t b a b a

example shows that the code cannot detect two qubit errors; indeed, the two bit-flip error
E1E2 = σx ⊗ σx ⊗ σI ⊗ σI ⊗ σI
is indistinguishable from E9 = σI ⊗σI ⊗σI ⊗σz ⊗σI because both E1E2 and E9 anti-commute
with the same subset of stabilizers, {M1, M4}, and give the same error syndrome. Therefore,
the 5-qubit code can correct any single qubit error, but cannot correct two qubit errors.
If we know the correlated error, for example, E3 = σI ⊗ σI ⊗ σx ⊗ σI ⊗ σI in the system,
from Table 5.2, it is easy to see that the errors E3Ei have distinct error syndromes for
1 ≤ i ≤ 10. Therefore we can identify these errors and correct them.
5.6 Other Codes
Quantum Convolutional Codes is a family of codes which are used in order to protect
a stream of quantum information in a long distance communication. They are the gener-
alization to the quantum domain of their classical analogues, and hence inherit the most
important properties. They impose a specific form such that on-line encoding, decoding and
error correction become possible even in the presence of an infinitely long to-be-protected
stream of information. In convolutional codes, qubits are not considered as independent of
each other: the encoding operation cannot be decomposed into a tensor product of encod-
ing operations acting on a small number of qubits. By contrast, an (N, K)-block code can
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