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This paper presents the design of a given quantum unitary gate by perturbing a three-dimensional (3-D) quantum harmonic oscillator with a time-varying but spatially constant electromagnetic field. The idea is based on expressing the radiationperturbed Hamiltonian as the sum of the unperturbed Hamiltonian and O(e) and O(e 2 ) perturbations and then solving the Schrödinger equation to obtain the evolution operator at time T up to O(e 2 ), and this is a linear-quadratic function of the perturbing electromagnetic field values over the time interval [0, T ]. Setting the variational derivative of the error energy with respect to the electromagnetic field values with an average electromagnetic field energy constraint leads to the optimal electromagnetic field solution, a linear integral equation. The reliability of such a gate design procedure in the presence of heat bath coupling is analysed, and finally, an example illustrating how atoms and molecules can be approximated using oscillators is presented.

This paper deals with the approximate design of quantum unitary gates using perturbed harmonic oscillator dynamics. The harmonic oscillator dynamics is perturbed by a small time-varying electric field which leads to time-dependent Schrödinger equation. The corresponding unitary evolution after time T is obtained by approximately solving the time-dependent Schrödinger equation. The aim of this work is to minimize the discrepancy between a given unitary gate and the gate obtained by evolving the oscillator in the weak electric field over [0, T ]. The proposed algorithm shows that the approximate design is able to realize the Hadamard gate and controlled unitary gate on three-qubit arrays with high accuracy.

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