  We construct a classical algorithm that designs quantum circuits for
algorithmic quantum simulation of arbitrary qudit channels on fault-tolerant
quantum computers within a pre-specified error tolerance with respect to
diamond-norm distance. The classical algorithm is constructed by decomposing a
quantum channel into a convex combination of generalized extreme channels by
optimization of a set of nonlinear coupled algebraic equations. The resultant
circuit is a randomly chosen generalized extreme channel circuit whose run-time
is logarithmic with respect to the error tolerance and quadratic with respect
to Hilbert space dimension, which requires only a single ancillary qudit plus
classical dits.
