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VOLUME 76 | ISSUE 11 | PAGE 799
How behavior of systems with sparse spectrum can be predicted on a quantum computer
PACS: 03.67.Lx
Call a spectrum of Hamiltonian H sparse if each eigenvalue can be quickly restored within \varepsilon from its rough approximation within \varepsilon_1 by means of some classical algorithm. It is shown how a behavior of system with sparse spectrum up to time T={(1-\rho)}/{14\varepsilon} can be predicted on a quantum computer with the time complexity t={4}/{(1-\rho)\varepsilon_1} plus the time of classical algorithm, where ρ is the fidelity. The quantum knowledge of Hamiltonian eigenvalues is considered as the new Hamiltonian WH whose action on each eigenvector of H gives the corresponding eigenvalue. Speedup of an evolution for systems with the sparse spectrum is possible because for such systems the Hamiltonian WH can be quickly simulated on the quantum computer. For an arbitrary system (even in the classical case) its behavior cannot be predicted on a quantum computer even for one step ahead. By this method we can also restore the history with the same efficiency.