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 qubit flip errors (X errors) and the dual of the Hamming code, the [7,3,4] code, to correct for 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

{\begin{bmatrix}H&0\\0&H\end{bmatrix}}

where H is the parity-check matrix of the Hamming code and is given by

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 $[[7,1,3]]$ Steane code is the first in the family of quantum Hamming codes, codes with parameters $[[2^{r}-1,2^{r}-1-2r,3]]$ for integers $r\geq 3$ . It is also a quantum color code.

Expression in the stabilizer formalism

In a quantum error correcting code, the codespace (the set of all logical states) is a subspace of the overall Hilbert space. In an $n$ -qubit stabilizer code, we can describe this subspace by its Pauli stabilizing group, the set of all $n$ -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 two dimensional subspace of the $2^{7}$ dimensional Hilbert space. If $|0_{L}\rangle$ and $|1_{L}\rangle$ are the logical $|0\rangle$ and $|1\rangle$ states encoded in the Steane code, then an arbitrary codestate is of the form $|\psi \rangle =\alpha |0_{L}\rangle +\beta |1_{L}\rangle$ .

In the stabilizer formalism, the Steane code has 6 generators:

{\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, the first generator tells us that applying an identity gate to the first three gates and an X gate to each of the other qubits will preserve any codestate. In other words, for any arbitrary codestate $|\psi \rangle =\alpha |0_{L}\rangle +\beta |1_{L}\rangle$ , we find $(I\otimes I\otimes I\otimes X\otimes X\otimes X\otimes X)|\psi \rangle =|\psi \rangle$ . For brevity, the tensor products are often omitted in notation.