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Approaching the Quantum Singleton Bound with Approximate Error Correction
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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? 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.
Forward citations
Cited by 2 Pith papers
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List Decoding Expander-Based Codes up to Capacity in Near-Linear Time
Near-linear-time list decoding and list recovery up to capacity are achieved for expander-based AEL and Tanner codes using a graph-regularity rigidity framework.
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Explicit Codes approaching Generalized Singleton Bound using Expanders
AEL expander amplification is shown to preserve a strengthened average-radius list decoding property with erasures, yielding explicit codes with constant alphabet and optimal list size near the generalized Singleton bound.
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