A fault-tolerant encoding for qubit-controlled collective spins

Quantum error correction (QEC) is indispensable for scalable quantum computing, but implementing it with minimal hardware overhead remains a central challenge. Large spin systems with collective degrees of freedom offer a promising route to reducing the control complexity of qubit architectures while retaining a large Hilbert space for fault-tolerant encoding. However, existing proposals for logical gates and QEC in spin ensembles generally rely on inefficient higher-order interactions. Here we introduce spin-N-Cat codes, which encode logical qubits in superpositions of spin-coherent states and generalize bosonic Cat codes to the modular subspaces of permutationally symmetric spin ensembles. The code corrects collective and individual dephasing, excitation, and decay errors. We also present an efficient physical realization in central-spin systems, such as a quantum dot, where encoding, decoding, and a universal, fault-tolerant, and bias-preserving gate set are implemented using only first-order interactions. Numerical simulations demonstrate high logical fidelity under dephasing and excitation-decay noise, independent of noise bias, and that full QEC cycles are feasible with realistic microscopic parameters. For the large collective spins available in quantum dots, this translates into a substantial extension of coherence time. Our results establish spin-N-Cat codes as a scalable, hardware-efficient approach to QEC in spin-based quantum architectures.