Biased spin-cat qubits encoded in metastable states of trapped ions for Fault-tolerant Quantum computation

Metastable states of bosonic trapped ions provide a large electronic-spin manifold suitable for encoding quantum information. We consider encoding spin-cat states in the D_5/2 manifold of the 40Ca⁺ ion. The fundamental source of errors in this system is optical pumping arising from coupling to the P_3/2 manifold.  We show that optical pumping errors acting on spin-cats can be converted into effective dephasing errors via a measurement-free error-correction protocol. We employ sympathetic cooling to enable the necessary non-unitary recovery operation, where the motional mode assumes the role of an ancillary qubit in syndrome-based error correction. We experimentally implement this technique and demonstrate the recovery operation for both coherent rotation errors and spontaneous emission errors. By repeating this error-conversion gadget, we realize a highly biased logical qubit encoded in a single ion, yielding a quadratic reduction in the overhead required for Fault-Tolerant quantum computation. We also present a theoretical comparison between this engineered error-bias and an erasure-based encoding for long-term fault tolerance. Together these results establish a new paradigm where the qubit-number overhead in trapped-ion systems can be reduced at the cost of increased control complexity at the level of individual ions.