Quantum Error-Correcting Code

A quantum error-correcting code is a subspace of the Hilbert space of qubits that encodes quantum states so that a specified set of errors can be detected or corrected. If is the orthogonal projection onto , then a set of error operators is correctable exactly when the Knill-Laflamme conditions

hold for constants and all , (Knill and Laflamme 1997).

A quantum code encoding logical qubits into physical qubits and having distance is denoted . A stabilizer code is the common eigenspace of an abelian group contained in the -qubit Pauli group, with . This construction converts many questions about quantum error correction into questions about finite groups and linear algebra (Gottesman 1997).