Steane code

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

[ H 0 0 H ] {\displaystyle {\begin{bmatrix}H&0\\0&H\end{bmatrix}}}

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

H = [ 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 ] . {\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 [ [ 7 , 1 , 3 ] ] {\displaystyle [[7,1,3]]} Steane code is the first in the family of quantum Hamming codes, codes with parameters [ [ 2 r − 1 , 2 r − 1 − 2 r , 3 ] ] {\displaystyle [[2^{r}-1,2^{r}-1-2r,3]]} for integers r ≥ 3 {\displaystyle r\geq 3} . It is also a quantum color code.

In a quantum error-correcting code, the codespace is the subspace of the overall Hilbert space where all logical states live. In an n {\displaystyle n} -qubit stabilizer code, we can describe this subspace by its Pauli stabilizing group, the set of all n {\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 2 {\displaystyle 2} -dimensional subspace of its 2 7 {\displaystyle 2^{7}} -dimensional Hilbert space.

In the stabilizer formalism, the Steane code has 6 generators:

I I I X X X X I X X I I X X X I X I X I X I I I Z Z Z Z I Z Z I I Z Z Z I Z I Z I Z . {\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, I I I X X X X {\displaystyle IIIXXXX} is just shorthand for I ⊗ I ⊗ I ⊗ X ⊗ X ⊗ X ⊗ X {\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 X {\displaystyle X} gate on each of the last four qubits. The tensor products are often omitted in notation for brevity.

The logical X {\displaystyle X} and Z {\displaystyle Z} gates are

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

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

| 0 ⟩ L = 1 8 [ | 0000000 ⟩ + | 1010101 ⟩ + | 0110011 ⟩ + | 1100110 ⟩ + | 0001111 ⟩ + | 1011010 ⟩ + | 0111100 ⟩ + | 1101001 ⟩ ] | 1 ⟩ L = X L | 0 ⟩ L . {\displaystyle {\begin{aligned}|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{aligned}}}

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

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