Abstract: The decoy state method makes the secure key rate of practical secure quantum key distribution (QKD) on the linear scale of channel transmittance. The twin-field (TF) QKD, such as the sending-or-not-sending (SNS) protocol, can have a higher secure key rate on square root scale of channel transmittance. Because of the special structure of SNS protocol, its performance can be largely improved if the error rejection method through two-way classical communication is applied. Using the quantum de Finetti theorem and methods for statistical fluctuation with intercorrelation, the finite key effects can be taken strictly and effectively. As a result, making use of all these methods, the SNS protocol can produce a secure key rate tens of times of the theoretical linear bound of relayless QKD. So far, the SNS protocol of TFQKD has been extensively applied in many important experiments, including the first TFQKD with real optical fibre, the experiment of 509 kms, the experiment of 555 kms and 600 kms. Very recently, the SNS protocol has been successfully applied to the intercity quantum key distribution between Jinan and Qingdao, over a distance of 511 kms.
