t b a b a

derive from classical linear codes and represent a systematic way of building a quantum error
correcting code.
In order to construct a CSS code you need to have two classical linear codes, C1[n, k1]
and C2[n, k2] such that C2 ⊂ C1 and C1, C ⊥ 2 both correct t errors. The resulting code is a
quantum code CSS(C1/C2) capable of correcting errors on t qubits, which encodes k1 − k2
logical qubits in n physical qubits, so this code is [n, k1 − k2].
The encoding is in a vector space spanned by all states constructed by taking a codeword
x ∈ C1 and then adding to it the whole of C2:
| x + C2〉 = 1
� |C2|
�
y∈C2
| x + y〉,
where + is bitwise addition modulo 2 and |C2| is the number of elements in C2. Suppose
x ′ ∈ C1 and x − x ′ ∈ C2. then it is easy to see that | x + C2〉 =| x ′ + C2〉, and thus the state
| x + C2〉 depends only upon the coset of C1/C2. Furthermore, if x and x ′ belong to different
cosets of C2, then for no y, y ′ ∈ C2 does x + y = x ′ + y ′ , and therefore | x + C2〉 and | x ′ + C2〉
are orthonormal states.
An important example of CSS code is the Steane code which is constructed using the
classic [7,4,3] Hamming code with parity check matrix:
⎡
⎢ 0
⎢
H1 = ⎢ 0
⎢
⎣
0
1
0
1
1
0
1
0
1
1
1 ⎥
1
⎥
⎦
1 0 1 0 1 0 1
71
⎤