3.4 Five-Qubit Code
33
3.4 Five-Qubit Code
The five-qubit code [88] was the first optimal quantum code exhibited in the literature.
Such code was constructed by Laflamme et al.. The stabilizer operators for the fivequbit code are
• X ⊗ Z ⊗ Z ⊗ X ⊗ I
• I ⊗ X ⊗ Z ⊗ Z ⊗ X
• X ⊗ I ⊗ X ⊗ Z ⊗ Z
• Z ⊗ X ⊗ I ⊗ X ⊗ Z
• X ⊗ X ⊗ X ⊗ X ⊗ X
• Z ⊗ Z ⊗ Z ⊗ Z ⊗ Z
The five-qubit code has parameters [[5, 1, 3]] and it is a maximum distance separable (MDS) code, because the parameters attain the quantum Singleton bound with
equality.
3.5 Stabilizer Codes
Let us consider the Pauli matrices {I, X, Y, Z }. Then it is easy to see that the set G 1 =
{±I, ±i I, ±X, ±i X, ±Y, ±iY, ±Z , ±i Z} endowed with the operation of matrix
multiplication is a group. Based on this fact, we can state the following definition.
Definition 3.5.1 The Pauli group on 1 qubit is defined by the group (G 1 , · m ), where
· m is the product of matrices.
More generally, we can define the general Pauli group on n qubits.
Definition 3.5.2 The general Pauli group on n qubits, denoted by G n , is the group
consisting of all n-fold tensor products of Pauli matrices with coefficients ±1 or ±i.
Exercise 3.5.1 Show that G n given in Definition 3.5.2 is a group.
It is interesting to note that until now we have dealt with quantum bits or tensor
product of quantum bits. With this theory in mind one can construct good quantum
error-correcting codes but only in the binary alphabet. Because of this limitation, it
is necessary to have a theory to construct quantum codes in nonbinary alphabets. A
giant step toward this was the work by Calderbank, Rains, Shor and Sloane [25].
In the past two decades, many authors [9–11, 53, 83–86, 132, 133] tried to obtain
different types of models in order to incorporate quantum code over nonbinary alphabets. Finally, in 2006, a brilliant work by Ketkar, Klappenecker, Kumar and Sarvepalli
[80] did this task. In fact, the authors generalized in several ways the formalism of
33
3.4 Five-Qubit Code
The five-qubit code [88] was the first optimal quantum code exhibited in the literature.
Such code was constructed by Laflamme et al.. The stabilizer operators for the fivequbit code are
• X ⊗ Z ⊗ Z ⊗ X ⊗ I
• I ⊗ X ⊗ Z ⊗ Z ⊗ X
• X ⊗ I ⊗ X ⊗ Z ⊗ Z
• Z ⊗ X ⊗ I ⊗ X ⊗ Z
• X ⊗ X ⊗ X ⊗ X ⊗ X
• Z ⊗ Z ⊗ Z ⊗ Z ⊗ Z
The five-qubit code has parameters [[5, 1, 3]] and it is a maximum distance separable (MDS) code, because the parameters attain the quantum Singleton bound with
equality.
3.5 Stabilizer Codes
Let us consider the Pauli matrices {I, X, Y, Z }. Then it is easy to see that the set G 1 =
{±I, ±i I, ±X, ±i X, ±Y, ±iY, ±Z , ±i Z} endowed with the operation of matrix
multiplication is a group. Based on this fact, we can state the following definition.
Definition 3.5.1 The Pauli group on 1 qubit is defined by the group (G 1 , · m ), where
· m is the product of matrices.
More generally, we can define the general Pauli group on n qubits.
Definition 3.5.2 The general Pauli group on n qubits, denoted by G n , is the group
consisting of all n-fold tensor products of Pauli matrices with coefficients ±1 or ±i.
Exercise 3.5.1 Show that G n given in Definition 3.5.2 is a group.
It is interesting to note that until now we have dealt with quantum bits or tensor
product of quantum bits. With this theory in mind one can construct good quantum
error-correcting codes but only in the binary alphabet. Because of this limitation, it
is necessary to have a theory to construct quantum codes in nonbinary alphabets. A
giant step toward this was the work by Calderbank, Rains, Shor and Sloane [25].
In the past two decades, many authors [9–11, 53, 83–86, 132, 133] tried to obtain
different types of models in order to incorporate quantum code over nonbinary alphabets. Finally, in 2006, a brilliant work by Ketkar, Klappenecker, Kumar and Sarvepalli
[80] did this task. In fact, the authors generalized in several ways the formalism of