Supervised learning of time-independent Hamiltonians for gate design

We present a general framework to approach the problem of finding time-independent dynamics generating target unitary evolutions. More specifically, given a target unitary gate G over a set of qubits, and a parametrized Hamiltonian of the form H (?) = ?i?isi with si Hermitian operators over the qubits, we want to find a set of values ?0 such that exp(iH (?0)) = G. We show that this problem is equivalently stated as a set of conditions over the spectrum of H (?), reducing the problem to an inverse eigenvalue problem. We show how to solve this problem in some physically relevant instances, like for example to find time-independent dynamics implementing Toffoli and Fredkin gates without the need for ancillary gates or effective evolutions.