6.1 Preliminaries
127
Similarly, if C ≤ F
n
q 2 is an additive code, then C
⊥ a denotes the trace-alternating
dual of C, where the trace-alternating form of two vectors v, w ∈ F
n
q 2 is defined as
v|w a = tr q/ p
v · w
q
− v
q
· w
β 2q − β 2
,
where (β, β
q
) is a normal basis of F
2
q over F q .
6.2 Error Groups and Asymmetric Codes
To start, we need to define the error model adopted to asymmetric quantum noise
channel.
Let H be the Hilbert space H = C
q
n = C
q
⊗ . . . ⊗ C
q . Let |x be the vectors of an
orthonormal basis of C
q , where the labels x are elements of F q . Consider a, b ∈ F q ;
the unitary operators X (a) and Z (b) on C
q are defined by X (a)|x =|x + a and
Z (b)|x = w
tr q/ p (bx)
|x, respectively, where w = exp(2πi/ p) is a pth root of unity.
Let a = (a 1 , . . . , a n ) ∈ F
n
q and b = (b 1 , . . . , b n ) ∈ F
n
q . Let
X (a) = X (a 1 ) ⊗ . . . ⊗ X (a n )
and
Z (b) = Z (b 1 ) ⊗ . . . ⊗ Z (b n )
be the tensor products of n error operators. It is easy to see that
E n = {X (a)Z (b) | a, b ∈ F
n
q }
is an error basis on the complex vector space C
q
n . The set
G n = {w
c X (a)Z (b) | a, b ∈ F
n
q , c ∈ F p }
is the error group associated with E n . For a quantum error e = w
c X (a)Z (b) ∈ G n ,
the X -weight is defined as
wt X (e) = #{i : 1 ≤ i ≤ n|a i = 0},
whereas the Z -weight is defined by
wt Z (e) = #{i : 1 ≤ i ≤ n|b i = 0}.
Recall that the symplectic (or quantum) weight wt(e) is defined similarly as in
the error model for quantum codes (see Definition 3.5.7):