Let’s now consider a channel that can produce both bit flips and phase
flips. A code that combines bit and phase flip coding should protect against
these errors. A simple way to do this is to first encode for phase flips:
|0i → |+ + +i; |1i → |− − −i
and then encode using the bit flip code:
|+i →
1
√
2
(|000i + |111i) ; |−i →
1
√
2
(|000i − |111i) ,
so that we have the final 9-qubit encoding
|0i →
1
2
√
2
(|000i + |111i)
⊗3
; |1i →
1
2
√
2
(|000i − |111i)
⊗3
. (10.17)
Such a code is called a concatenated code, and this particular 9-qubit code was
first proposed by Peter Shor. The circuit to achieve this encoding is obtained
by concatenating the circuits for the phase flip and the bit flip encoding, as
shown in Figure 10.10. The syndrome generators are easy to construct: bit-
α|0i + β|1i
• •
H
• •
|0i
|0i
|0i
H
• •
|0i
|0i
|0i
H
• •
|0i
|0i
FIGURE 10.10: Encoding circuit for the 9-qubit Shor code
flips in each block can be detected by measuring (Z
1
Z
2
, Z
2
Z
3
), (Z
4
Z
5
, Z
5
Z
6
)
and (Z
7
Z
8
, Z
8
Z
9
). Further, phase flips between blocks can be distinguished
by measuring X
1
X
2
X
3
X
4
X
5
X
6
and X
4
X
5
X
6
X
7
X
8
X
9
.
206 Introduction to Quantum Physics and Information Processing
Exercise 10.2. Construct the circuit for error correction for this case.
Note that with eight stabilizers, we have a possibility of correcting for 2
8
different errors, but we have only tried to look at bit/phase flips of 9 qubits,
which is 3 × 9 + 1 = 28! Thus this scheme is highly redundant. More efficient
schemes using fewer encoding qubits have been proposed, and Shor’s 9-qubit
code is of purely historical interest now. The reason it is important to study
this code is that it shows that it is possible to simultaneously correct for both
bit flips and phase flips. Now it turns out that this will actually correct for
arbitrary single-qubit errors, since, as we are about to show, any such error
can be thought of as a combination of just bit flips and phase flips
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