Achieving high-fidelity and robust qubit manipulations is a crucial requirement for realizing fault-tolerant quantum computation. Here, we demonstrate a single-hole spin qubit in a germanium quantum dot and characterize its control fidelity using gate set tomography. The maximum control fidelities reach 97.48%, 99.81%, 99.88% for the I, X/2 and Y/2 gate, respectively. These results reveal that off-resonance noise during consecutive I gates in gate set tomography sequences severely limits qubit performance. Therefore, we introduce geometric quantum computation to realize noise-resilient qubit manipulation. The geometric gate control fidelities remain above 99% across a wide range of Rabi frequencies. The maximum fidelity surpasses 99.9%. Furthermore, the fidelities of geometric X/2 and Y/2 (I) gates exceed 99% even when detuning the microwave frequency by ± 2.5 MHz (± 1.2 MHz), highlighting the noise-resilient feature. These results demonstrate that geometric quantum computation is a potential method for achieving high-fidelity qubit manipulation reproducibly in semiconductor quantum computation.

MAX phases are a family of ternary carbide or nitride ceramics possessing a layered crystal structure and, due to their chemical bonds having a mixed covalent-ionic-metallic nature, have unique properties combining those of metals and ceramics. In this review, the formation mechanisms of MAX phases from elemental and compound powders are reviewed in detail, as the formation mechanisms are closely related to the unique properties of well-synthesized MAX phases. The stability of MAX phases in some harsh external environments is significantly influenced by the defect population, allowing the mechanisms of defect formation and migration to strongly influence their self-healing performance and radiation tolerance. The properties of MAX phases can be tailored by creating solid solutions, which have lattice distortions, and texturing which results in the preferential orientation of plate-like grains.