36
3 Quantum Error-Correcting Codes
We must know how the errors act on multiple qudits. In other words, it is necessary
to know if the tensor products of a finite number of nice error basis is also a nice
error basis.
Proposition 3.5.2 Let β 1 and β 2 be two sets of nice error bases on C
q . Then the set
β = {E 1 ⊗ E 2 |E 1 ∈ β 1 , E 2 ∈ β 2 }
is also a nice error basis.
By applying induction, we know that Proposition 3.5.2 also holds for a finite
number of tensor products. Assuming that a = (a 1 , a 2 , . . . , a n ) is a vector in F
n
q , we
denote X (a) = X (a 1 ) ⊗ X (a 2 ) ⊗ · · · ⊗ X (a n ) and Z(a) = Z (a 1 ) ⊗ Z (a 2 ) ⊗ · · · ⊗
Z (a n ) for tensor products of n error operators.
Corollary 3.5.1 Assume the notation above. Then the set
ε n = {X (a)Z(b)|a, b ∈ F
n
q }
is a nice error basis on the complex vector space C
q
n .
Hence, we have a complete characterization (the model) of the errors that can corrupt
the quantum digits.
In the sequence we define the concept of stabilizer code. We start with the group
G n generated by the matrices of ε n :
G n = {ωX (a)Z(b)|a, b ∈ F
n
q , c ∈ F p },
which is called error group associated with ε n . A stabilizer code is the joint eigenspace
with eigenvalue 1 of some subgroup of G n , as we see in the following.
Definition 3.5.6 Let S be a subgroup of the error group G n . A stabilizer code Q =
{0} is a subspace of C
q
n satisfying the equality
Q =
E∈S
{|v ∈ C
q
n : E|v = |v}.
We need to define the weight of an element in the error group G n . To this end, let
a, b be two vectors in F
n
q and consider the vector (a|b) ∈ F
2n
q .
Definition 3.5.7 The symplectic weight swt((a|b)) of (a|b) is the number of nonzero
ordered pairs of the form (a i , b i ), where i = 1, 2, . . . , n, i.e.,
swt((a|b)) = |{i |(a i , b i ) = (0, 0)}|.
Definition 3.5.8 Let E = ω
c
X (a)Z(b) be an element in the error group G n . Then
the weight wt(E) of E is defined as wt(E) = swt((a|b)).
3 Quantum Error-Correcting Codes
We must know how the errors act on multiple qudits. In other words, it is necessary
to know if the tensor products of a finite number of nice error basis is also a nice
error basis.
Proposition 3.5.2 Let β 1 and β 2 be two sets of nice error bases on C
q . Then the set
β = {E 1 ⊗ E 2 |E 1 ∈ β 1 , E 2 ∈ β 2 }
is also a nice error basis.
By applying induction, we know that Proposition 3.5.2 also holds for a finite
number of tensor products. Assuming that a = (a 1 , a 2 , . . . , a n ) is a vector in F
n
q , we
denote X (a) = X (a 1 ) ⊗ X (a 2 ) ⊗ · · · ⊗ X (a n ) and Z(a) = Z (a 1 ) ⊗ Z (a 2 ) ⊗ · · · ⊗
Z (a n ) for tensor products of n error operators.
Corollary 3.5.1 Assume the notation above. Then the set
ε n = {X (a)Z(b)|a, b ∈ F
n
q }
is a nice error basis on the complex vector space C
q
n .
Hence, we have a complete characterization (the model) of the errors that can corrupt
the quantum digits.
In the sequence we define the concept of stabilizer code. We start with the group
G n generated by the matrices of ε n :
G n = {ωX (a)Z(b)|a, b ∈ F
n
q , c ∈ F p },
which is called error group associated with ε n . A stabilizer code is the joint eigenspace
with eigenvalue 1 of some subgroup of G n , as we see in the following.
Definition 3.5.6 Let S be a subgroup of the error group G n . A stabilizer code Q =
{0} is a subspace of C
q
n satisfying the equality
Q =
E∈S
{|v ∈ C
q
n : E|v = |v}.
We need to define the weight of an element in the error group G n . To this end, let
a, b be two vectors in F
n
q and consider the vector (a|b) ∈ F
2n
q .
Definition 3.5.7 The symplectic weight swt((a|b)) of (a|b) is the number of nonzero
ordered pairs of the form (a i , b i ), where i = 1, 2, . . . , n, i.e.,
swt((a|b)) = |{i |(a i , b i ) = (0, 0)}|.
Definition 3.5.8 Let E = ω
c
X (a)Z(b) be an element in the error group G n . Then
the weight wt(E) of E is defined as wt(E) = swt((a|b)).