Modified Zalka–Wiesner scheme for quantum systems simulation

The method of quantum many particle systems simulation is built, which rests on the modification of known simulating quantum algorithm of Zalka and Wiesner. It is proved that for any positive $\epsilon$ there exists the algorithm, which gives the state of the quantum system in the instant $t$ after $t^{1+\epsilon}$ iterations, each of which is the action of simple Hamiltonian that comes from the main system Hamiltonian by the simple transformation depending on $\epsilon$. In general, the algorithm requires exponential memory of the total number of particles, though it makes possible to find the state in almost real time mode. This method of simulation exceeds the explicit scheme of finite differences and its accuracy is close to the implicit one. Its adventage is that the process of photon emission-absorption can be easily included to the model. Numerical experiments for one dimension problems are performed on this scheme that give interference picture for the colliding of two Gaussians and Rabi oscillations for a particle in two holes potential.