  Quantum key distribution with solid-state single-photon emitters is gaining
traction due to their rapidly improving performance and compatibility with
future quantum network architectures. In this work, we perform fibre-based
quantum key distribution with a quantum dot frequency-converted to telecom
wavelength, achieving count rates of 1.6 MHz with
$g^{\left(2\right)}\left(0\right) = 3.6 \%$. We demonstrate positive key rates
up to 175 km in the asymptotic regime. We then show that the community standard
analysis for non-decoy state QKD drastically overestimates the acquisition time
required to generate secure finite keys. Our improved analysis using the
multiplicative Chernoff bound reduces the required number of received signals
by a factor of $10^8$ over existing work, with the finite key rate approaching
the asymptotic limit at all achievable distances for acquisition times of one
hour. Over a practical distance of 100 km we achieve a finite key rate of 13
kbps after one minute of integration time. This result represents major
progress towards the feasibility of long-distance single-emitter QKD networks.
