  In this dissertation, I present a general method for studying quantum error
correction codes (QECCs). This method not only provides us an intuitive way of
understanding QECCs, but also leads to several extensions of standard QECCs,
including the operator quantum error correction (OQECC), the
entanglement-assisted quantum error correction (EAQECC). Furthermore, we can
combine both OQECC and EAQECC into a unified formalism, the
entanglement-assisted operator formalism. This provides great flexibility of
designing QECCs for different applications. Finally, I show that the
performance of quantum low-density parity-check codes will be largely improved
using entanglement-assisted formalism.
