32
3 Quantum Error-Correcting Codes
from a noise process E with operation elements {E i }. Assume that F is a quantum
operation with operation elements F j which are linear combinations of the E i . Then
R also corrects for the effects of the noise process F on C.
3.3 The Steane Code
The Steane [[7, 1, 3]] seven qubit code is an example of the application of the
Calderbank–Shor–Steane (CSS) quantum code construction that will be presented in
Sect. 3.6. For more details on how to encode an arbitrary state or to learn syndrome
measurement strategies for the Steane code, see [24, 158, 159].
The basis states for this code are given in the sequence:
|0 L :=
1
√
8
[|0000000 + |1010101 + |0110011 + |1100110
+|0001111 + |1011010 + |0111100 + |1101001]
and
|1 L :=
1
√
8
[|1111111 + |0101010 + |1001100 + |0011001
+|1110000 + |0100101 + |1000011 + |0010110]
The stabilizer for the seven qubit code due to Steane is
• I ⊗ I ⊗ I ⊗ X ⊗ X ⊗ X ⊗ X
• I ⊗ X ⊗ X ⊗ I ⊗ I ⊗ X ⊗ X
• X ⊗ I ⊗ X ⊗ I ⊗ X ⊗ I ⊗ X
• I ⊗ I ⊗ I ⊗ Z ⊗ Z ⊗ Z ⊗ Z
• I ⊗ Z ⊗ Z ⊗ I ⊗ I ⊗ Z ⊗ Z
• Z ⊗ I ⊗ Z ⊗ I ⊗ Z ⊗ I ⊗ Z
The code has parameters [[7, 1, 3]], that is, it can correct an arbitrary error in a single
qubit and utilizes seven qubits in the encoding process. The classical self-orthogonal
code utilized in the encoding process is the [7, 4, 3] Hamming code with parity check
matrix
H =
⎡
⎣
0 0 0 1 1 1 1
0 1 1 0 0 1 1
1 0 1 0 1 0 1
⎤
⎦ .