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

. It is also a quantum color code.

Expression in the stabilizer formalism

-qubit stabilizer code-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.

-dimensional Hilbert space.

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

gate on each of the last four qubits. The tensor products are often omitted in notation for brevity.

gates are

states of the Steane code are

.

References