Spin Kerr-cat qubits

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 $Igeq 1$ 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 mathbb{Z}₂-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 1/f noise, as well as relaxation of the qubit due to breaking of the mathbb{Z}₂ symmetry by charge-noise-induced fluctuations of the quadrupolar tensor. Using measured parameters for antimony ${}¹²³Sb$ donors in silicon, we estimate that a coherence time of T₂^*=100 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 approx 4, a gate fidelity of 99\% could be achieved for spin Kerr-cat qubits encoded in {}¹²³Sb nuclear spins, neglecting errors that impact the electron while it is being shuttled and read out.