40
3 Quantum Error-Correcting Codes
Since the classical syndrome H 1 e 1 is known, then C 1 tells us the vector error e b .
To recover the state, we apply the NOT gate in all the qubits corrupted by the error,
which leads to the state
1
√
|C 2 |
x∈C 2
(−1)
(c+x)·e p |c + x.
Our next task is to detect the qubits corrupted by the phase flip error.
Phase flip Detection. Starting from the state
1
√
|C 2 |
x∈C 2
(−1)
(c+x)·e p |c + x,
the Hadamard gate
H =
1
√
2
1 1
1 −1
is applied to each qubit producing the state
1
√ |C 2 |2 n
v∈F
n
2
x∈C 2
(−1)
(c+x)·(e p +v)
|v,
which can be rewritten as
1
√
|C 2 |2 n
w∈F
n
2
x∈C 2
(−1)
(c+x)·w
|w + e p ,
where w = v + e p (notice that e p + e p is the zero vector). If w ∈ C
⊥
2 then w · x = 0
for all x ∈ C 2 ; so
x∈C 2
(−1)
xw
= 1 + 1 + · · · + 1
|C 2 | times
= |C 2 |.
On the other hand, if w /
∈ C
⊥
2 then it is easy to see that
x∈C 2
(−1)
xw
= 0.
Hence, the state can be rewritten as
1
√
2 n /|C 2 |
w∈C
⊥
2
(−1)
c·w
|w + e p ,