- Source: Steane code
The Steane code is a tool in quantum error correction introduced by Andrew Steane in 1996. It is a CSS code (Calderbank-Shor-Steane), using the classical binary [7,4,3] Hamming code to correct for both qubit flip errors (X errors) and phase flip errors (Z errors). The Steane code encodes one logical qubit in 7 physical qubits and is able to correct arbitrary single qubit errors.
Its check matrix in standard form is
[
H
0
0
H
]
{\displaystyle {\begin{bmatrix}H&0\\0&H\end{bmatrix}}}
where H is the parity-check matrix of the Hamming code and is given by
H
=
[
1
0
0
1
0
1
1
0
1
0
1
1
0
1
0
0
1
0
1
1
1
]
.
{\displaystyle H={\begin{bmatrix}1&0&0&1&0&1&1\\0&1&0&1&1&0&1\\0&0&1&0&1&1&1\end{bmatrix}}.}
The
[
[
7
,
1
,
3
]
]
{\displaystyle [[7,1,3]]}
Steane code is the first in the family of quantum Hamming codes, codes with parameters
[
[
2
r
−
1
,
2
r
−
1
−
2
r
,
3
]
]
{\displaystyle [[2^{r}-1,2^{r}-1-2r,3]]}
for integers
r
≥
3
{\displaystyle r\geq 3}
. It is also a quantum color code.
Expression in the stabilizer formalism
In a quantum error-correcting code, the codespace is the subspace of the overall Hilbert space where all logical states live. In an
n
{\displaystyle n}
-qubit stabilizer code, we can describe this subspace by its Pauli stabilizing group, the set of all
n
{\displaystyle n}
-qubit Pauli operators which stabilize every logical state. The stabilizer formalism allows us to define the codespace of a stabilizer code by specifying its Pauli stabilizing group. We can efficiently describe this exponentially large group by listing its generators.
Since the Steane code encodes one logical qubit in 7 physical qubits, the codespace for the Steane code is a
2
{\displaystyle 2}
-dimensional subspace of its
2
7
{\displaystyle 2^{7}}
-dimensional Hilbert space.
In the stabilizer formalism, the Steane code has 6 generators:
I
I
I
X
X
X
X
I
X
X
I
I
X
X
X
I
X
I
X
I
X
I
I
I
Z
Z
Z
Z
I
Z
Z
I
I
Z
Z
Z
I
Z
I
Z
I
Z
.
{\displaystyle {\begin{aligned}&IIIXXXX\\&IXXIIXX\\&XIXIXIX\\&IIIZZZZ\\&IZZIIZZ\\&ZIZIZIZ.\end{aligned}}}
Note that each of the above generators is the tensor product of 7 single-qubit Pauli operations. For instance,
I
I
I
X
X
X
X
{\displaystyle IIIXXXX}
is just shorthand for
I
⊗
I
⊗
I
⊗
X
⊗
X
⊗
X
⊗
X
{\displaystyle I\otimes I\otimes I\otimes X\otimes X\otimes X\otimes X}
, that is, an identity on the first three qubits and an
X
{\displaystyle X}
gate on each of the last four qubits. The tensor products are often omitted in notation for brevity.
The logical
X
{\displaystyle X}
and
Z
{\displaystyle Z}
gates are
X
L
=
X
X
X
X
X
X
X
Z
L
=
Z
Z
Z
Z
Z
Z
Z
.
{\displaystyle {\begin{aligned}X_{L}&=XXXXXXX\\Z_{L}&=ZZZZZZZ.\end{aligned}}}
The logical
|
0
⟩
{\displaystyle |0\rangle }
and
|
1
⟩
{\displaystyle |1\rangle }
states of the Steane code are
|
0
⟩
L
=
1
8
[
|
0000000
⟩
+
|
1010101
⟩
+
|
0110011
⟩
+
|
1100110
⟩
+
|
0001111
⟩
+
|
1011010
⟩
+
|
0111100
⟩
+
|
1101001
⟩
]
|
1
⟩
L
=
X
L
|
0
⟩
L
.
{\displaystyle {\begin{aligned}|0\rangle _{L}=&{\frac {1}{\sqrt {8}}}[|0000000\rangle +|1010101\rangle +|0110011\rangle +|1100110\rangle \\&+|0001111\rangle +|1011010\rangle +|0111100\rangle +|1101001\rangle ]\\|1\rangle _{L}=&X_{L}|0\rangle _{L}.\end{aligned}}}
Arbitrary codestates are of the form
|
ψ
⟩
=
α
|
0
⟩
L
+
β
|
1
⟩
L
{\displaystyle |\psi \rangle =\alpha |0\rangle _{L}+\beta |1\rangle _{L}}
.
References
Steane, Andrew (1996). "Multiple-Particle Interference and Quantum Error Correction". Proc. R. Soc. Lond. A. 452 (1954): 2551–2577. arXiv:quant-ph/9601029. Bibcode:1996RSPSA.452.2551S. doi:10.1098/rspa.1996.0136. S2CID 8246615.
Kata Kunci Pencarian:
- Steane code
- CSS code
- Quantum error correction
- Hamming(7,4)
- Steane
- Andrew Steane
- Glossary of quantum computing
- ZX-calculus
- Stabilizer code
- Toric code
