t b a b a

Since the first qubit was flipped, it will disagree with the third; if the second qubit had been
flipped, the first and third would be the same. Therefore, we have narrowed down the error
to the first qubit and we can fix it simply by flipping it back. To handle possible bit-flips on
the other blocks of three, we can make the same comparisons inside the blocks.
However a phase-flip will leave the identity of the 0 and 1 alone, but alter their relative
phase. For example, such a phase-flip flips the sign of the first block of qubits, changing
| 000〉+ | 111〉 into | 000〉− | 111〉, and vice versa. By comparing the sign of the first block
of three with the second block of three, we can see that a sign error has occurred in one of
these blocks. Then by comparing the signs of the first and third block of three, we narrow
the sign error down to the first block, and flip the sign back to what it should be. Again,
we do not actually measure the sign, which will destroy the state. Instead, we only compare
signs between the blocks.
If we have both a bit-flip and a phase-flip on the same qubit, it is easy to see that going
through both processes above, we can detect and correct them. Thus, the Shor code could
correct any one-qubit error.
5.4 CSS Codes
Calderbank-Shor-Steane codes, or CSS codes for short, are quantum codes which let us
identify and correct large qubit errors, i.e., errors described by Pauli matrices. CSS codes
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