Quantum gate synthesis by small perturbation of a free particle in a box with electric field

A quantum unitary gate is realized in this paper by perturbing a free charged particle in a one-dimensional box with a time- and position-varying electric field. The perturbed Hamiltonian is composed of a free particle Hamiltonian plus a perturbing electric potential such that the Schr$\ddot{o}$dinger evolution in time $T$, the unitary evolution operator of the unperturbed system after truncation to a finite number of energy levels, approximates a given unitary gate such as the quantum Fourier transform gate. The idea is to truncate the half-wave Fourier sine series to $M$ terms in the spatial variable $\mathbf{x}$ before extending the potential as a Dyson series in the interaction picture to compute the evolution operator matrix elements up to the linear and quadratic integral functionals of ${\mathbf{V}}_n(t{)}^{\mathrm{\prime }}$s. As a result, we used the Dyson series with the Frobenius norm to reduce the difference between the derived gate energy and the given gate energy, and we determined the temporal performance criterion by plotting the noise-to-signal energy ratio (NSER). A mathematical explanation for a quantum gate’s magnetic control has also been provided. In addition, we provide a mathematical explanation for a quantum gate that uses magnetic control.