  Cavity quantum electrodynamics (QED), wherein a quantum emitter is coupled to
electromagnetic cavity modes, is a powerful platform for implementing quantum
sensors, memories, and networks. However, due to the fundamental tradeoff
between gate fidelity and execution time, as well as limited scalability, the
use of cavity-QED for quantum computation was overtaken by other architectures.
Here, we introduce a new element into cavity-QED - a free charged particle,
acting as a flying qubit. Using free electrons as a specific example, we
demonstrate that our approach enables ultrafast, deterministic and universal
discrete-variable quantum computation in a cavity-QED-based architecture, with
potentially improved scalability. Our proposal hinges on a novel excitation
blockade mechanism in a resonant interaction between a free-electron and a
cavity polariton. This nonlinear interaction is faster by several orders of
magnitude with respect to current photon-based cavity-QED gates, enjoys wide
tunability and can demonstrate fidelities close to unity. Furthermore, our
scheme is ubiquitous to any cavity nonlinearity, either due to light-matter
coupling as in the Jaynes-Cummings model or due to photon-photon interactions
as in a Kerr-type many-body system. In addition to promising advancements in
cavity-QED quantum computation, our approach paves the way towards ultrafast
and deterministic generation of highly-entangled photonic graph states and is
applicable to other quantum technologies involving cavity-QED.
