Motivated by an experiment on a superconducting quantum processor [X. Mi et al., Science 378, 785 (2022).], we study level pairings in the many-body spectrum of the random-field Floquet quantum Ising model. The pairings derive from Majorana zero and 𝜋 modes when writing the spin model in Jordan-Wigner fermions. Both splittings have log-normal distributions with random transverse fields. In contrast, random longitudinal fields affect the zero and 𝜋 splittings in drastically different ways. While zero pairings are rapidly lifted, the 𝜋 pairings are remarkably robust, or even strengthened, up to vastly larger disorder strengths. We explain our results within a self-consistent Floquet perturbation theory and study implications for boundary spin correlations. The robustness of 𝜋 pairings against longitudinal disorder may be useful for quantum information processing.
