Hybrid quantum information processing with spin and motional states of trapped ions

Hybrid approach to quantum information processing that involves both motional and spin states of trapped ions offers access to a larger Hilbert space and hardware efficient encoding of quantum information compared to the qubit only schemes. Here I present an experimental implementation of controlled-beamsplitter and controlled-SWAP gates that involves quantum states of two bosonic modes and a control qubit in a system of trapped Yb-171 ions. I discuss how to apply the controlled-SWAP gate to measure overlap between infinite dimensional quantum states of harmonic oscillators and show how this overlap test can be used for quantum enhanced machine learning algorithms.