Polar Codes for Quantum Key Distribution

The polar codes are a class of linear block error-correcting codes transmitted over symmetric binaryinput discrete memoryless channels [1, 2, 3]. As their length N, N = 2n, increases, their performance tends to the Shannon limit. Subsequent publications (e.g., [4, 5]) have addressed practical aspects of the polar codes such as the size length, N, of a polar code that can support a specific feasible performance profile and the impact of the available arithmetic precision on the performance of the polar decoder. As shown in Refs. [6, 7], the polar codes can be used in the reconciliation stage of the quantum key distribution (QKD) protocol. The QKD protocol [8] creates shared secrets by using a quantum channel for “data” that suffers massive deletions (50 % or more) and high bit-error rates (typically between 1 % and 4%, and in theory as high as 11%) and it resolves the bit-value discrepancies through information exchanged over a classical channel that supports data integrity, origin authentication, and protection against replays. The first sound error-correcting protocol to be used by QKD, Cascade [9, 10], is interactive. Cascade went out of fashion because it was believed that it has latency problems. As a result, the use of other, noninteractive decoding schemes, such as the “Low-density parity-check”, LDPC, code [11–13] and polar codes [1–7] were proposed. It should be noted, however, that Cascade is performing a return [14, 15] and there are claims [15] that Cascade currently has no real latency problems. The objective of the QKD protocol’s reconciliation stage is to correct errors in the quantum channel data in such a way that the expected secrecy yield is as high as possible. This translates into maximizing the following: