  High-dimensional degrees of freedom of photons can encode more quantum
information than their two-dimensional counterparts. While the increased
information capacity has advantages in quantum applications (such as quantum
communication), controlling and manipulating these systems has been
challenging. Here we show a method to perform lossless arbitrary
high-dimensional Pauli-X gates for single photon. The X-gate consists of a
cyclic permutation of qudit basis vectors, and, together with the Z gate, forms
the basis for performing arbitrary transformations. We propose an
implementation of such gates on the orbital angular momentum of photons. The
proposed experimental setups only use two basic optical elements such as
mode-sorters and mode-shifters -- thus could be implemented in any system where
these experimental tools are available. Furthermore the number of involved
interferometers scales logarithmically with the dimension, which is important
for practical implementation.
