Steane code

Short description: Code for quantum correction

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

$\left[\begin{array}{cc}H& 0\\ 0& H\end{array}\right]$

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

$H=\left[\begin{array}{ccccccc}1& 0& 0& 1& 0& 1& 1\\ 0& 1& 0& 1& 1& 0& 1\\ 0& 0& 1& 0& 1& 1& 1\end{array}\right].$

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 $r\ge 3$. 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 $n$-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:

$\begin{array}{rl}& IIIXXXX\\ & IXXIIXX\\ & XIXIXIX\\ & IIIZZZZ\\ & IZZIIZZ\\ & ZIZIZIZ.\end{array}$

Note that each of the above generators is the tensor product of 7 single-qubit Pauli operations. For instance, $IIIXXXX$ is just shorthand for $I\otimes I\otimes I\otimes X\otimes X\otimes X\otimes X$, 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

$\begin{array}{rl}{X}_L& =XXXXXXX\\ Z_L& =ZZZZZZZ.\end{array}$

The logical $|0⟩$ and $|1⟩$ states of the Steane code are

$\begin{array}{rl}|0{⟩}_L=& \frac{1}{\sqrt{8}}[|0000000⟩+|1010101⟩+|0110011⟩+|1100110⟩\\ & +|0001111⟩+|1011010⟩+|0111100⟩+|1101001⟩]\\ |1{⟩}_L=& X_L|0{⟩}_L.\end{array}$

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

References

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