Efficient Reconciliation of Continuous Variable Quantum Key Distribution with Multiplicatively Repeated Non-Binary LDPC Codes

Continuous variable quantum key distribution bears the promise of simple quantum key distribution directly compatible with commercial off the shelf equipment. However, for a long time its performance was hindered by the absence of good classical postprocessing capable of distilling secret-keys in the noisy regime. Advanced coding solutions in the past years have partially addressed this problem enabling record transmission distances of up to 165~km, and 206~km over ultra-low loss fiber. In this paper, we show that a very simple coding solution with a single code is sufficient to extract keys at all noise levels. This solution has performance competitive with prior results for all levels of noise, and we show that non-zero keys can be distilled up to a record distance of 192~km assuming the standard loss of a single-mode optical fiber, and 240~km over ultra-low loss fibers. Low-rate codes are constructed using multiplicatively repeated non-binary low-density parity-check codes over a finite field of characteristic two. This construction only makes use of a $(2, k)$-regular non-binary low-density parity-check code as mother code, such that code design is in fact not required, thus trivializing the code construction procedure. The construction is also inherently rate-adaptive thereby allowing to easily create codes of any rate. Rate-adaptive codes are of special interest for the efficient reconciliation of errors over time or arbitrary varying channels, as is the case with quantum key distribution. In short, these codes are highly efficient when reconciling errors over a very noisy communication channel, and perform well even for short block-length codes. Finally, the proposed solution is known to be easily amenable to hardware implementations, thus addressing also the requirements for practical reconciliation in continuous variable quantum key distribution.