Quantum error-correcting code (QECC)

Denoting Hilbert spaces by the letter \(\mathsf{H}\), a quantum code \((U,\cal{E})\) is a partial isometry \(U:\mathsf{H}_{\text{logical}}\to\mathsf{H}_{\text{physical}}\) with a set of correctable errors \(\cal{E}\) with the following property: there exists a quantum operation \(\cal{D}\) such that for all \(E\in\cal{E}\) and states \(|\psi\rangle\in\mathsf{H}_{\text{logical}}\), \begin{align} {\cal D} (EU|\psi\rangle\langle\psi|U^{\dagger}E^{\dagger}) = c(E,|\psi\rangle)|\psi\rangle\langle\psi| \end{align} for some constant \(c\).

Equivalently, correction capability is determined by the Knill-Laflamme conditions, which may admit infinite terms due to non-normalizability of ideal code states in the case of codes with infinite-dimenional physical spaces. A code that satisfies these conditions approximately, i.e., up to some small quantifiable error, is called an approximate code. These conditions can also be formulated in terms of a dual Heisenberg picture, where correctability is checked for some algebra of observables [1].