Constructing Qudits from Infinite Dimensional Oscillators by Coupling to Qubits

TL;DR

This paper demonstrates how to construct finite-dimensional qudits from infinite-dimensional harmonic oscillators coupled to qubits, enabling universal quantum operations with standard Jaynes-Cummings interactions.

Contribution

It extends the construction of finite-dimensional Hilbert spaces from qubit-oscillator systems and proves the universality of certain pulse sets for quantum control.

Findings

Finite-dimensional qudits can be analytically constructed from oscillator-qubit systems.

First-order sideband and carrier pulses form a universal set for quantum operations.

The approach offers a hardware-efficient method for quantum information processing.

Abstract

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 quantum harmonic oscillator is coupled to a qubit, for example via a Jaynes-Cummings interaction, it is well known that the total 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 analytically constructed from the lowest energy states of the qubit-oscillator system. We extend this idea and show how a $d$-dimensional Hilbert space can be analytically constructed, which is closed under a finite set of unitary operations resulting solely from manipulating standard Jaynes-Cummings Hamiltonian terms. Moreover, we prove that the first-order sideband pulses and carrier pulses comprise a universal set for quantum operations on the 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 information processing.