Let us estimate the probability for the above technique to yield an uncor-
rupted state, considering a channel characterized by flipping of a qubit with
probability p < 1/2. We list in Table 10.3 the probability of occurrence of
various corrupted states in order of decreasing probability.
The probability that our procedure corrects errors is therefore
P(correct) = (1 − p)
3
+ 3p(1 − p)
2
= 1 − 3p
2
+ 2p
3
. (10.7)
and the probability that we have an erroneous state is
P(incorrect) = p
2
(1 − p) + p
3
= 3p
2
− 2p
3
< P(correct). (10.8)
Quantum Error Correction 203
TABLE 10.3: Probability of occurrence of corrupted states in a bit-flip chan-
nel.
Number of flips states probability
0 |ψ
0
i (1 − p)
3
1
X
1
|ψ
0
i
X
2
|ψ
0
i
X
3
|ψ
0
i
3p(1 − p)
2
2
X
1
X
2
|ψ
0
i
X
2
X
3
|ψ
0
i
X
1
X
3
|ψ
0
i
3p
2
(1 − p)
3 X
1
X
2
X
3
|ψ
0
i p
3
Box 10.1: Error Correction and Fidelity
People in the error-correcting business are not satisfied with this, and try
to work out schemes that are better by comparing fidelities. We will see in
Section 11.3.2 that the fidelity of two states is defined by their degree of
overlap. If we start with a pure state |ψi, errors cause it to become a mixed
state with probability p of transforming by X. This is represented by the
density matrix
ρ
bf
= pX|ψihψ|X + (1 − p)|ψihψ|. (10.9)
The fidelity of state transmission without error correction is given by
F =
p
hψ|ρ
bf
|ψi (10.10)
=
q
phψ|X|ψi
2
+ (1 − p) (10.11)
This has a minimum value of
√
1 − p, when the first term is zero. If we make
use of the above protocol for error correction then for the 3-qubit encoded
state,
ρ
corrected
=
(1 − p)
3
+ 3p(1 − p)
2
|ψihψ|
+ (ρ for 2 or more bit flips) , (10.12)
and the fidelity, using only the first two terms, is
F ≥
p
(1 − p)
3
+ 3p(1 − p)
2
,
the same as the above.
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