Hybrid continuous-discrete-variable quantum computing: a guide to utility

TL;DR

Some of the advantages of this modality are discussed, and a number of potential applications that can make use of it are laid out; these include applications from physics, chemistry, and computer science.

Abstract

Quantum computing has traditionally centered around the discrete variable paradigm. A new direction is the inclusion of continuous variable modes and the consideration of a hybrid continuous-discrete approach to quantum computing. In this paper, we discuss some of the advantages of this modality, and lay out a number of potential applications that can make use of it; these include applications from physics, chemistry, and computer science. We also briefly overview some of the algorithmic and software considerations for this new paradigm.

Figures (9)

• Figure 1: Applications of hybrid CV-DV quantum computing in Natural sciences and beyond. Part of the chemistry panel is adapted with permission from vu2025computational. Copyright@2025 American Chemical Society.
• Figure 2: A prototypical mixed-fermi boson model: the Hubbard-Holstein model. Electrons (arrows) are coupled to local Einstein oscillators, can hop between sites, and are subject to an on-site Coulomb repulsion. Note that the dimension of the Hilbert space associated to each site is 4.
• Figure 3: An illustration of the IVR process. A laser pulse excites a local vibrational mode, the energy of which is redistributed to the other vibrational modes over time.
• Figure 4: The two extreme limits of the Bose-Hubbard model with $N = L = 5$ (unit filling). In the superfluid phase, depending on the filling and the ratio $U/J$ all the bosons can occupy (condense) to a particular lattice site. In the lower panel, we show the conjectured phase diagram at zero temperature where the tip of Mott lobes (red circles) correspond to the BKT-type phase transition found in the 2d classical XY model. The particle density is fixed for a given lobe and changes by $1$ as we increase $\mu$.
• Figure 5: Inspired by the sequence of theories in kogut1979introduction, we show a roadmap towards simulating QCD with quantum simulators (left). The complexity scale presented is not rigorous, rather it identifies relative algorithmic difficulties and quantum resource requirements in implementing respective theories. Representative field content for generic field-theoretic constructions in (1+1)D and (3+1)D are shown (middle). An example field configuration is shown in a particular gauge-invariant sector identified by $\{g_i\}$ (middle-top) for the $\mathbb{Z}_2$ gauge theory. Magnetic term (green), electric term (red), matter field ($\psi_x$) on vertex and gauge fields $U_{x,\mu}$ on edges are identified (middle-bottom). The quantum simulation framework for lattice field theories and some of the approaches to encode gauge fields and matter fields are identified (right).