Abstract
The use of noise-robust qubit encodings provides a way of extending the lifetime of quantum information at the hardware level. In this work, we introduce the spin Kerr-cat encoding, which leverages a clock transition in the spectrum of quadrupolar nuclei (having spin length ) to achieve a first-order suppression of noise leading to qubit dephasing. The basis states of the spin Kerr-cat qubit are given by the two lowest levels of a -symmetric nuclear-spin Hamiltonian and are well approximated by spin cat states. We compute the dephasing time of the spin Kerr-cat qubit under a model of noise, as well as relaxation of the qubit due to breaking of the symmetry by charge-noise-induced fluctuations of the quadrupolar tensor. Using measured parameters for antimony () donors in silicon, we estimate that a coherence time of s could be achieved with this encoding. We propose a two-qubit gate mediated by hopping electrons and estimate that with an enhancement of measured quadrupolar splittings by a factor of , a gate fidelity of could be achieved for spin Kerr-cat qubits encoded in nuclear spins, neglecting errors that impact the electron while it is being shuttled and read out.