CSS code

In quantum error correction, Calderbank–Shor–Steane (CSS) codes, named after their inventors, Robert Calderbank, Peter Shor¹ and Andrew Steane,² are a special type of stabilizer code constructed from classical linear codes with some special properties. Examples of CSS codes include the Shor code, Steane code, the toric code, and more general surface codes.

Let ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ be two (classical) ${\displaystyle [n,k_{1}]}$ and ${\displaystyle [n,k_{2}]}$ linear codes such, that ${\displaystyle C_{2}\subset C_{1}}$ and ${\displaystyle C_{1},C_{2}^{\perp }}$ both have minimal distance ${\displaystyle \geq 2t+1}$, where ${\displaystyle C_{2}^{\perp }}$ is the dual code to ${\displaystyle C_{2}}$. Then define ${\displaystyle {\text{CSS}}(C_{1},C_{2})}$, the CSS code of ${\displaystyle C_{1}}$ over ${\displaystyle C_{2}}$ as an ${\displaystyle [n,k_{1}-k_{2},d]}$ code, with ${\displaystyle d\geq 2t+1}$ as follows:

Define for $${\displaystyle x\in C_{1}:{|}x+C_{2}\rangle :={\frac {1}{\sqrt {|C_{2}|}}}\sum _{y\in C_{2}}{|}x+y\rangle ,}$$ where ${\displaystyle +}$ is bitwise addition modulo 2. Then ${\displaystyle {\text{CSS}}(C_{1},C_{2})}$ as quantum correcting code ${\displaystyle [[n,k_{1}-k_{2},d]]}$ defined as ${\displaystyle \{{|}x+C_{2}\rangle \mid x\in C_{1}\}}$.³

In the stabilizer code formalism, all CSS codes have stabilizers composed of tensor products of Pauli matrices such that each stabilizer contains either only Pauli X operations or only Pauli Z operations. The Shor code and the Steane code are examples of this condition. The five-qubit error correcting code is not a CSS code because it mixes X and Z in its stabilizers.⁴

As with classical linear codes, the limit of how many qubits can be corrected is also given by the Gilbert–Varshamov bound.³