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Steane code

## Summary

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

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

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

${\displaystyle H={\begin{bmatrix}1&0&0&1&0&1&1\\0&1&0&1&1&0&1\\0&0&1&0&1&1&1\end{bmatrix}}.}$

The ${\displaystyle [[7,1,3]]}$ 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 ${\displaystyle r\geq 3}$. 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 ${\displaystyle n}$ -qubit stabilizer 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 its generators.

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 the stabilizer formalism, the Steane code has 6 generators:

{\displaystyle {\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, ${\displaystyle IIIXXXX}$  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

{\displaystyle {\begin{aligned}X_{L}&=XXXXXXX\\Z_{L}&=ZZZZZZZ.\end{aligned}}}

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

{\displaystyle {\begin{aligned}|0\rangle _{L}=&|0000000\rangle +|1010101\rangle +|0110011\rangle +|1100110\rangle \\&+|0001111\rangle +|1011010\rangle +|0111100\rangle +|1101001\rangle \\|1\rangle _{L}=&X_{L}|0\rangle _{L}.\end{aligned}}}

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

## References

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