present carefully the background
concerning linear codes mentioned in Lemma 3.6.1. We prefer not to present Theory
of Linear Codes before in order to maintain the quantum theory without interruption.