TheSteane code is a tool inquantum error correction introduced byAndrew Steane in 1996. It is aCSS code (Calderbank-Shor-Steane), using the classical binary [7,4,3]Hamming code to correct for bothqubit 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.

Itscheck matrix instandard form is

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

The${\displaystyle [[7,1,3]]}$ Steane code is the first in the family of quantum Hamming codes, codes with parameters${\displaystyle [[2^r-1,2^r-1-2r,3]]}$ for integers${\displaystyle r\ge 3}$. 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. In an${\displaystyle n}$-qubitstabilizer code, we can describe this subspace by its Pauli stabilizing group, the set of all${\displaystyle n}$-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 itsgenerators.

Since the Steane code encodes one logical qubit in 7 physical qubits, the codespace for the Steane code is a${\displaystyle 2}$-dimensional subspace of its${\displaystyle 2^7}$-dimensional Hilbert space.

In thestabilizer 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,${\displaystyle IIIXXXX}$ is just shorthand for${\displaystyle I\otimes I\otimes I\otimes X\otimes X\otimes X\otimes X}$, that is, an identity on the first three qubits and an${\displaystyle X}$ gate on each of the last four qubits. The tensor products are often omitted in notation for brevity.

The logical${\displaystyle X}$ and${\displaystyle Z}$ gates are

The logical${\displaystyle |0⟩}$ and${\displaystyle |1⟩}$ states of the Steane code are

Arbitrary codestates are of the form${\displaystyle |\psi ⟩=\alpha |0{⟩}_L+\beta |1{⟩}_L}$.

• Steane, Andrew (1996). "Multiple-Particle Interference and Quantum Error Correction".Proc. R. Soc. Lond. A.452 (1954):2551–2577.arXiv:quant-ph/9601029.Bibcode:1996RSPSA.452.2551S.doi:10.1098/rspa.1996.0136.S2CID 8246615.

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