  We derive a formula for the time-energy costs of general quantum channels
proposed in [Phys. Rev. A 88, 012307 (2013)]. This formula allows us to
numerically find the time-energy cost of any quantum channel using positive
semidefinite programming. We also derive a lower bound to the time-energy cost
for any channels and the exact the time-energy cost for a class of channels
which includes the qudit depolarizing channels and projector channels as
special cases.
