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
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