How behavior of systems with sparse spectrum can be predicted on a quantum computer

Abstract

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.