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 over a set of qubits, and a parametrized Hamiltonian of the form with σi Hermitian operators over the qubits, we want to find a set of values λ0 such that . We show that this problem is equivalently stated as a set of conditions over the spectrum of , 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.
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