3.2 The Shor Code
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Definition 3.2.2 Let |v = a|0 + b|1 be a qubit. The phase flip channel acts on
|v as follows:
• |v
channel
−−−→ Z |v = a|0 − b|1 with probability p;
• |v
channel
−−−→ |v with probability 1 − p.
The procedure to recover the encoded state is to turn the phase flip channel into a
bit flip channel. In order to do this, let us consider the qubit basis |++ = (|0 + |1)/2
and |−− = (|0 − |1)/2. Since Z |++ = |−− and Z |−− = |++, Z acts as a bit flip in
such vectors. We then perform the encoding:
|0
encode
−−−→ |0 L := | + +++
and
|1
encode
−−−→ |1 L := | − −−−.
In this manner we can protect at least one qubit against phase flip errors. From this
moment, the encoding, detection and the recovery process is the same as the bit flip
channel with respect to the basis |++ and |−−.
3.2.3 The Shor Code
Here we present the Shor code, the first quantum error-correcting code to protect an
arbitrary single qubit against an arbitrary quantum error. The code is constructed by
means of concatenation of qubits as we can see in the following.
The construction of this code is based on the three qubit bit flip and the three qubit
phase flip codes presented in Sects. 3.2.1 and 3.2.2, respectively.
The stages of construction of the Shor code is given in the sequence.
(1) The first stage is to utilize the three qubit phase flip code to encode the qubit,
that is, |0
encode
−−−→ | + +++ and |1
encode
−−−→ | − −−−.
(2) Each of these qubits (namely, |++ and |−−) are encoded by applying the
three qubit phase flip code, i.e., |++
encode
−−−→ (|000 + |111)/
√
2 and |−−
encode
−−−→
(|000 − |111)/
√
2.
Thus, the resulting code is the Shor code given by
|0
encode
−−−→ |0 L :=
(|000 + |111)(|000 + |111)(|000 + |111)
2
√
2