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
Steane code is the first in the family of quantum Hamming codes, codes with parameters
It is also a quantum color code.
In a quantum error-correcting code, the codespace is the subspace of the overall Hilbert space where all logical states live.
-qubit stabilizer code, we can describe this subspace by its Pauli stabilizing group, the set of all
-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
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.
The tensor products are often omitted in notation for brevity.
states of the Steane code are Arbitrary codestates are of the form