t b a b a

α | 0 + β | 1
| 0
| 0
noisy
channel
H H
H H
H H
| 0
| 0
Figure 5.1: Encoding and decoding circuits for the phase-flip code.
5.3 The Shor Code
α | 0 + β | 1
The Shor code is a simple code which can protect against the effects of an arbitrary error on
a single qubit. The code is a combination of the three qubit phase-flip and bit-flip codes. It
encode the logic qubits as follows:
| 0〉 →| 0L〉 = (| 000〉+ | 111〉)(| 000〉+ | 111〉)(| 000〉+ | 111〉)
| 1〉 →| 1L〉 = (| 000〉− | 111〉)(| 000〉− | 111〉)(| 000〉− | 111〉)
The Shor code is able to protect against phase-flip and bit-flip errors on any qubit.
Suppose a bit-flip error occurs on some qubit , say the first one, switching | 0〉 and | 1〉.
Then by comparing the first two qubits, we find they are different, which is not allowed by
any valid codeword. Therefore an error should have occurred which flipped either the first
or second qubit. Note that we do not actually measure the first and second qubits, since this
would destroy the superposition in the codeword; we can make a measurement Z1Z2 which
only measures the difference between them. Now we compare the first and third qubits.
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| 0
| 0