Steane code

Expression in the stabilizer formalism

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 \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{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 = & \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} Arbitrary codestates are of the form | \psi \rangle = \alpha | 0 \rangle_L + \beta | 1 \rangle_L.

References