  The recently proposed phase-matching quantum key distribution offers means to
overcome the linear key rate-transmittance bound. Since the key information is
encoded onto the phases of coherent states, the misalignment between the two
remote reference frames would yield errors and significantly degrade the key
generation rate from the ideal case. In this work, we propose a
reference-frame-independent design of phase-matching quantum key distribution
by introducing high-dimensional key encoding space. With encoded phases
spanning the unit circle, the error statistics at arbitrary fixed phase
reference difference can be recovered and treated separately, from which the
misalignment angle can be identified. By naturally extending the binary
encoding symmetry and complementarity to high dimensions, we present a security
proof of this high-dimensional phase-matching quantum key distribution and
demonstrate with simulation that a 17-dimensional protocol is completely immune
to any degree of fixed misalignment and robust to slow phase fluctuations. We
expect the high-dimensional protocol to be a practical
reference-frame-independent design for general phase-encoding schemes where
high-dimensional encoding is relatively easy to implement.
