Steane code explained

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

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

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

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

[[ 7 , 1 , 3 ]] Steane code is the first in the family of quantum Hamming codes, codes with parameters [[ 2 ^{r-1,} 2 ^{r-1-2r,} 3 ]] for integers r \geq 3 . It is also a quantum color code.

Expression in the stabilizer formalism

See main article: 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:

\begin{align} & IIIXXXX \\ & IXXIIXX \\ & XIXIXIX \\ & IIIZZZZ \\ & IZZIIZZ \\ & ZIZIZIZ. \end{align} Note that each of the above generators is the tensor product of 7 single-qubit Pauli operations. For instance, IIIXXXX is just shorthand for I ⊗ I ⊗ I ⊗ X ⊗ X ⊗ X ⊗ X , 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 \begin{align} X _{L} & = XXXXXXX \\ Z _{L} & = ZZZZZZZ. \end{align}

The logical

| 0 \rangle and | 1 \rangle states of the Steane code are\begin{align} | 0 \rangle _{L} = & (1)/(\sqrt{ 8) } [| 0000000 \rangle + | 1010101 \rangle + | 0110011 \rangle + | 1100110 \rangle \\ & + | 0001111 \rangle + | 1011010 \rangle + | 0111100 \rangle + | 1101001 \rangle ] \\| 1 \rangle_L = & X_L | 0 \rangle_L. \endArbitrary codestates are of the form | \psi \rangle = \alpha | 0 \rangle _{L} + \beta | 1 \rangle _{L} .

References

• Steane . Andrew . Andrew Steane . Multiple-Particle Interference and Quantum Error Correction . Proc. R. Soc. Lond. A . 452 . 1996 . 2551–2577 . 10.1098/rspa.1996.0136 . 1954. quant-ph/9601029 . 1996RSPSA.452.2551S . 8246615 .