Fault-tolerant qubit encoding using a spin-7/2 qudit

The implementation of error correction protocols is a central challenge in the development of practical quantum information technologies. Recently, multilevel quantum resources such as harmonic oscillators and qudits have attracted interest in this context because they offer the possibility of additional Hilbert space dimensions in a spatially compact way. Here, we propose a quantum memory, implemented on a spin-7/2 nucleus hyperfine coupled to an electron spin-1/2 qubit, which provides first-order $X, Y$, and $Z$ error correction using significantly fewer quantum resources than the equivalently effective qubit-based protocols. Our encoding may be efficiently implemented in existing experimentally realized molecular electron-nuclear quantum spin systems. The strategy can be extended to higher-order error protection on higher-spin nuclei.