Free-electron interactions with photonic GKP states: Universal control and quantum error correction

We show that the coherent interaction between free electrons and photons can be used for universal control of continuous-variable photonic quantum states in the form of Gottesman-Kitaev-Preskill (GKP) qubits. Specifically, we find that electron energy combs enable nondestructive measurements of the photonic state and can induce arbitrary gates. Moreover, a single electron interacting with multiple photonic modes can create highly entangled states such as Greenberger-Horne-Zeilinger states and cluster states of GKPs.