CSS code

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]