Dimensional Reductions for the Computation of Time-Dependent Quantum Expectations

We consider dimension reduction techniques for the Liouville-von Neumann equation for the evaluation of the expectation values in a mixed quantum system. We describe several existing methods that have appeared in the literature, showing the failure of them when the system is scaled up. We introduce a new method termed DEC (Direct Expectation values via Chebyshev) based on evaluation of a trace formula combined with a direct expansion in modified Chebyshev polynomials. This reduction is highly efficient and does not destroy any information. We demonstrate the practical application of the scheme for a nuclear spin system and compare with popular alternatives. In nuclear spin dynamics the main goal for simulations is being able to simulate a system with as many spins as possible, for this reason it is very important to have an efficient method that scales the least with respect to the number of particle. This method may be applied to autonomous quantum problems where the desired outcome of quantum simulation, rather than being a full description of the system, is only the expectation value of some observables.