  In a recent work (arXiv:2207.01954), we showed that a uniformly coupled chain
could be symmetrically extended by engineered spin chains in such a way that we
could choose part of the spectrum of the overall system. When combined with an
encoding that avoids the uncontrolled eigenvalues, this resulted in the
possibility of achieving a range of tasks such as perfect quantum state
transfer. In this paper, we apply the same strategy to a much broader range of
initial systems - arbitrary chains, and even coupled networks of spins - while
providing guarantees on the existence of solutions.
