  We investigate the effect of stochastic control errors in the time-dependent
Hamiltonian on isolated quantum dynamics. The control errors are formulated as
time-dependent stochastic noise in the Schrodinger equation. For a class of
stochastic control errors, we establish a threshold theorem that provides a
sufficient condition to obtain the target state, which should be determined in
noiseless isolated quantum dynamics, as a relation between the number of
measurements and noise strength. The theorem guarantees that if the sum of the
noise strengths is less than the inverse of computational time, the target
state can be obtained through a constant-order number of measurements. If the
opposite is true, the number of measurements to guarantee obtaining the target
state increases exponentially with computational time. Our threshold theorem
can be applied to any isolated quantum dynamics such as quantum annealing and
adiabatic quantum computation.
