  The Hamiltonian model of quantum error correction code in the literature is
often constructed with the help of its stabilizer formalism. But there have
been many known examples of nonadditive codes which are beyond the standard
quantum error correction theory using the stabilizer formalism. In this paper,
we suggest the other type of Hamiltonian formalism for quantum error correction
code without involving the stabilizer formalism, and explain it by studying the
Shor nine-qubit code and its generalization. In this Hamiltonian formulation,
the unitary evolution operator at a specific time is a unitary basis
transformation matrix from the product basis to the quantum error correction
code. This basis transformation matrix acts as an entangling quantum operator
transforming a separate state to an entangled one, and hence the entanglement
nature of the quantum error correction code can be explicitly shown up.
Furthermore, as it forms a unitary representation of the Artin braid group, the
quantum error correction code can be described by a braiding operator.
Moreover, as the unitary evolution operator is a solution of the quantum
Yang--Baxter equation, the corresponding Hamiltonian model can be explained as
an integrable model in the Yang--Baxter theory. On the other hand, we
generalize the Shor nine-qubit code and articulate a topic called quantum error
correction codes using Greenberger-Horne-Zeilinger states to yield new
nonadditive codes and channel-adapted codes.
