  Variations of phase-matching measurement-device-independent quantum key
distribution (PM-MDI QKD) protocols have been investigated before, but it was
recently discovered that this type of protocol (under the name of twin-field
QKD) can beat the linear scaling of the repeaterless bound on secret key rate
capacity. We propose a variation of PM-MDI QKD protocol, which reduces the
sifting cost and uses non-phase-randomized coherent states as test states. We
provide a security proof in the infinite key limit. Our proof is conceptually
simple and gives tight key rates. We obtain an analytical key rate formula for
the loss-only scenario, confirming the square root scaling and also showing the
loss limit. We simulate the key rate for realistic imperfections and show that
PM-MDI QKD can overcome the repeaterless bound with currently available
technology.
