  Call a spectrum of Hamiltonian sparse if each eigenvalue can be quickly
restored with accuracy $\epsilon$ from its rough approximation in within
$\epsilon_1$ by means of some classical algorithm. It is shown how a behavior
of system with sparse spectrum up to time $T=\frac{1-\rho}{14\epsilon}$ can be
predicted with fidelity $\rho$ on quantum computer in time
$t=\frac{4}{(1-\rho)\epsilon_1}$ plus the time of classical algorithm. The
quantum knowledge of Hamiltonian $H$ eigenvalues is considered as a wizard
Hamiltonian $W_H$ which action on any eigenvector of $H$ gives the
corresponding eigenvalue. Speedup of evolution for systems with sparse spectrum
is possible because for such systems wizard Hamiltonians can be quickly
simulated on a quantum computer. This simulation, generalizing Shor trick, is a
part of presented algorithm.
  In general case the action of wizard Hamiltonian cannot be simulated in time
smaller than the dimension of main space which is exponential of the size of
quantum system. For an arbitrary system (even for classical) its behavior
cannot be predicted on quantum computer even for one step ahead. This method
can be used also for restoration of a state of an arbitrary primary system in
time instant $-T$ in the past with the same fidelity which requires the same
time.
