  It is well known that no quantum error correcting code of rate $R$ can
correct adversarial errors on more than a $(1-R)/4$ fraction of symbols. But
what if we only require our codes to *approximately* recover the message? We
construct efficiently-decodable approximate quantum codes against adversarial
error rates approaching the quantum Singleton bound of $(1-R)/2$, for any
constant rate $R$. Moreover, the size of the alphabet is a constant independent
of the message length and the recovery error is exponentially small in the
message length. Central to our construction is a notion of quantum list
decoding and an implementation involving folded quantum Reed-Solomon codes.
