  There are two main factors limiting the performance of quantum key
distribution --- channel transmission loss and noise. Previously, a linear
bound was believed to put an upper limit on the rate-transmittance performance.
Remarkably, the recently proposed twin-field and phase-matching quantum key
distribution schemes have been proven to overcome the linear bound. In
practice, due to the intractable phase fluctuation of optical signals in
transmission, these schemes suffer from large error rates, which renders the
experimental realization extremely challenging. Here, we close this gap by
proving the security based on a different principle --- encoding symmetry. With
the symmetry-based security proof, we can decouple the privacy from the channel
disturbance, and eventually remove the limitation of secure key distribution on
bit error rates. That is, the phase-matching scheme can yield positive key
rates even with high bit error rates up to 50%. In simulation, with typical
experimental parameters, the key rate is able to break the linear bound with an
error rate of 13%. Meanwhile, we provide a finite-data size analysis for the
scheme, which can break the bound with a reasonable data size of $10^{12}$.
Encouraged by high loss- and error-tolerance, we expect the approach based on
symmetry-protected privacy will provide a new insight to the security of
quantum key distribution.
