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 where H is the parity-check matrix of the Hamming code and is given by The 7,1,3 Steane code is the first in the family of quantum Hamming codes, codes with parameters 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 is the subspace of the overall Hilbert space where all logical states live. 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 2-dimensional subspace of its 2^7-dimensional Hilbert space. In the stabilizer formalism, the Steane code has 6 generators: Note that each of the above generators is the tensor product of 7 single-qubit Pauli operations. For instance, IIIXXXX is just shorthand for, that is, an identity on the first three qubits and an X gate on each of the last four qubits. The tensor products are often omitted in notation for brevity. The logical X and Z gates are The logical | 0 \rangle and | 1 \rangle states of the Steane code are Arbitrary codestates are of the form.