  Though the theory of quantum error correction is intimately related to the
classical coding theory, in particular, one can construct quantum error
correction codes (QECCs) from classical codes with the dual containing
property, this does not necessarily imply that the computational complexity of
decoding QECCs is the same as their classical counterparts. Instead, decoding
QECCs can be very much different from decoding classical codes due to the
degeneracy property. Intuitively, one expect degeneracy would simplify the
decoding since two different errors might not and need not be distinguished in
order to correct them. However, we show that general quantum decoding problem
is NP-hard regardless of the quantum codes being degenerate or non-degenerate.
This finding implies that no considerably fast decoding algorithm exists for
the general quantum decoding problems, and suggests the existence of a quantum
cryptosystem based on the hardness of decoding QECCs.
