30
3 Quantum Error-Correcting Codes
and
|1
encode
−−−→ |1 L :=
(|000 − |111)(|000 − |111)(|000 − |111)
2
√
2
.
We explain now how the Shor code can correct phase flip and bit flip errors on
any single qubit. In fact, the analysis we perform here is in the same spirit with as
the second procedure of measurement shown in Sect. 3.2. We present a scheme in
order to clarify the understanding of how the code can recover the initial state.
First Case—Correcting bit flip errors. Assume w.l.o.g. that an error has
occurred in the seventh qubit. We then perform the measurement of the observable
Z 7 Z 8 , finding the eigenvalue −1. After this, we follow by measuring Z 8 Z 9 obtaining
therefore the eigenvalue +1; so the seventh qubit is the corrupted one. Applying bit
flip again in the seventh qubit one has the initial state. Proceeding similarly, we can
detect and recover any (single) state which was corrupted by bit flip errors, by means
of the measurement of the observables Z 1 Z 2 , Z 2 Z 3 , Z 4 Z 5 , Z 5 Z 6 , Z 7 Z 8 and Z 8 Z 9 .
Second Case—Correcting phase flip errors. Assume that an error occurs in the
second qubit for example. Due to the properties of tensor product, the first block
of three qubits |000 + |111 becomes |000 − |111 and |000 − |111 becomes
|000 + |111. In other words, the two basis states now read as
|0 L
channel
− −−− →
(|000 − |111)(|000 + |111)(|000 + |111)
2
√
2
and
|1 L
channel
− −−− →
(|000 + |111)(|000 − |111)(|000 − |111)
2
√
2
.
After this, we compare the sign of the first and the second blocks of qubits, i.e.,
|000 − |111 is compared with |000 + |111 (has distinct sign) and |000 + |111
is compared with |000 − |111 (has distinct sign). Here, we consider that the
block has the same sign in the cases (|000 + |111)(|000 + |111) and (|000 −
|111)(|000 − |111) and they have different signs in the cases (|000 + |111)
(|000 − |111) and (|000 − |111)(|000 + |111). Next, we perform a comparison between the sign of the second and the third blocks of qubits, i.e., (|000 +
|111)(|000 + |111) (has the same sign) and (|000 − |111)(|000 − |111) (has
the same sign). Thus, we know that the phase flip has corrupted one of the three first
qubits. To recover the initial encoded state it suffices to flip the sign of the first block
of three qubits.
Such procedure to detect phase flip errors presented above is similar to perform
the measurement of the observables 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 .
3 Quantum Error-Correcting Codes
and
|1
encode
−−−→ |1 L :=
(|000 − |111)(|000 − |111)(|000 − |111)
2
√
2
.
We explain now how the Shor code can correct phase flip and bit flip errors on
any single qubit. In fact, the analysis we perform here is in the same spirit with as
the second procedure of measurement shown in Sect. 3.2. We present a scheme in
order to clarify the understanding of how the code can recover the initial state.
First Case—Correcting bit flip errors. Assume w.l.o.g. that an error has
occurred in the seventh qubit. We then perform the measurement of the observable
Z 7 Z 8 , finding the eigenvalue −1. After this, we follow by measuring Z 8 Z 9 obtaining
therefore the eigenvalue +1; so the seventh qubit is the corrupted one. Applying bit
flip again in the seventh qubit one has the initial state. Proceeding similarly, we can
detect and recover any (single) state which was corrupted by bit flip errors, by means
of the measurement of the observables Z 1 Z 2 , Z 2 Z 3 , Z 4 Z 5 , Z 5 Z 6 , Z 7 Z 8 and Z 8 Z 9 .
Second Case—Correcting phase flip errors. Assume that an error occurs in the
second qubit for example. Due to the properties of tensor product, the first block
of three qubits |000 + |111 becomes |000 − |111 and |000 − |111 becomes
|000 + |111. In other words, the two basis states now read as
|0 L
channel
− −−− →
(|000 − |111)(|000 + |111)(|000 + |111)
2
√
2
and
|1 L
channel
− −−− →
(|000 + |111)(|000 − |111)(|000 − |111)
2
√
2
.
After this, we compare the sign of the first and the second blocks of qubits, i.e.,
|000 − |111 is compared with |000 + |111 (has distinct sign) and |000 + |111
is compared with |000 − |111 (has distinct sign). Here, we consider that the
block has the same sign in the cases (|000 + |111)(|000 + |111) and (|000 −
|111)(|000 − |111) and they have different signs in the cases (|000 + |111)
(|000 − |111) and (|000 − |111)(|000 + |111). Next, we perform a comparison between the sign of the second and the third blocks of qubits, i.e., (|000 +
|111)(|000 + |111) (has the same sign) and (|000 − |111)(|000 − |111) (has
the same sign). Thus, we know that the phase flip has corrupted one of the three first
qubits. To recover the initial encoded state it suffices to flip the sign of the first block
of three qubits.
Such procedure to detect phase flip errors presented above is similar to perform
the measurement of the observables 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 .