Entangling Qubits in a One-Dimensional Harmonic Oscillator

We present a method for generating entanglement between qubits associated with a pair of particles interacting in a one-dimensional harmonic potential. By considering the effect of the interaction on the energy spectrum of the system, we show that, under certain approximations, a ``power-of-SWAP" operation is performed on the initial two-qubit quantum state without requiring any time-dependent control. Initialization errors and deviations from our approximation are shown to have a negligible effect on the final state. Using a GPU-accelerated iteration scheme to find numerical solutions to the two-particle time-dependent Schr\"{o}dinger equation, we demonstrate that it is possible to generate maximally entangled Bell states between the two qubits with high fidelity for a range of possible interaction potentials.