  We present a general theory of entanglement-assisted quantum convolutional
coding. The codes have a convolutional or memory structure, they assume that
the sender and receiver share noiseless entanglement prior to quantum
communication, and they are not restricted to possess the
Calderbank-Shor-Steane structure as in previous work. We provide two
significant advances for quantum convolutional coding theory. We first show how
to "expand" a given set of quantum convolutional generators. This expansion
step acts as a preprocessor for a polynomial symplectic Gram-Schmidt
orthogonalization procedure that simplifies the commutation relations of the
expanded generators to be the same as those of entangled Bell states (ebits)
and ancilla qubits. The above two steps produce a set of generators with
equivalent error-correcting properties to those of the original generators. We
then demonstrate how to perform online encoding and decoding for a stream of
information qubits, halves of ebits, and ancilla qubits. The upshot of our
theory is that the quantum code designer can engineer quantum convolutional
codes with desirable error-correcting properties without having to worry about
the commutation relations of these generators.
