3.2 The Shor Code
27
• P 0 := |000000| + |111111| (associated with no occurrence of error);
• P 1 := |100100| + |011011| (error in the first qubit);
• P 2 := |010010| + |101101| (error in the second qubit);
• P 3 := |001001| + |110110| (error in the third qubit);
Let us see how the detection process works. Assume without loss of generality
(w.l.o.g.) that an error corrupted the second qubit; then the state
|v enc = a|000 + b|111
becomes
|w = a|010 + b|101.
Hence (according to Postulate 2.1), applying P 2 to |w we have
p(2) = =w|P 2 |w
= (a
∗
010| + b
∗
101|)|010010|(a|010 + b|101)
+ (a
∗
010| + b
∗
101|)|101101|(a|010 + b|101)
= |a|
2
+ |b|
2
= 1.
Therefore, we know that the error occurred in the second qubit.
Remark 3.2.1 In this detection process, it is interesting to observe that the corrupted
state a|010 + b|101 is not affected by the syndrome measurement. In fact, the
syndrome contains only information about the corrupted qubit, but no information
about the state being measured (a and b are not known). This is excellent, since none
of the measurements applied for the decoding operation destroys the superpositions
of quantum states that must be preserved by applying the encoding process.
To recover the original encoded state |v enc , note that since the error has occurred in
the second qubit and since the channel flips the qubit, then it suffices to flip to second
qubit again. Thus, the encoded state |v enc is recovered. It is clear that this procedure
holds in general, independently in which qubit the error has occurred. If no error
occurs in this process, by applying the operator P 0 we have p(0) = =w|P 0 |w = 1,
that is, we know that (probability one) no error has occurred. Proceeding similarly,
we can recover the original encoded state in all cases.
3.2.1.2 Second Procedure of Measurement
We next present an alternative way to proceed with the measurement process. Assume
that we replace the four measurements operators P 0 , P 1 , P 2 , P 3 by the observables
Z 1 Z 2 := Z ⊗ Z ⊗ I and Z 2 Z 3 := I ⊗ Z ⊗ Z , both with eigenvalues −1 and +1. To
perform the measurement, we first apply Z 1 Z 2 and, in the sequence, the observable
Z 2 Z 3 .
27
• P 0 := |000000| + |111111| (associated with no occurrence of error);
• P 1 := |100100| + |011011| (error in the first qubit);
• P 2 := |010010| + |101101| (error in the second qubit);
• P 3 := |001001| + |110110| (error in the third qubit);
Let us see how the detection process works. Assume without loss of generality
(w.l.o.g.) that an error corrupted the second qubit; then the state
|v enc = a|000 + b|111
becomes
|w = a|010 + b|101.
Hence (according to Postulate 2.1), applying P 2 to |w we have
p(2) = =w|P 2 |w
= (a
∗
010| + b
∗
101|)|010010|(a|010 + b|101)
+ (a
∗
010| + b
∗
101|)|101101|(a|010 + b|101)
= |a|
2
+ |b|
2
= 1.
Therefore, we know that the error occurred in the second qubit.
Remark 3.2.1 In this detection process, it is interesting to observe that the corrupted
state a|010 + b|101 is not affected by the syndrome measurement. In fact, the
syndrome contains only information about the corrupted qubit, but no information
about the state being measured (a and b are not known). This is excellent, since none
of the measurements applied for the decoding operation destroys the superpositions
of quantum states that must be preserved by applying the encoding process.
To recover the original encoded state |v enc , note that since the error has occurred in
the second qubit and since the channel flips the qubit, then it suffices to flip to second
qubit again. Thus, the encoded state |v enc is recovered. It is clear that this procedure
holds in general, independently in which qubit the error has occurred. If no error
occurs in this process, by applying the operator P 0 we have p(0) = =w|P 0 |w = 1,
that is, we know that (probability one) no error has occurred. Proceeding similarly,
we can recover the original encoded state in all cases.
3.2.1.2 Second Procedure of Measurement
We next present an alternative way to proceed with the measurement process. Assume
that we replace the four measurements operators P 0 , P 1 , P 2 , P 3 by the observables
Z 1 Z 2 := Z ⊗ Z ⊗ I and Z 2 Z 3 := I ⊗ Z ⊗ Z , both with eigenvalues −1 and +1. To
perform the measurement, we first apply Z 1 Z 2 and, in the sequence, the observable
Z 2 Z 3 .