Magic state distillation and injection is a promising strategy toward universal fault tolerant quantum computation, especially in architectures based on the bosonic Gottesman-Kitaev-Preskill (GKP) grid codes where non-Clifford gates remain challenging to implement. Here we address GKP magic state preparation by studying a non-Gaussian unitary mediated by a self-Kerr nonlinearity which realizes a logical gate ${\sqrt{H}}_L$ for square grid GKP codes. This gate does not directly involve an auxiliary qubit and is compatible with finite energy constraints on the code. Fidelity can be further enhanced using the small-Big-small error correction protocol and postselection, making the scheme robust against a single photon loss event. We finally propose a circuit QED implementation to operate the self-Kerr nonlinearity.

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• Quantum algorithms & computation
• Quantum error correction
• Quantum gates
• Quantum information with atoms & light
• Superconducting qubits