**Universal quantum simulation for physical demonstrations and applications**

The transformation of the classical intractable mathematical problems using quantum algorithms requires a quantum computer of critical long-term performance. In the short term, the development of a quantum computer more motivated and really interested in the unique ability to model a complex quantum dynamics. Quantum simulators are important, especially in physics, and their potential is to model the behavior it is hard to Hamiltonian systems, including solutions of problems related to the large number of systems of linear equations. Yet only a few tens or hundreds of qubits quantum logic gates in a quantum Turing machine is sufficient to achieve the computing power comparable to modern classical supercomputers. This explains the interest and optimism in evaluating the prospects for the development of quantum simulators in the foreseeable future.
In this lecture, I present historical review of research quantum simulators, beginning with the idea of the universal quantum simulator, proposed by R. Feynman and quantum generalization computer Turing proposed Deytchem for quantum computing. Next, we address the nature of quantum algorithms to understand the principle of the universal quantum simulator, based on the representation Lee Trotta-Suzuki and the assumption of separated Hamiltonians. Simulation of n-quantum bits and k-local Hamiltonians is quite feasible by constructing the corresponding quantum current. Modeling of time-dependent Hamiltonian dynamics is very challenging, but if successful will make any adiabatic quantum state. Finally, we discuss the experimental advances in the development and use of quantum simulation.