Class of quantum error correcting codes
For the document presentation language, see
CSS .
In
quantum error correction , CSS codes , named after their inventors,
Robert Calderbank ,
Peter Shor
[1]
and
Andrew Steane ,
[2] are a special type of
stabilizer code constructed from classical codes with some special properties. An example of a CSS code is the
Steane code .
Construction
Let
C
1
{\displaystyle C_{1}}
and
C
2
{\displaystyle C_{2}}
be two (classical)
n
,
k
1
{\displaystyle [n,k_{1}]}
,
n
,
k
2
{\displaystyle [n,k_{2}]}
codes such, that
C
2
⊂
C
1
{\displaystyle C_{2}\subset C_{1}}
and
C
1
,
C
2
⊥
{\displaystyle C_{1},C_{2}^{\perp }}
both have
minimal distance
≥
2
t
+
1
{\displaystyle \geq 2t+1}
, where
C
2
⊥
{\displaystyle C_{2}^{\perp }}
is the code
dual to
C
2
{\displaystyle C_{2}}
. Then define
CSS
(
C
1
,
C
2
)
{\displaystyle {\text{CSS}}(C_{1},C_{2})}
, the CSS code of
C
1
{\displaystyle C_{1}}
over
C
2
{\displaystyle C_{2}}
as an
n
,
k
1
−
k
2
,
d
{\displaystyle [n,k_{1}-k_{2},d]}
code, with
d
≥
2
t
+
1
{\displaystyle d\geq 2t+1}
as follows:
Define for
x
∈
C
1
:
|
x
+
C
2
⟩
:=
{\displaystyle x\in C_{1}:{|}x+C_{2}\rangle :=}
1
/
|
C
2
|
{\displaystyle 1/{\sqrt {{|}C_{2}{|}}}}
∑
y
∈
C
2
|
x
+
y
⟩
{\displaystyle \sum _{y\in C_{2}}{|}x+y\rangle }
, where
+
{\displaystyle +}
is bitwise addition modulo 2. Then
CSS
(
C
1
,
C
2
)
{\displaystyle {\text{CSS}}(C_{1},C_{2})}
is defined as
{
|
x
+
C
2
⟩
∣
x
∈
C
1
}
{\displaystyle \{{|}x+C_{2}\rangle \mid x\in C_{1}\}}
.
References
Nielsen, Michael A. ;
Chuang, Isaac L. (2010).
Quantum Computation and Quantum Information (2nd ed.). Cambridge: Cambridge University Press.
ISBN
978-1-107-00217-3 .
OCLC
844974180 .
External links