Quantum error correction of qudits beyond break-even

Hilbert space dimension is a key resource for quantum information processing(1,2). Not only is a large overall Hilbert space an essential requirement for quantum error correction, but a large local Hilbert space can also be advantageous for realizing gates and algorithms more efficiently(3, 4, 5, 6-7). As a result, there has been considerable experimental effort in recent years to develop quantum computing platforms using qudits (d-dimensional quantum systems with d > 2) as the fundamental unit of quantum information(8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18-19). Just as with qubits, quantum error correction of these qudits will be necessary in the long run, but so far, error correction of logical qudits has not been demonstrated experimentally. Here we report the experimental realization of an error-corrected logical qutrit (d = 3) and ququart (d = 4), which was achieved with the Gottesman-Kitaev-Preskill bosonic code(20). Using a reinforcement learning agent(21,22), we optimized the Gottesman-Kitaev-Preskill qutrit (ququart) as a ternary (quaternary) quantum memory and achieved beyond break-even error correction with a gain of 1.82 +/- 0.03 (1.87 +/- 0.03). This work represents a novel way of leveraging the large Hilbert space of a harmonic oscillator to realize hardware-efficient quantum error correction.