Abstract

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 $\mathcal{G}$ over a set of qubits, and a parametrized Hamiltonian of the form $\mathcal{H}\left(\lambda \right)={\sum }_i{\lambda }_i{\sigma }_i$ with σ_i Hermitian operators over the qubits, we want to find a set of values λ₀ such that $\text{exp}\left(i\mathcal{H}\left(\lambda 0\right)\right)=\mathcal{G}$. We show that this problem is equivalently stated as a set of conditions over the spectrum of $\mathcal{H}\left(\lambda \right)$, 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.

© 2019 The Author(s)