Using a self-Kerr nonlinearity for magic state preparation in grid codes

Magic state distillation and injection is a promising strategy towards 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 root-of-Hadamard gate 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 practical error-correction schemes such as the small-Big-small (SBS) error correction protocol and post-selection, making the scheme robust against a single photon loss event. Finally, we discuss practical circuit QED implementation ideas to realize the self-Kerr nonlinearity.