Bit flips alone are a very limited kind of error a qubit could undergo.
Consider phase flips, which have no classical equivalent. A phase-flip quantum
channel is defined as one that only allows single phase flips, that is, with
probability p, |1i → −|1i. Under the action of this channel, a generic state
transforms as
|ψi = α|0i + β|1i → α|0i − β|1i. (10.13)
If we represent |ψi in the X basis spanned by |±i =
1
√
2
[|0i ± |1i],
|ψi = α
0
|+i + β
0
|−i, (10.14)
then phase flip causes |+i → |−i and |−i → |+i. Thus, the phase-flip case
is unitarily equivalent to the bit-flip case since we can change basis to the
X-basis by applying a H transform. Error correction can be followed just as
in the bit-flip case, except that we now transform everything to the X basis
by using the H gate at appropriate places. The 3-qubit encoding that will
correct phase flip errors should then be
|0i → |+ + +i, |1i → |− − −i, (10.15)
which is achieved by the circuit in Figure 10.9.
|ψi
• •
H
|0i
H
|
¯
ψi = α|+ + +i + β|− − −i
|0i
H
FIGURE 10.9: Encoding circuit for 3-qubit phase-flip code.
Syndrome measurement and recovery is now identical to the bit-flip case,
except that we work in the X basis by applying an H gate to each qubit. The
stabilizers are the operators
ˆ
O
0
I
= H
⊗3
Z
1
Z
2
H
⊗3
= X
1
X
2
,
ˆ
O
0
II
= H
⊗3
Z
2
Z
3
H
⊗3
= X
2
X
3
. (10.16)
Measuring these operators distinguishes the syndromes. This is like com-
paring the signs of the corresponding qubit values. Finally, recovery is per-
formed by applying HXH = Z to the appropriate qubit.
Exercise 10.1. Construct the circuit for detecting phase flip syndromes and for
correcting them
Leave a Reply