Approaching the Quantum Singleton Bound with Approximate Error Correction

It is well known that no quantum error correcting code of rate $R$ can correct adversarial errors on more than a $(1-R)\mathrm{/}4$ 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 $(1-R)\mathrm{/}2$, 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.