Simulating Hamiltonians in quantum networks: Efficient schemes and complexity bounds

We address the problem of simulating pair-interaction Hamiltonians in n-node quantum networks where the subsystems have arbitrary, possibly different, dimensions. We show that any pair interaction can be used to simulate any other by applying sequences of appropriate local control sequences. Conditions on time optimal simulation are formulated in terms of spectral majorization of matrices characterizing the coupling parameters. Efficient schemes for decoupling and time reversal can be constructed from orthogonal arrays provided that the dimensions of all nodes are equal to the same prime power. Moreover, we consider a specific system of n harmonic oscillators with bilinear interaction. In this case, decoupling can efficiently be achieved using the combinatorial concept of difference schemes. For this type of interaction we present optimal schemes for inversion.