Qubit-oscillator-based gate implementations for approximate Gottesman-Kitaev-Preskill codes

We consider hybrid qubit-oscillator systems together with Gaussian, multi-qubit as well as qubit-controlled Gaussian unitaries. We propose implementations of logical gates for approximate Gottesman-Kitaev-Preskill codes in this model using two oscillators and three qubits. We show that these gate implementations become exact in the limit of large squeezing: The logical gate error is upper bounded by a linear function of the squeezing parameter, and depends polynomially on the number of encoded qubits. For certain Cliffords, our constructions overcome a shortcoming of well-known Gaussian implementations which have a constant logical gate error even in the absence of noise.