Error correction and logical entangling gates with ⁴⁰Ca⁺ single-ion logical qubits

Quantum error correction (QEC) is essential for quantum computers to perform useful algorithms, but large-scale fault-tolerant computation remains out of reach due to demanding requirements on operation fidelity and the number of controllable quantum bits (qubits). While traditional QEC schemes involve encoding each logical qubit into many physical qubits, in this work we experimentally demonstrate a complementary approach by encoding a logical qubit in the multi-level Hilbert space of a single atomic ion. By performing QEC, we reduce errors by a factor of up to 2.2 and extend the qubit's useful lifetime by up to 1.5 times compared to an unencoded qubit. Additionally, we introduce and demonstrate a novel entangling gate for multi-level trapped-ion systems, which can create both high-dimensional, fully entangled states of two trapped ions and entangled logical states of two single-ion logical qubits. Single-atom error correction schemes such as this one may prove useful either as components of larger QEC codes or when used alone in few-qubit devices such as quantum network nodes.