  Quantum computing with discrete variable (DV, qubit) hardware is approaching
the large scales necessary for computations beyond the reach of classical
computers. However, important use cases such as quantum simulations of physical
models containing bosonic modes, and quantum error correction are challenging
for DV-only systems. Separately, hardware containing native continuous-variable
(CV, oscillator) systems has received attention as an alternative approach, yet
the universal control of such systems is non-trivial. In this work, we show
that hybrid CV-DV hardware offers a great advantage in meeting these
challenges, offering a powerful computational paradigm that inherits the
strengths of both DV and CV processors. We provide a pedagogical introduction
to CV-DV systems and the multiple abstraction layers needed to produce a full
software stack connecting applications to hardware. We present a variety of new
hybrid CV-DV compilation techniques, algorithms, and applications, including
the extension of quantum signal processing concepts to CV-DV systems and
strategies to simulate systems of interacting spins, fermions, and bosons. To
facilitate the development of hybrid CV-DV processor systems, we introduce
formal Abstract Machine Models and Instruction Set Architectures -- essential
abstractions that enable developers to formulate applications, compile
algorithms, and explore the potential of current and future hardware for
realizing fault-tolerant circuits, modules, and processors. Hybrid CV-DV
quantum computations are beginning to be performed in superconducting, trapped
ion, and neutral atom platforms, and large-scale experiments are set to be
demonstrated in the near future. We present a timely and comprehensive guide to
this relatively unexplored yet promising approach to quantum computation and
providing an architectural backbone to guide future development.
