  Quantum simulation of dynamics is an important goal in the NISQ era, within
which quantum error mitigation may be a viable path towards modifying or
eliminating the effects of noise. Most studies on quantum error mitigation have
been focused on the resource cost due to its exponential scaling in the circuit
depth. Methods such as probabilistic error cancellation rely on discretizing
the evolution into finite time steps and applying the mitigation layer after
each time step, modifying only the noise part without any
Hamiltonian-dependence. This may lead to Trotter-like errors in the simulation
results even if the error mitigation is implemented ideally, which means that
the number of samples is taken as infinite. Here we analyze the aforementioned
errors which have been largely neglected before. We show that, they are
determined by the commutating relations between the superoperators of the
unitary part, the device noise part and the noise part of the open dynamics to
be simulated. We include both digital quantum simulation and analog quantum
simulation setups, and consider defining the ideal error mitigation map both by
exactly inverting the noise channel and by approximating it to the first order
in the time step. We take single-qubit toy models to numerically demonstrate
our findings. Our results illustrate fundamental limitations of applying
probabilistic error cancellation in a stepwise manner to continuous dynamics,
thus motivating the investigations of truly time-continuous error cancellation
methods.
