3.2 The Shor Code
31
Third Case—Phase flip and bit flip on the same qubit. This case is a direct
application of the previous ones. More precisely, it suffices to apply the procedure
shown in the First Case to recover the qubit affected by the bit flip action of the
channel after applying the Second Case to correct the phase flip error occurred.
These facts are true because both error-correction process are independent.
Therefore, the stabilizer for the Shor nine qubit code is
• Z ⊗ Z ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I
• I ⊗ Z ⊗ Z ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I
• I ⊗ I ⊗ I ⊗ Z ⊗ Z ⊗ I ⊗ I ⊗ I ⊗ I
• I ⊗ I ⊗ I ⊗ I ⊗ Z ⊗ Z ⊗ I ⊗ I ⊗ I
• I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ Z ⊗ Z ⊗ I
• I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ Z ⊗ Z
• X ⊗ X ⊗ X ⊗ X ⊗ X ⊗ X ⊗ I ⊗ I ⊗ I
• I ⊗ I ⊗ I ⊗ X ⊗ X ⊗ X ⊗ X ⊗ X ⊗ X
The Shor code has parameters [[9, 1, 3]], that is, the code utilizes nine qubits to
encode a qubit, and it is capable of correcting one arbitrary quantum error.
Until now we have seen only errors of the types phase and bit flip. But the question
is: Is the Shor code capable of correcting an arbitrary error? The answer for this
question is yes!
To see this, note that the Pauli matrices I, X, Y, Z span M 2 (C), the vector space
of the matrices of order 2 with complex entries. Since X Z = −iY then the matrices
I, X, Z , X Z also span M 2 (C). Thus, given an error matrix E in one qubit we can write
E = a 1 I + a 2 X + a 3 Z + a 4 X Z, where a i ∈ C for all i = 1, 2, 3, 4. Therefore, if |v
is a qubit, we have the quantum state E|v = a 1 |v + a 2 X |v + a 3 Z |v + a 4 X Z|v.
By the measurement of the error syndrome the state E|v collapses to one of
the states |v, X |v, Z |v or X Z|v. Since these operators are invertible, we then
apply the inverse operator to recover the initial quantum state. In other words, if the
code is capable of correcting errors of the type bit flip, phase flip and bit-phase flip
combined in a given qubit, then the code is capable of correcting all arbitrary errors
of such qubit. This is an interesting feature of quantum codes: if the code C corrects
a suitable discrete subset of errors then C corrects all types of (continuum) errors.
This fact is essential in quantum error correction; based on this property, it is possible
to construct efficient quantum codes against arbitrary quantum errors.
Theorem 3.2.1 ([121, Theorem 10.1]) (Quantum error-correction conditions) Let
C be a quantum code, and let P be the projector onto C. Assume that E is a quantum
operation with operation elements {E i }. Then there exists an error-correction operation R correcting E on C if and only if P E
†
i E j P = a i j P for some Hermitian matrix
A = [a i j ] with complex entries. The operation elements {Ei} are called errors for
the noise E. If R exists then {Ei} is called a correctable set of errors.
This previous discussion can be summarized in the next result.
Theorem 3.2.2 ([121, Theorem 10.2]) Let C be a quantum code and let R be the
error-correction operation constructed in the proof of Theorem 3.2.1 to recover
31
Third Case—Phase flip and bit flip on the same qubit. This case is a direct
application of the previous ones. More precisely, it suffices to apply the procedure
shown in the First Case to recover the qubit affected by the bit flip action of the
channel after applying the Second Case to correct the phase flip error occurred.
These facts are true because both error-correction process are independent.
Therefore, the stabilizer for the Shor nine qubit code is
• Z ⊗ Z ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I
• I ⊗ Z ⊗ Z ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I
• I ⊗ I ⊗ I ⊗ Z ⊗ Z ⊗ I ⊗ I ⊗ I ⊗ I
• I ⊗ I ⊗ I ⊗ I ⊗ Z ⊗ Z ⊗ I ⊗ I ⊗ I
• I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ Z ⊗ Z ⊗ I
• I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ Z ⊗ Z
• X ⊗ X ⊗ X ⊗ X ⊗ X ⊗ X ⊗ I ⊗ I ⊗ I
• I ⊗ I ⊗ I ⊗ X ⊗ X ⊗ X ⊗ X ⊗ X ⊗ X
The Shor code has parameters [[9, 1, 3]], that is, the code utilizes nine qubits to
encode a qubit, and it is capable of correcting one arbitrary quantum error.
Until now we have seen only errors of the types phase and bit flip. But the question
is: Is the Shor code capable of correcting an arbitrary error? The answer for this
question is yes!
To see this, note that the Pauli matrices I, X, Y, Z span M 2 (C), the vector space
of the matrices of order 2 with complex entries. Since X Z = −iY then the matrices
I, X, Z , X Z also span M 2 (C). Thus, given an error matrix E in one qubit we can write
E = a 1 I + a 2 X + a 3 Z + a 4 X Z, where a i ∈ C for all i = 1, 2, 3, 4. Therefore, if |v
is a qubit, we have the quantum state E|v = a 1 |v + a 2 X |v + a 3 Z |v + a 4 X Z|v.
By the measurement of the error syndrome the state E|v collapses to one of
the states |v, X |v, Z |v or X Z|v. Since these operators are invertible, we then
apply the inverse operator to recover the initial quantum state. In other words, if the
code is capable of correcting errors of the type bit flip, phase flip and bit-phase flip
combined in a given qubit, then the code is capable of correcting all arbitrary errors
of such qubit. This is an interesting feature of quantum codes: if the code C corrects
a suitable discrete subset of errors then C corrects all types of (continuum) errors.
This fact is essential in quantum error correction; based on this property, it is possible
to construct efficient quantum codes against arbitrary quantum errors.
Theorem 3.2.1 ([121, Theorem 10.1]) (Quantum error-correction conditions) Let
C be a quantum code, and let P be the projector onto C. Assume that E is a quantum
operation with operation elements {E i }. Then there exists an error-correction operation R correcting E on C if and only if P E
†
i E j P = a i j P for some Hermitian matrix
A = [a i j ] with complex entries. The operation elements {Ei} are called errors for
the noise E. If R exists then {Ei} is called a correctable set of errors.
This previous discussion can be summarized in the next result.
Theorem 3.2.2 ([121, Theorem 10.2]) Let C be a quantum code and let R be the
error-correction operation constructed in the proof of Theorem 3.2.1 to recover