Approaching the Quantum Singleton Bound with Approximate Error Correction

Thiago Bergamaschi, Louis Golowich, Sam Gunn

Abstract: 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?In this work, 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. Specifically, for every R ∈ (0,1) and γ>0, we construct codes of rate R, message length k, and alphabet size 2O(1/γ5), that are efficiently decodable against a (1−R−γ)/2 fraction of adversarial errors and recover the message up to inverse-exponential error 2−Ω(k).At a technical level, we use classical robust secret sharing and quantum purity testing to reduce approximate quantum error correction to a suitable notion of quantum list decoding. We then instantiate our notion of quantum list decoding by (i) introducing folded quantum Reed-Solomon codes, and (ii) applying a new, quantum version of distance amplification.