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{{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 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 [[canonical form|standard form]] is
: <math>
\begin{bmatrix}
H & 0 \\
0 & H
\end{bmatrix}
</math>

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

: <math>
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}.
</math>

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

== Expression in the stabilizer formalism ==
{{Main|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 <math>n</math>-qubit [[stabilizer code]], we can describe this subspace by its Pauli stabilizing group, the set of all <math>n</math>-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 [[generator (mathematics)|generators]].

Since the Steane code encodes one logical qubit in 7 physical qubits, the codespace for the Steane code is a <math>2</math>-dimensional subspace of its <math>2^7</math>-dimensional Hilbert space.

In the [[stabilizer formalism]], the Steane code has 6 generators:
:<math>
\begin{align}
& IIIXXXX \\
& IXXIIXX \\
& XIXIXIX \\
& IIIZZZZ \\
& IZZIIZZ \\
& ZIZIZIZ.
\end{align}
</math>
Note that each of the above generators is the tensor product of 7 single-qubit Pauli operations. For instance, <math>IIIXXXX</math> is just shorthand for <math>I \otimes I \otimes I \otimes X \otimes X \otimes X \otimes X</math>, that is, an identity on the first three qubits and an <math>X</math> gate on each of the last four qubits. The tensor products are often omitted in notation for brevity.

The logical <math>X</math> and <math>Z</math> gates are
:<math>
\begin{align}
X_L & = XXXXXXX \\
Z_L & = ZZZZZZZ.
\end{align}
</math>

The logical <math>| 0 \rangle</math> and <math>| 1 \rangle</math> states of the Steane code are
:<math>
\begin{align}
| 0 \rangle_L = & \frac{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.
\end{align}
</math>
Arbitrary codestates are of the form <math>| \psi \rangle = \alpha | 0 \rangle_L + \beta | 1 \rangle_L</math>.

==References==
*{{cite journal |last=Steane |first=Andrew |authorlink=Andrew Steane |title=Multiple-Particle Interference and Quantum Error Correction |journal=Proc. R. Soc. Lond. A |volume=452 | year=1996 |pages=2551–2577 |doi=10.1098/rspa.1996.0136 |issue=1954|arxiv=quant-ph/9601029 |bibcode=1996RSPSA.452.2551S |s2cid=8246615 }}
{{Quantum computing}}

[[Category:Quantum information science]]

Steane code - Revision history

140.180.240.242 at 00:18, 30 April 2024