Constructing Qudits from Infinite Dimensional Oscillators by Coupling to Qubits

An infinite dimensional system such as a quantum harmonic oscillator offers a potentially unbounded Hilbert space for computation, but accessing and manipulating the entire state space requires a physically unrealistic amount of energy. When such a simple harmonic oscillator is coupled to a qubit, for example via a Jaynes-Cummings interaction, it is well known that the Hilbert space can be separated into independently accessible subspaces of constant energy, but the number of subspaces is still infinite. Nevertheless, a closed four-dimensional Hilbert space can be formed from the lowest energy states of the qubit + oscillator. We extend this idea, and show for arbitrary N how a 2(N+1)-dimensional Hilbert space can be constructed, which is closed under a certain set of unitary operations resulting solely from manipulating standard Jaynes-Cummings Hamiltonian terms. Moreover, these unitaries are a universal set for quantum operations on this “qubit-oscillator qudit”. This work suggests that the combination of a qubit and a bosonic system may serve as hardware-efficient quantum resources for quantum computation and memory.