  Quantum key distribution (QKD offers a long-term solution to establish
information-theoretically secure keys between two distant users. In practice,
with a careful characterization of quantum sources and the decoy-state method,
measure-device-independent quantum key distribution (MDI-QKD) provides secure
key distribution. While short-distance fibre-based QKD has already been
available for real-life implementation, the bottleneck of practical QKD lies on
the limited transmission distance. Due to photon losses in transmission, it was
believed that the key generation rate is bounded by a linear function of the
channel transmittance, $O(\eta)$, without a quantum repeater, which puts an
upper bound on the maximal secure transmission distance. Interestingly, a new
phase-encoding MDI-QKD scheme, named twin-field QKD, has been suggested to beat
the linear bound, while another variant, named phase-matching quantum key
distribution (PM-QKD), has been proven to have a quadratic key-rate
improvement, $O(\sqrt{\eta})$. In reality, however, the intrinsic optical mode
mismatch of independent lasers, accompanied by phase fluctuation and drift,
impedes the successful experimental implementation of the new schemes. Here, we
solve this problem with the assistance of the laser injection technique and the
phase post-compensation method. In the experiment, the key rate surpasses the
linear key-rate bound via 302 km and 402 km commercial-fibre channels,
achieving a key rate over 4 orders of magnitude higher than the existing
results in literature. Furthermore, with a 502 km ultralow-loss fibre, our
system yields a secret key rate of 0.118 bps. We expect this new type of QKD
schemes to become a new standard for future QKD.
