Constructing Qudits from Infinite Dimensional Oscillators by Coupling to\n Qubits

Abstract

An infinite dimensional system such as a quantum harmonic oscillator offers a\npotentially unbounded Hilbert space for computation, but accessing and\nmanipulating the entire state space requires a physically unrealistic amount of\nenergy. When such a quantum harmonic oscillator is coupled to a qubit, for\nexample via a Jaynes-Cummings interaction, it is well known that the total\nHilbert space can be separated into independently accessible subspaces of\nconstant energy, but the number of subspaces is still infinite. Nevertheless, a\nclosed four-dimensional Hilbert space can be analytically constructed from the\nlowest energy states of the qubit-oscillator system. We extend this idea and\nshow how a $d$-dimensional Hilbert space can be analytically constructed, which\nis closed under a finite set of unitary operations resulting solely from\nmanipulating standard Jaynes-Cummings Hamiltonian terms. Moreover, we prove\nthat the first-order sideband pulses and carrier pulses comprise a universal\nset for quantum operations on the qubit-oscillator qudit. This work suggests\nthat the combination of a qubit and a bosonic system may serve as\nhardware-efficient quantum resources for quantum information processing.\n